Keywords: VANET, Broadcasting protocol, Autonomic computing, Message priority level, Density evaluation, Optimization VANET, protocoles de diffusion, Informatique auto-régulée, niveau de priorité de messages, évaluation de densité, Optimisation

The broadcasting communication mode is widely used in Vehicular Ad hoc Networks (VANETs). It is used for sending emergency messages, road-traffic information or to help routing protocols to determine routes. This communication mode is known to be hard to achieve efficiently since it depends on the network density. Indeed, broadcasting methods may cause network congestion if they are not well designed. This paper introduces a novel Autonomic Dissemination Method (ADM) which delivers messages in accordance with given message classes and network density levels. The proposed approach is based on two steps: an offline optimization process and an online adaptation to the network characteristics. ADM allows each node to dynamically adapt its broadcasting strategy not only with respect to the network density, but also according to the class of the message to send: emergency (high-priority), road-traffic (medium-priority) or either comfort message (low-priority). The ultimate goal of ADM is to make effective use of radio resources when there are many messages to send simultaneously. This approach increases the efficiency of the broadcast process in terms of message delivery ratio, latency and interferences reduction. The autonomic computing paradigm improves the robustness of protocols.

Introduction

A Vehicular Ad hoc Network (VANET) is a collection of vehicles communicating through wireless connections. In such a network, each vehicle acts simultaneously as a client node and as a wireless router, allowing multi-hop packet forwarding. Each node has a limited coverage area that contains the neighbours it can directly communicate with. The range of this area can vary from one hundred meters to a few kilometres (depending on the wireless technology onboard, signal propagation constraints, external radio interferences, or environmental features) allowing VANET to provide vehicle-to-vehicle communications over long distance. The continuous movement of vehicles at different speeds and directions induces a highly dynamic network topology. New nodes constantly join the network while others quit when they fall out of the radio range of inner nodes. VANETs are often presented as specific MANETs (Mobile Ad hoc Networks) dedicated to inter-vehicle communications but with the same purposes, challenges, and complexity. Another characteristic of these networks is an heterogeneous node density due to road traffic conditions.

This paper focuses on broadcast communications, which are widely used for sending emergency messages, road traffic reports and information to improve drivers' or passengers' comfort, etc. Typically, a broadcast operation arises when a node produces or detects a significant information that has to be shared with all nodes. As the broadcasted message has to reach every node, the source of the broadcast sends the message to its whole neighbourhood, regardless of the receiver identities. If no collision or interference occurs, each adjacent node within the transmitter's communication range may receive the message. To achieve a multi-hop broadcast, intermediate nodes have to relay incoming messages to allow a wide dissemination. This hop-to-hop transmission would lead to a full network coverage and information delivery. Performing an efficient multi-hop broadcast in VANETs is however a difficult task. The protocols should take into account the specificities of the radio channel, the high mobility of nodes and the network density.

During a broadcast communication, every node receiving a message decides to relay the message to the other nodes within its coverage area or not. This decision is taken in a distributed way, but each node's decision has a real impact on the efficiency of the overall dissemination strategy: in highdensity networks, too many relay nodes would quickly increase the number of collisions, leading to a saturation of the bandwidth and a significant increase of the latency. Conversely, if there are not enough relay nodes in low-density networks the message may not be widely disseminated. It is also important to adapt the broadcasting strategy to the message priority level. For instance, emergency messages such accident alerts, should be delivered as fast as possible to minimize the latency in the source node's neighbourhood. Conversely, it does not matter if comfort information are broadcasted with a more important latency, since it is less urgent.

This paper investigates the problem of designing and evaluating an autonomic and robust broadcasting protocol, which provides each node with the adequate strategy to determine if an incoming message has to be forwarded or not depending on its priority level and the network density. This protocol, named Autonomic Dissemination Method (ADM), follows the autonomic computing paradigm: each node continuously updates its perception of the environment, adding new informations in its knowledge-base and performs a closed control loop thanks to an autonomic manager. Moreover, ADM allows each node to dynamically adapt its broadcast strategy not only with respect to the network density, but also according to the priority level of the message to send. The ultimate goal is to make effective use of radio resources when there are many messages to send simultaneously.

The remaining of this paper is organized as follows: in Sections 2 and 3, we present a state of the art of existing broadcasting methods in wireless ad hoc networks and describe the autonomic computing paradigm. Section 4 formulates the broadcasting problem in VANETs as a multiobjective problem and introduces an optimization methodology. The proposed Autonomic Dissemination Method (ADM) is detailed in Section 5. Sections 6 and 7 present the performance evaluation of ADM respectively in a scenario with a single density level and in a network with heterogeneous density levels.

Multi-hop Broadcasting Protocols

In the literature, wireless ad hoc broadcasting methods are classified into two categories: deterministic and stochastic methods (see Figure 1) [START_REF] Williams | Comparison of broadcasting techniques for mobile ad hoc networks[END_REF] [START_REF] Jaffar | Broadcasting methods in mobile ad hoc networks: Taxonomy and current state of the art[END_REF].

Broadcasting methods

Deterministic methods

Stochastic methods

Simple flooding

Neighbour knowledge-based

Probabilistic scheme

Counter-based

Location-based

Figure 1: Categories of broadcasting methods

Deterministic Methods

A broadcasting method is deterministic if its process is predictable. This group includes simple flooding and neighbour knowledge-based methods.

Simple flooding

The Simple flooding is the simplest broadcasting method. Every packet is relayed exactly once by each node. Any redundant copy of the packet received later is ignored. Thus, in a network consisting of n nodes, n copies of the packet will be sent. A drawback of this method is that it does not take into account the network density. In high density networks, the Simple flooding algorithm would generate many redundant copies of broadcasted packets, leading to the overuse of the radio resources.

Neighbor Knowledge-based Methods

The neighbour knowledge-based methods are founded on a comparison of neighbours lists. Thanks to "Hello" packets, each node discovers its neighbours. The Distributed Vehicular Broadcast (DV-CAST) [START_REF] Tonguz | Dv-cast: A distributed vehicular broadcast protocol for vehicular ad hoc networks[END_REF] and the Flooding with Self Pruning [START_REF] Lim | Multicast tree construction and flooding in wireless ad hoc networks[END_REF] allow each node to build a list of 1-hop neighbours while the Scalable Broadcast Algorithm (SBA) [START_REF] Peng | On the reduction of broadcast redundancy in mobile ad hoc networks[END_REF] draws up a 2-hop neighbours list. These lists are included in the broadcast packets so that the receiver (r) can compare the sender's list to its own list. This comparison allows to determine the additional nodes that may receive the message if r forwards it. For static or low mobility networks, it is a fair good method. But when the node's velocity is high, the information about the neighbours become quickly inaccurate. The Multi-Point Relay (MPR) technique [START_REF] Nguyen | Analysis of mpr selection in the olsr protocol[END_REF] is a neighbour knowledge-based broadcasting method. To reduce the number of redundant packets in the network, each node chooses several nodes among its neighbours that will relay its communications. The selected nodes are called MPRs. The MPRs are selected among the 1-hop neighbours so that they enable to reach all 2hop neighbours. The goal is to have the smallest list of MPRs in the network, which optimizes communications. This method requires a bidirectional link. When a node sends a packet, all its neighbours receive it, but only the MPRs of the source node will relay the message. That means, each node will have a list of all the nodes that have chosen it as their "repeater" (MPRs selectors' list).

Stochastic Methods

The stochastic methods statistically assess the gain that could be obtained if the packets are relayed by a given node. They include probabilistic scheme, counter-based and location-based methods.

Probabilistic Methods

To reduce the number of collisions and to enable better use of the radio channel, probabilistic methods limit the number of relays by setting up the probability for each node to relay the packets. For a given network density, there exists p s , a probability threshold value (0 ≤ p s ≤ 1), that would allow all nodes receive the packets, while reducing the number of unnecessary repetitions and causing few collisions. Any other value p > p s would not lead to better coverage, but may downgrade the quality of the communication. Since p s varies locally in the network, the main challenge of probabilistic methods is to determine its correct value. Some approaches to dynamically assign value to p s are proposed in the literature. They combine probabilistic methods with some other techniques for assessing the network density (eg counter-based or distance-based methods).

Counter-based Methods

The principle of the counter-based methods is simple: the more a node receives copies of the same packet, the less likely it is useful to relay this packet. Upon reception of the first copy, the node initializes a counter C to 1 and sets a timeout RAD (Random Access Delay). During the waiting period, C is incremented upon reception of a new copy of the packet. When the RAD expires, C is compared to a threshold value C t . If C is less than C t , the packet is broadcasted. Otherwise, it is dropped. Like probabilistic methods, one challenge is to find an appropriate value for C t . Ni et al. [START_REF] Ni | The broadcast storm problem in a mobile ad hoc network[END_REF] demonstrate that the additional area covered by the broadcasting process decreases significantly when the number of redundant copies increases.

Yassein et al. [START_REF] Bani Yassein | New counter based broadcast scheme using local neighborhood information in manets[END_REF] proposed the Smart Counter Based Broadcast Algorithm which adapts C t according to the network density. Thanks to "Hello" packets, the nodes build neighbor lists. The size of these lists allows to dynamically adjust C t . More recently, Karthikeyan et al. [START_REF] Karthikeyan | Optimum density based model for probabilistic flooding protocol in mobile ad hoc network[END_REF] proposed a protocol that defines two categories of nodes according to their number of neighbours, with respect to a given threshold τ . Each node decides to relay each packet depending on its own category and the category of the last hop of this packet

Location-based Methods

Before relaying a message, the node evaluates the additional coverage area that will result from this retransmission. The location-based methods do not consider whether nodes exist within that additional area or not. AckPBSM [START_REF] Ros | Reliable and Efficient Broadcasting in Vehicular Ad Hoc Networks[END_REF] and POCA [START_REF] Nakorn | POCA: Position-Aware Reliable Broadcasting in VANET[END_REF] use this approach and set lower RAD to nodes that are far from the source (or last-hop relay) node. To evaluate the extra coverage area, the node can use the distance between itself and each node that has previously relayed the message (distance-based scheme) or the geographical coordinates (location-based scheme). In both distance-based and location-based schemes, a RAD timeout is set and the message is relayed if the additional coverage area is higher than a fixed threshold.

3 Autonomic Computing in Vehicular Ad hoc Networks

Autonomic Computing Paradigm

Traditionally, networks and systems management is a manually controlled process. Thus, it is necessary to have a human intervention of one or several operators in order to manage all the aspects in relation to the dynamic evolution of a system or a network. The creation of self-management systems with limited human interventions was the vision to bring autonomy within IT environment in order to cope with increasing complexity and excessive maintenance costs [START_REF] Group | How much is an hour of downtime worth to you?[END_REF]. Such autonomic systems are able to be self-organized. Networks become a collection of interconnected self-governed entities where human intervention is limited to high-level directives and system management details are transparent for the administrators. The first initiative dealing with this paradigm is inspired by biological systems and in particular, the autonomic nervous system. Indeed, the term Autonomic Computing is partly owed to the autonomic nervous system [START_REF] Horn | Autonomic Computing: IBM's Perspective on the State of Information Technology[END_REF]. This new management concept makes it possible thanks to a holistic approach where all research fields are implicated to contribute in the evolution towards a global autonomy in networks.

Although the objectives list of the self-management concept was extended since 2001 (year of this new paradigm birth), the main objectives for autonomic systems are Self-configuring, Selfoptimizing, Self-healing and Self-protecting [START_REF] Ganek | The dawning of the autonomic computing era[END_REF]. To achieve those objectives, autonomic systems have a detailed knowledge of their internal state as well as their environment [START_REF] Sterritt | Towards an autonomic computing environment[END_REF] using a continuous monitoring of eventual changes that could affect their components. Detecting changes induce the autonomic system to adjust its resources and the monitoring continues to determine if the new measures satisfy the desired performance. That is the closed control loop of self-management systems. It enables autonomic systems to make adequate decisions while conforming to global objectives without human interventions thanks to measurements collected from its resources. This closed control loop is implemented by autonomic managers, which control managed resources using sensors and effectors manageability interfaces [START_REF]An Architectural Blueprint for Autonomic Computing[END_REF].

Autonomic vehicular communications

Adapting the Autonomic Computing paradigm to transportation systems and in particular to VANET networks in order to enhance the performance of communications within such changing environment is a challenging task. In [START_REF] Hsu | Challenges, approaches, and solutions in intelligent transportation systems[END_REF] and [START_REF] Li | Vehicular networks and applications: Challenges, requirements and service opportunities[END_REF], the authors describe the challenges, approaches and solutions in intelligent transportation systems. Indeed, they introduce the cooperative communication concept in vehicular networks. These networks should be self-managed thanks to a self-configuration function using decision elements and control loops. Monitoring and policing information will be used within cooperative VANET communications in conformance with the Autonomic Computing concepts in order to enhance vehicle safety. The research work presented in [START_REF] Wódczak | Autonomic cooperative networking for vehicular communications[END_REF] describes the self-management capability of vehicles in order to perform autonomic cooperative communications and routing within VANETs. The author presents the architecture of an autonomic cooperative node (i.e vehicle) based on the Generic Autonomic Network Architecture (GANA). This autonomic node includes different Decision Elements (DE) controlling a Managed Entity (ME) in order to enhance Vehicle-to-Vehicle (V2V) communications performance. Research challenges concerning Inter Vehicular Communication (IVC) are presented in [START_REF] Dressler | Research challenges in intervehicular communication: lessons of the 2010 dagstuhl seminar[END_REF] according to four areas. The one dealing with IVC communication principles and patterns discussed emerging IVC applications such as safety traffic and described how V2V communications could be used for selforganized traffic control. The author in [START_REF] Insaurralde | Autonomic management for the next generation of autonomous underwater vehicles[END_REF] introduces the autonomic management of Autonomous Underwater Vehicles (AUVs) in order to provide these vehicles with self-maintenance during their missions. An autonomic AUV control architecture is proposed. The objective of this architecture is to achieve the self-management capabilities described by the Autonomic Computing paradigm.

Definition and Optimization of Broadcasting Problem 4.1 Broadcasting in VANETs: an Optimization Problem

Designing an efficient protocol requires to meet several objectives that could be antagonistic: for instance, transmitting messages to the maximum of nodes while avoiding the overuse of the radio channel; delivering packets as quickly as possible, knowing that this speed may cause radio interferences. In a nutshell, this is clearly a multi-objective optimization problem for which each solution is a set of parameters that define a broadcasting strategy.

A broadcasting strategy may consist of the following parameters:

• the probability to relay a packet (P ). It is inherited from the classical probabilistic methods.

When a node receives the first copy of a broadcast packet, it decides to relay it or not, depending on the value of P . The following three parameters are applicable only if the node decides to relay the packet.

• the number of repetitions (N r). In low density networks, when a node broadcasts a packet, it is not unusual that it has no neighbour in its coverage area that will receive the message and relays it. Sending several times the packet, particularly in a context of mobility, the node increases the chance that the packet will be received and relayed. N r is also useful when the first transmitted packet is lost due to a collision or poor radio propagation.

• the delay between two successive repetitions (Dr). When a node transmits several times the same packet (N r > 1), it is important to determine the frequency with which the copies of the same packet will be transmitted. A very short delay could result in many collisions, whereas a very long delay may slow down the broadcast.

• the packets lifetime. It allows a limited packets spread within the network and/or for a long period of time. In this paper we use the maximum number of hops allowed for each packet, T T L (Time To Live). This parameter can be replaced by geographical coordinates or transmission time.

Adding new parameters to classical probabilistic methods expands the search space, making it difficult to determine optimized parameters values for different network contexts. We use a genetic algorithm to perform an effective investigation of the search space.

The performance of dissemination strategies defined by P , N r, Dr and T T L are evaluated by a network simulator. These strategies are compared on four criteria:

• the average Number of Collisions (N C);

• the Propagation Time (P T ). It is the time between the transmission of a packet and the time it is received by all nodes of the studied area.

• the total number of Retransmissions during the simulation (R).

• the Full Reception ratio (F R). It refers to the guarantee that the broadcast packets will be received by all nodes (the reachability). A simulation in which all nodes receive the packet is considered as successful. Whereas, if the network conditions (propagation or topology) do not allow the reception of the packet by all nodes, the simulation is considered as a failure. F R is the ratio of the number of successes on the total number of repetitions of each scenario executions.

N C and R enable to measure the radio channel usage: high values indicate that the evaluated strategy is likely to interfere with other communications in the network. The calculation of N C, P T and R takes into account only successful simulations.

Determining the best broadcasting strategy can be seen as a multiobjective optimization problem that aims to find the solution -→ x (or a solution set

- → X = { -→ x 0 , ..., -→ x n }) such that:        N C( - → x ) is to be minimized P T ( - → x ) is to be minimized R( - → x ) is to be minimized F R( - → x ) is to be maximized
The next section describes the methodology used to solve this optimization problem.

Methodology

We optimize the parameters P , N r Dr and T T L using an approach that combines an optimizer, a network simulator and a trace analyzer. Figure 2 illustrates the interaction of these three modules. P , N r Dr and T T L are optimized using HOPES (Hybrid Optimization Platform using Evolutionary Algorithms and Simulations). HOPES combines an optimizer, a network simulator and a trace analyzer.

Generating a set of solutions (P 0 population)

Transmission of the parameters Evaluation Evaluation

Stop ? P 0 ?

Replacement

Selection

Crossover

Mutation

S o l u t i o n ( s )

Yes No

Yes No

Optimzation Engine

P a r a me t e r s o f t h e p r o t o c o l

Ot h e r p a r a me t e r s :

• propagtion model The optimizer used is our proposed genetic algorithm aGAME [START_REF] Abdou | Adaptive multi-objective genetic algorithm using multi-pareto-ranking[END_REF]. The decision variables of the problem are P , N r, Dr and T T L. They are the different genes defining a solution (a broadcast strategy). The genetic algorithm is used to traverse the search space effectively. The optimization process starts with the random generation of the initial population (P 0 ). The evaluation stage is split into two steps: the first one performed by the network simulator and the second one by the genetic algorithm.

Broadcast parameters (P , N r, Dr and T T L) are transmitted to the network simulator which integrates them with other parameters in order to better reproduce the conditions of the evaluated network. The trace files generated during the simulation are then transmitted to the analyzer module. It parses the files in order to extract the evaluation criteria values (N C, P T , R, and F R) and presents the obtained results according to the genetic algorithm required format.

When the genetic algorithm receives the results of the trace analyzer module, it proceeds to the second step of the evaluation in order to classify solutions and assign values of fitness. To penalize solutions which do not guarantee the full reception of transmitted packets, a constraint is associated with the problem: F R must be greater than or equal to a reachability threshold (F R s ).

The remaining operations of the genetic algorithm are performed independently of the problem. At each iteration, the three modules are involved in the evaluation task. The second test of the optimization module denoted "P 0 ?" checks whether the current population is the initial population. The overall optimization process leads to a set of non-dominated solutions, corresponding to dissemination strategies adapted to the considered network density. This study is repeated for several densities by changing the corresponding parameter in the simulation module.

From the results of this offline optimization phase, ADM builds a knowledge-base that establishes a correspondence between density levels and broadcasting strategies. Density levels are represented by number of neighbouring intervals. Each node can therefore choose, depending on the density of the network in which it is located, the appropriate dissemination strategy. Then, depending on the probability of retransmission associated with the chosen strategy, the node decides to relay the packet or not. If the decision is to relay the communication, it applies the other corresponding parameters (N r, Dr and T T L).

5 An Autonomic Robust Broadcasting Method

Overview

We propose in this paper an extension of our Smart-flooding protocol [START_REF] Abdou | Using an evolutionary algorithm to optimize the broadcasting methods in mobile ad hoc networks[END_REF] thanks to an autonomic robust broadcasting method called ADM (Autonomic Dissemination Method). In our approach, we adapt the broadcasting strategy used by the Smart-flooding protocol according to, not only the density level of VANETs but also the priority level of the message that has to be disseminated. Indeed, ADM is based on the closed control loop implemented by an autonomic manager within a mobile node (vehicle) that we consider as a managed resource according to autonomic computing concepts presented in section 3.1. The resulting architecture, enabling broadcasting strategy optimisation according to VANET's environment characteristics and changes occurrence, is detailed in the following section.

Architecture

We adopt self-management of radio communications to ensure the robustness of broadcasting protocol. Indeed, each node (i.e. vehicle) is considered as an autonomic element thanks to an autonomic manager that enables broadcasting decisions making according to message priority level (see section 5.3) and takes into account environment changes in terms of density level (see section 5.4). To achieve those goals, the autonomic manager implements the MAPE-K closed control loop (see Figure 3) and communicates with the mobile node called Managed Element using Sensors and Effectors manageability interfaces. Each autonomic node within a VANET continuously monitors its environment and network traffic by listening to the radio channel and provides the Monitor function (M) of the Autonomic Manager with network traffic information's thanks to the Sensors manageability interface. In the context of our ADM, the Monitor determines if the received packet is a broadcasting one thanks to its destination address. In case of a broadcasting message, the Monitor provides the Analyze function (A) with this information to follow the control loop process. The Analyze function determines, not only the priority level of the message according to the packet header information, but also the current density level of the node environment thanks to the node local view table stored within the Knowledge-base (K). After density level evaluation that we detail in section 5.4, the Plan function (P) will use the density and priority values provided by the Analyze function to retrieve the adequate broadcasting strategy from the Knowledge base thanks to the strategy table created by the optimization offline phase (see section 4.2). Then, the Plan function will provide the Execute function (E) with the broadcasting parameters (P , N r, Dr and T T L) in order to change the behavior of the mobile node managed resource by executing the corresponding actions of broadcasting strategy thanks to the Effectors manageability interface.

Managed Element

Execute (E)

The architecture of ADM enables the Autonomic Manager to determine how to adapt the dissemination strategy based on information reported by the Sensors manageability interface. Each of the four functions (MAPE) of the Autonomic Manager has a specific role, but all share the same Knowledge-base. The latter contains a set of dissemination strategies optimized for various contexts corresponding to different density and priority levels.

Priority level of broadcasted messages

Several recent works in VANETs and their applications highlight the need to classify messages into several classes [START_REF] Anna | Smart vehicles, technologies and main applications in vehicular ad hoc networks[END_REF] [25] [START_REF] Suthaputchakun | Priority based inter-vehicle communication in vehicular ad-hoc networks using ieee 802.11e[END_REF]. Processing these messages depends on many criteria such as their emergency level, their impact on the road traffic management or the desired reachability.

We define three messages classes for broadcast operations in VANETs (corresponding to three priority levels) and we define for each class a broadcast policy to satisfy. These classes may be based on a single or a dual objective, and may also consider other broadcast characteristics, i.e. the covered nodes ratio evolution over time.

These policies mainly illustrate the adaptability of the protocol to the messages contents and can be easily redefined or extended with additional classes.

• High-Level priority messages (for short HL) correspond to emergency messages, e.g. safety message or accident detection. They have to be delivered as quickly as possible since they may require a prompt reaction from the driver. For these messages, ADM tries to minimize the required propagation time, so that vehicles that are close to the broadcast source may receive the message with a very short delay. Indeed, safety messaging is a near-space application where vehicles in close proximity exchange information to increase safety awareness [START_REF] Anna | Smart vehicles, technologies and main applications in vehicular ad hoc networks[END_REF]. These applications have strict latency constraints. In addition to the reduction of the propagation, ADM will try to maximize the full reception ratio.

• Medium-Level priority messages (M L) correspond to road-traffic messages e.g. traffic jam report. They suppose less-critical information, where the driving reflexes are not part of the equation and only attention is required. These messages should cover a high ratio of nodes, while the broadcast operation requires to reduce the number of radio interferences. According to [START_REF] Anna | Smart vehicles, technologies and main applications in vehicular ad hoc networks[END_REF] traffic monitoring applications require gathering information from vehicles that span multiple kilometres.

• Low-Level priority messages (LL) correspond to comfort messages, e.g. weather information, tourist attraction or point of interest. They are optional messages whose delivery must not alter the dissemination of emergency and alert messages. The use of the radio resources has to be optimized, though reducing the number of collisions as well as the number of retransmissions, for an acceptable node coverage ratio.

Density Level Evaluation

In classical approaches, the density around a node i is often calculated by counting the number of nodes (N i ) located within the coverage area of i. These methods are based on the assumption that all nodes have uniform and identical coverage areas. This is usually the case when radio propagation model is deterministic, such as free-space or Two-ray ground reflection. However, for a more realistic model where packet losses are distributed according to the distance between the transmitter and receiver, this definition is impractical. ADM evaluates the local density for each autonomic node based on the number of active neighbours from which it received the packets. During communication, each node builds a view of its neighbourhood. This view depends on the neighbour list having transmitted or relayed packets. Each autonomic node maintains a history in which it associates with each received packet a list of nodes having sent or relayed it. Upon reception of a packet first copy, its identifier and the source / relay address are recorded within the autonomic manager Knowledge base in a table called local view. When a redundant copy is received, the address of the new relay is appended to the local view table list of addresses (L) corresponding to the packet. Each address is recorded only once for each packet, so receiving multiple copies issued by one neighbour does not lengthen the list of addresses for the concerned packet. When the table is full, the oldest information is replaced by the new one according to FIFO (First In First Out) principle. The current number of neighbours for each autonomic node i is equal to the average number of transmitters for all the packets stored in L (see Equation 1).

N i = n i=1 |L(i)| n ( 1
)
where n is the total number of packets in the local view table and |L(i)| the number of nodes that issued / relayed the i th packet in the table.

Calibration of ADM

ADM's broadcasting strategies consist of four parameters P , N r, Dr and T T L. By calibration we mean the offline optimization of these parameters with respect to network density and message classes corresponding to three priority levels. The robustness of this novel protocol is especially based of this tuning process.

Scenarios

We studied the behaviour of VANETs when varying the network density. For this purpose, we simulated a convoy of vehicles lined up on 10 km. The simulations were carried out using the ns2 network simulator (ns2.34). We used the Shadowing Pattern propagation model [START_REF] Dhoutaut | Impact of radio propagation models in vehicular ad hoc networks simulations[END_REF] which is a realistic and probabilistic model that can produce statistical error distributions, such as slow and fast fading, while being easy enough to be carried out on medium to large simulations. We varied the density by changing the inter-vehicle distance. This leads to increase or to decrease the average number of neighbours each vehicle may have. The optimization process was repeated for different density levels. In this paper we present the tuning process for four density levels illustrated by the following scenarios:

• Urban scenario: an inter-vehicle distance of 25 m.

• Suburban scenario: 75 m between two consecutive vehicles.

• Highway scenario: an inter-vehicle distance of 200 m.

• Rural scenario: most of the time, in rural areas, a vehicle might have no neighbour within its coverage area. To simulate a sparse vehicle distribution, we configured the Shadowing pattern propagation model so that each packet has a probability of 0.2 to be received. This is like having, for each vehicle, a 20% chance to have a neighbour when it sends a packet.

Offline Optimization Process

For each scenario described in Section 6.1, we determined the parameters values of ADM not only with respect to the density level but also according to the class of messages. For this calibration process, we used the HOPES platform (see Section 4.2) with the right network topology and density values in the simulator module. At the end of the optimization process, for each scenario HOPES produced few dozens of solutions that provide a good compromise between the four objective functions N C, P T , R and F R. This means that for a given density level, all the solutions obtained after the optimization process have very good overall performances. However, they do not result in the same behaviour in the network. For instance, some solutions will allow delivering packets as quick as possible, with a risk of overusing the radio channel. Conversely, other solutions will enable reducing the number of redundant packets to ensure wide coverage of the network. Basically, since the objective functions of a multiobjective problem are antagonistic, trying to improve one of the objective functions (even among good solutions) generally leads to the deterioration in at least one other objective. Therefore, we considered the importance of each objective function with respect to the type of messages to send. This method helps to refine the solutions sets. We used a multiple-criteria decision-making approach based on preferences.

Our preference rules are simple and take into account the type of the message to send. For emergency (high-priority level) messages, the ADM protocol should focus on solutions that enable to deliver packets as quickly as possible (lowest value of P T ), especially in the close vicinity of the transmitter (tens or even hundreds of meters). For road-traffic (medium-priority level) messages, solutions that cover the largest number of vehicles (maximizing F R) while reducing the risk of interference (minimizing C) are preferred. And finally, for comfort (low-priority level) messages, the goal is to avoid impacting other higher priority communications (minimize the values of N C and R).

Optimized Broadcasting Parameters for Various Density and Priority Levels

The broadcasting parameters for the three priority levels and the objective functions values are summarized in Tables 2 to 5 (note that Dr and P T are expressed in seconds). In high-density networks, the probability to relay the packets are low (see Table 2). When N r is equal to 1, the Dr cell (the delay between successive repetitions) has been darkened since this parameter is only applicable when N r > 1. For high-priority messages (in the high density network), relaying each packet only once, with a probability of about 0.3 allows rapid dissemination of the message. However, this probability value generates a large number of collisions. This drawback is mended for medium-priority level messages. To reduce the number of collisions and increase the reachability (F R), we selected a solution with a lower probability and a number of repetitions equal to 2. Moreover, as the repetitions are not made in burst the risk of interference is reduced.

Broadcasting parameters

For low-priority level messages, it is worth noting that the results only concern the packets that have been received by all vehicles. In other words, 86.8% of packets that are received spread quickly (due to low competition in the access to the radio channel), but 13.2% of them are not completely delivered.

Following the same reasoning, we obtain the broadcasting parameters for suburban and highway scenarios (Tables 3 and4 For the scenario of the rural area, the low density level of the network implies the need to retransmit each packet many times (see Table 5). Indeed, in this scenario, VANETs behave like delay tolerant networks (DTNs) [START_REF] Paridel | VVID: A delay tolerant data dissemination architecture for VANETs using V2V and V2I communication[END_REF]. In such a context, since the radio channel is rarely used, even if ADM is able to differentiate broadcasting strategies according to the class of a message, in practice these classes scarcely impact the communication process. The main constraints that must be met are: having a probability P close to 1 and a high number of repetition N r.

Broadcasting parameters

Performance The ultimate goal of this optimization step is to provide the Knowledge-base of the Autonomic Manager (see Figure 3) with broadcasting strategies. In the Knowledge-base, each set of parameters will be associated with a density level expressed in number of active neighbours. Figure 4 presents the average number of neighbours for several scenarios. If a vehicle V has n neighbours and:

• n ∈ ]0; 2], V will consider that it is in a rural area.

• n ∈ ]2; 6], V will consider that it is in a highway.

• n ∈ ]6; 12], V will consider that it is in a suburban area.

• n > 23, V will consider that it is in a urban area.

Obviously this list is not exhaustive. For instance, between the suburban and the urban scenarios, we may have several other scenarios. Due to a lack of space, some scenarios studied and shown in Figure 4 are not discussed in this paper.

Comparative Study

To assess the performance of ADM, we compare this protocol to Simple flooding and Smart-flooding. In this section, we will just focus on the suburban scenario (134 vehicles lined up on 10 km, with an inter-vehicle distance of 75 meters). The case of a heterogeneous-density network (especially with mobile nodes) is described in Section 7.

Let us recall that the design of ADM had three major goals. (g1) swiftness: delivering safety messages as soon as possible; (g2) network coverage: reaching the maximum nodes for road traffic messages; (g3) effective use of radio channel for comfort messages. These objectives must be met even if the traffic load increases (for instance when messages are sent simultaneously).

We vary the number of source nodes from 3 to 30. With 30 source nodes in a convoy of vehicles over 10 km, a message is issued approximately every 330 meters. Taking into account the communication range (for WiFi broadcast messages), each node may have within its coverage area 4 or 5 neighbours which simultaneously issue a message. At the second and third hop, the number of simultaneously issued messages, within each node's coverage area, greatly increases. This may tend to quickly congest the radio channel. as fast as can be done, whatever the number of source nodes. Figure 5 shows that the performances of ADM meet the first goal (g1). Compared to Smart-flooding and Simple-flooding, ADM is less sensitive to the number of sources than the other two protocols. Ultimately, even with 30 source nodes, the average delay of emergency (high-priority level) messages is less than 250 ms (for a 10 km line), which is acceptable. It is worth recall that a driver's reaction time to traffic warning signals can be in order of 700 ms [START_REF] Ma | Design and analysis of a robust broadcast scheme for vanet safety-related services[END_REF]. The goal g2 is assessed in Figure 6 which shows the delivery ratio. A packet is considered as "delivered" if it is received by all nodes. Road-traffic messages (ADM M L ) are always received by all nodes. ADM M L ensures this result because it slightly decreases the probability of retransmission (see Table 3), which leads to the reduction of radio interferences. It should be noted that increasing the probability would degrade the performance of ADM and would get close Simple flooding's results.

And finally, Figures 7 and8 show that our third goal (g3) is met: the comfort messages use little radio channel by limiting the total number of repetitions for each packet (Figure 7). In addition, the fact that two potential successive repetitions of the same packet are spaced out (Dr value in Table 3) reduces the number of collisions (Figure 8). In this section, we evaluate the performance of ADM in a network where the density varies according to geographical locations. The aim is to assess the ability of ADM to adapt to density changes thanks to different broadcast strategies provided by the Knowledge-base of the corresponding Autonomic Manager.

Topology and Mobility Models

Figure 9 shows the simulated network topology which consists of three main areas. The first zone is the main road where the average speed is 130 km/h. In the second area the average speed is 90 km/h. And finally the third area tally with an urban network where the average speed is 50 km/h. These speeds correspond to the maximum speed in France respectively on highways, on back roads and in urban areas. We used a mobility model that redirects vehicles at every intersection to keep the average density (average number of neighbours) required in each area (see Figure 9). In addition, the low velocity within the third zone leads to increase the density in this part of the network. Packets are sent every 5 seconds. This allows to evaluate the robustness of ADM with respect to the network traffic. At each sending phase, there is a concurrent access to the radio channel because there are several source vehicles (between 3 and 30 sources depending on the scenario).

Comparative Study

The results obtained in a network with heterogeneous density corroborate the trends we observed in the suburban scenario (see Section 6.4).

The performances of communication protocols in mobile and heterogeneous-density networks depend on their ability to dynamically adapt to changes in their environment. The results in Figures 10 to 12 clearly show that the lack of adaptation mechanism to the density level leads to poor performance of the Simple Flooding. Its propagation time when there are more than 18 simultaneous source nodes is at least 1 second (see Figure 10). This delay can be detrimental for emergency messages. Moreover, one can observe that in case of concurrent access to the radio channel, Simple Flooding is struggling to deliver packets across the network (see Figure 11). This low reachability ratio is due to the collisions caused by redundant packets, especially in high-density areas (see Figure 12).

Regarding the two protocols that are able to adapt to the density, we observe that ADM has better performance results than Smart-flooding. These differences are due to the fact that Smartflooding underestimates the network density by using a theoretical approach while ADM is based on this theory, but also uses experimental results (indeed, the results of the density evaluation in Figure 4 take into account both the number of neighbors and collisions). In general, ADM delivers emergency packets in less that 700 milliseconds in a relatively large area (even with 30 source nodes). This allows to be in conformance with the limits of drivers reaction upon alerts. Besides, for medium-priority messages (for instance road traffic regulation) that should be received by a maximum of nodes, in the scenario with 30 sources, ADM has a reachability ratio of almost 75%, while Smart-flooding has 66% and Simple flooding 53%.

Conclusion

We have proposed in this paper a new distributed autonomic broadcast protocol for VANETs named ADM (Autonomic Dissemination Method). Using pre-computed broadcasting strategies obtained thanks to an evolutionary algorithm, each node is able to dynamically adapt its own broadcast parameters to the network density and to the class of message corresponding to a priority level. The results of simulations carried out on both homogeneous and heterogeneous density-levels networks show that ADM outperforms two other broadcasting methods: the Smart-flooding protocol and the simple flooding method. These results also reveal the scalability of ADM when the number of simultaneous transmissions significantly increases and for different message classes. Despite we have considered three classes of message only, ADM can easily be adapted to include further other message classes, each with its own features and characteristics.

As a future work, we plan to extend ADM to dynamic multi-radio networks, stating that each vehicle is equipped with at least two radio acting on different channels. This perspective would surely increase the overall performance of broadcast operations in VANET, but also lead to several challenge, including channel assignment, solving potential load-balancing issues between radio cards, or deafness problems on mono-processor system when a node may hear incoming messages on one radio per time.

Figure 2 :

 2 Figure 2: The flowchart of HOPES

Figure 3 :

 3 Figure 3: Autonomic manager closed control loop

Figure 4 :

 4 Figure 4: Average Number of Neighbours for Each Density Level

Figure 5 :

 5 Figure 5: Propagation time Regarding the propagation time, ADM aims to deliver emergency messages (denoted ADM HL )as fast as can be done, whatever the number of source nodes. Figure5shows that the performances of ADM meet the first goal (g1). Compared to Smart-flooding and Simple-flooding, ADM is less sensitive to the number of sources than the other two protocols. Ultimately, even with 30 source nodes, the average delay of emergency (high-priority level) messages is less than 250 ms (for a 10 km line), which is acceptable. It is worth recall that a driver's reaction time to traffic warning signals can be in order of 700 ms[START_REF] Ma | Design and analysis of a robust broadcast scheme for vanet safety-related services[END_REF].

Figure 6 :

 6 Figure 6: Delivery ratio

Figure 7 :

 7 Figure 7: Number of retransmissions

Figure 8 :

 8 Figure 8: Number of collisions

Figure 9 :

 9 Figure 9: Network Topology

Figure 10 :

 10 Figure 10: Propagation time

Figure 11

 11 Figure 11: Delivery ratio

Figure 12 :

 12 Figure 12: Collisions

Table 1

 1 sums up the classes considered in this study.

	Message Classes	Examples	Strategies
	High-Level (HL)	Accident reports	(1) Minimize the propagation time
			(2) Maximize the reachability
	Medium-Level (ML) Traffic reports	(1) Maximize the reachability
			(2) Minimize the interferences
	Low-Level (LL)	Tourist attractions (1) Minimize the number of collisions
			(2) Minimize the number of retransmis-
			sions

Table 1 :

 1 Message priority levels

Table 2 :

 2 ADM Parameters and Performance Results for the Urban Scenario

	Performance Results

Table 3 :

 3 respectively). ADM Parameters and Performance Results for the Suburban Scenario

		Broadcasting parameters	Performance Results
	Message	P	N r	Dr	T T L	N C	P T	R	F R
	Classes
	High-Level (HL)	0.776	1		26	166 0.044 104 100%
	Medium-Level	0.519	2	0.951	16	93	0.121 139 100%
	(M L)
	Low-Level (LL)	0.291	2	0.276	27	35	0.209 82 75.8%
		Broadcasting parameters	Performance Results
	Message	P	N r	Dr	T T L	N C	P T	R	F R
	Classes
	High-Level (HL)	0.999	4	1.147	40	31	0.092 199 100%
	Medium-Level	0.916	2	0.729	28	24	0.124 90	100%
	(M L)
	Low-Level (LL)	0.649	2	1.933	34	10	1.414 66 82.8%

Table 4 :

 4 ADM Parameters and Performance Results for the Highway Scenario

Table 5 :

 5 ADM Parameters and Performance Results for the Rural Area Scenario

	Results


Elaine Pratt
email: pratt@lma.cnrs-mrs.fr

Xiang Alain Leger
email: leger@lma.cnrs-mrs.fr

Alain Léger

Xiang Zhang
email: xzhang@sjtu.edu.cn

Study of a transition in the qualitative behaviour of a simple oscillator with Coulomb friction

Keywords: Coulomb friction, equilibrium states, mass-spring systems, nonsmooth dynamics

come

Study of a transition in the qualitative behaviour of a simple oscillator with Coulomb friction

Introduction

This paper goes deeper into the investigation of the dynamics of a simple mass-spring system involving non regularized unilateral contact and Coulomb friction conditions. The investigation began by the set of equilibria which was immediately shown to have very specific features by comparison to those of classical dynamical systems. Two fundamental questions were addressed which continue to motivate theoretical works. The first one concerned stability. Classical stability analysis of the equilibrium states were performed both from the numerical [START_REF] Basseville | Investigation of the equilibrium states and their stability for a simple model with unilateral contact and Coulomb friction[END_REF] and theoretical [START_REF] Basseville | Stability of equilibrium states in a simple system with unilateral contact and Coulomb friction[END_REF] points of view, but it was soon observed that these analysis were not sufficient to take the actual dynamics into account. The observations led to a new notion of stability and to a conjecture specially suited to the dynamics of systems involving Coulomb friction [START_REF] Pratt | About a stability conjecture concerning unilateral contact with friction[END_REF]. The second question was closely related to the qualitative analysis of dynamical systems, and can be stated in the following way: when a system involving unilateral contact and dry friction is submitted to an oscillating excitation, are we able to predict its answer by performing a partition of the period-amplitude plane of the excitation into different kinds of qualitative behaviours? This was investigated in [START_REF] Léger | Qualitative analysis of a forced nonsmooth oscillator with contact and friction[END_REF] where it was shown that the period-amplitude plane is essentially divided into three ranges. One range, for external forces of sufficiently small amplitude, is a horizontal strip, in which there exist everywhere infinitely many equilibrium states, no periodic solutions, and where any initial data out of equilibrium leads to a trajectory which attains an equilibrium in finite time. In a second range there exist no equilibrium states but the whole range contains periodic solutions, the number and the period of which depend on the period of the excitation. The latter range has a relatively intricate upper boundary above which the last range is such that there still exist periodic solutions, but all initial data lead to a trajectory which looses contact, so that the periodic trajectories involve jumps and shocks. The present work continues the investigation of the period-amplitude plane, focussing on the transition between the range where there exist only equilibrium states and no periodic solutions and the range where there exist periodic solutions and no equilibrium states. We stress the fact that since the very beginning, all these analysis where possible due to a rigourous statement of the dynamics (see [START_REF] Schatzman | A class of differential equations of second order in time[END_REF] and [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF]) and to a proof that the corresponding Cauchy problem was well-posed [START_REF] Ballard | Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem[END_REF].

The paper is concerned with sliding trajectories and is organized as follows. Section 2 recalls the foundations of the present work, gives the dynamical system and the previous basic results concerning the partition of the period-amplitude plane. Section 3 extends the investigation presented in the previous works and gives the proofs of the results which had simply been observed numerically before. It is shown that if the period of the external force is large enough, there exist infinitely many periodic solutions in addition to a single equilibrium state. Moreover, any initial data which does not belong to a periodic solution leads to a trajectory which converges to a periodic one, either in finite time or at infinity depending on the period of the excitation. At the end of this section we study the special case where the period of the external force is equal to an intrinsic value corresponding to a sliding phase to the right. In Section 4, we consider smaller values of the period of the excitation, up to the limit of very high frequencies. Here there exists a single equilibrium state and no other periodic solutions. The investigation is here relatively intricate and requires the range to be shared into several subintervals. It is shown that a trajectory tends generically to the equilibrium at infinity. But for any small enough period there exists a unique initial data such that the equilibrium is reached in finite time. The whole analysis of these sections is performed using an excitation of a rectangular wave shape, in order to carry out analytical calculations as far as possible. So that the last section studies what remains of the transitions and of their qualitative meaning if the excitation is changed into a more classical harmonic one.

Preliminaries

The mechanical problem

We are studying a simple mass spring system which has been given in [START_REF] Klarbring | Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction[END_REF] for the analysis of the quasi-static evolution, and which is represented on Figure 1. A single mass m is connected to a rigid frame by two springs, is submitted to an external force of components (F t , F n ), and its motion in the plane is constrained by an obtacle which is the horizontal plane at the level zero. The contact with the obstacle involves unilateral contact and Coulomb friction's conditions. The corresponding contact and friction laws are kept all along this work without any regularization. Let K = (K t , K n , W ) be the stiffness matrix of the system of springs, µ the friction coefficient, (R t , R n ) the reaction of the obstacle. The trajectories are the solutions to the following problem Figure 1: The mass-spring system.

                       i) mü t = -K t u t + W u n + F t + R t , ii) mü n = -W u t + K n u n + F n + R n , t > 0, iii) u t (0) = u t0 , u+ t (0) = ut0 u n (0) = u n0 , u+ n (0) = un0 iv) u n ≤ 0, R n ≤ 0, u n R n = 0, v) µR n ≤ R t ≤ -µR n , |R t | < -µR n =⇒ ut = 0, |R t | = -µR n =⇒ ∃λ ≥ 0 s.t. ut = -λR t , vi) u n (τ ) = 0 =⇒ u+ n (τ ) = -e u- n (τ ), e ∈ [0, 1]. (1)
In the case of sliding motions, that is when the trajectories do not involve loss of contact and jumps, inequalities iv) in system (1) reduce to u n ≡ 0, R n ≤ 0 so that system (1) reads

               i) mü t = -K t u t + F t + R t , ii) 0 = -W u t + F n + R n , t > 0, iii) u t (0) = u t0 , ut (0) = ut0 iv) R n ≤ 0, v) µR n ≤ R t ≤ -µR n , |R t | < -µR n =⇒ ut = 0, |R t | = -µR n =⇒ ∃λ ≥ 0 s.t. ut = -λR t , (2)
where we observe that through equation (2-ii) we have a simple linear relation between u t and R n which allows to deal indifferently with u t or R n , and that in particular equations (2-i,ii,v) give:

either mü t = -(K t -µW )u t + F t -µF n , or mü t = -(K t + µW )u t + F t + µF n ,
for sliding respectively to the right and to the left, which we rewrite:

üt + ω 2 α u t = F t -µF n , üt + ω 2 β u t = F t + µF n ,
setting ω 2 α := (K t -µW )/m and ω 2 β := (K t + µW )/m. All along this work we set T α = π/ω α and T β = π/ω β for the half-periods of the sliding phases respectively to the right and to the left and we shall take m equal to 1.

Some preliminary results

The set of equilibria has been investigated in several previous articles according to the values of the parameters µ, K t , K n , W, F t , F n (e.g. [START_REF] Léger | Qualitative analysis of a forced nonsmooth oscillator with contact and friction[END_REF]). It has in particular been stated that the case when K t F n -W F t > 0 and K t -µW > 0 can be considered as the most general in the sense that it involves all the difficulties that can be encountered separately in the other cases. Let A be the quantity K t F n -W F t . We assume in the following A > 0 and K t -µW > 0. In this case, the set of equilibria represented in the {R t , R n } plane, which is in general the intersection of the equilibrium equation (a straight line in the present case) and the Coulomb cone, is the thick line on Figure 2. A rectangular wave tangential force, of amplitude ε and half-period T , oscillating between 0 and +ε, is now applied on the mass in addition to the constant force (F t , F n ). Then the set of equilibria changes periodically from the thick line of Figure 3 to the dotted one. Whether the corresponding intervals of normal components of the reaction at equilibrium have an empty intersection or not determines whether equilibrium states exist or not for such an oscillating load. The system under this periodic excitation behaves in a relatively complicated way, it may have equilibria, periodic solutions and non periodic solutions. A first investigation of the {T, ε} plane has been published in [START_REF] Léger | Qualitative analysis of a forced nonsmooth oscillator with contact and friction[END_REF]. It has in particular been stated that there exists a critical value ε 0 of the amplitude such that: Statement 1.

• i) For ε < ε 0 , there exist infinitely many equilibrium states for any T , and any initial data out of equilibrium leads to a trajectory which reaches one of the equilibrium states in finite time.

• ii) For ε > ε 0 , there no longer exist any equilibrium states, but there exist periodic solutions for any T , the number of these periodic solutions depends on T .

The critical value ε 0 of the amplitude is given by ε 0 = 2µA Kt+µW , where A is the quantity involving the loading and stiffness parameters defined above. The present work deals with the transition between these two ranges. We shall moreover restrict our attention to sliding solutions, which means that we shall refer to initial data leading to loss of contact only as a boundary for the validity of the present study. Three points must be proved, which have essentially been observed from symbolic calculations and presented in [START_REF] Léger | Qualitative analysis of a forced nonsmooth oscillator with contact and friction[END_REF]:

Statement 2. For ε = 2µA K t + µW
there exists a single equilibrium solution for any T . Moreover:

• i) when T < T α all the initial data different from the single equilibrium lead to trajectories which tend to the equilibrium,

• ii) when T > T α there exist infinitely many periodic solutions and any initial data which does not belong to a periodic solution leads to a trajectory which goes to a periodic solution,

• iii) when T = T α the set of periodic solutions is strictly larger than the set of periodic solutions obtained for T > T α . Any initial data which does not belong to a periodic solution leads to a trajectory which tends to the periodic solution of largest orbit at infinity.

Throughout this paper, the figures are obtained by symbolic calculations using the following values for the coefficients and parameters of the problem m = 1, µ = 0.5, and

F n = 2, F t = 1, K t = 2, W = 1, so that in particular A = 3.
(3)

3 The set of solutions when T is larger than T α

When T is strictly larger than T α

When the amplitude ε of the perturbation is equal to ε 0 = 2µA Kt+µW there exists a single equilibrium solution for any value of the half period T , let U e be this equilibrium position. Other specific positions will be useful: U ℓ is the equilibrium position in imminent sliding to the left when the perturbation is equal to ε 0 , U r is the equilibrium position in imminent sliding to the right with the perturbation equal to 0, U d is the position such that all trajectories issued from (u 0 , 0) with u 0 < U d loose contact.

Using equation (2-ii) and the fact that a trajectory shall no longer be a sliding trajectory (i.e. the mass looses contact) as soon as its reaction goes through the vertex of the cone, we easily get

                                   U e = F n W + -A + ε 0 W W (K t -µW ) = F n W - A W (K t + µW ) with normal reaction R e , U d = F n W - 2A W (K t + µW ) with normal reaction R d , U ℓ = F n W + -A + ε 0 W ) W (K t + µW ) ≡ F t + µF n + ε 0 ω 2 β with normal reaction R ℓ , U r = F n W - A W (K t -µW ) ≡ F t -µF n ω 2 α with normal reaction R r . (4)
We shall also use the quantity d defined as

d = U ℓ -U e = F n W + -A + ε 0 W W (K t + µW ) -U e = ε 0 K t + µW > 0,
so that when necessary the symmetric of U ℓ with respect to U e will be simply U e -d.

Remark 1. i) The numerical values chosen in (3) give U r = 0.

ii) The inequalities U r < U e -d < U e and U d < U e -d always hold. In particular inequality U d < U e -d results from the condition K t -µW > 0 which has been referred to as the generic case and which is consequently assumed to hold all through this paper.

Let us recall the following lemma which shall be needed througout this section (the proof can be found in [START_REF] Pratt | About a stability conjecture concerning unilateral contact with friction[END_REF]): Lemma 3.1. Let the loading be piecewise analytical and let {R n }(t) be the set of normal components of the reactions at time t corresponding to a strictly stuck equilibrium solution. Suppose that A > 0 and consider the trajectory of a sliding mass which satisfies problem [START_REF] Basseville | Stability of equilibrium states in a simple system with unilateral contact and Coulomb friction[END_REF]. If at an instant t * when the mass stops sliding its normal reaction R * n belongs to the interior of {R n }(t * ) then the mass shall remain in a strictly stuck equilibrium state as long as its normal reaction belongs to the interior of {R n }(t) (i.e. in particular for any subsequent time if the external force does not change).

For example, when the external load changes periodically from

(F t , F n ) to (F t + ε, F n ) with a period 2T , {R n }(t) = [R e , R ℓ ] for t ∈ [0, T [ and {R n }(t) = [R r , R e ] for t ∈ [T, 2T [ (see Figure 4-a).
We now use the notation u in place of u t (since we are studying sliding motions no confusion is possible). Let the external loading be such that ε = ε 0 defined at statement 1. Then the qualitative behavior of the solutions to system (2) is given by the two following propositions.

Proposition 3.2. Let T > T α , ε = ε 0 and K t -µW > 0 then any initial data (u 0 , u0 ) such that u 0 ∈ [U e -d, U e ]
and u0 = 0 leads to a periodic solution of period 2T .

Proof. The external load changes periodically from

(F t , F n ) to (F t + ε 0 , F n ) with a period 2T . It is equal to (F t + ε 0 , F n ) on [0, T [ and to (F t , F n ) on[T, 2T [. If u 0 ∈ [U e -d, U e ] then R n (T α ) belongs to [R e ,
               ü + ω 2 α u = F t + ε -µF n , t ∈ [0, T α ], u(0) = u 0 , u(0) = 0, u(t) = u(T α ), u(t) = 0, t ∈ [T α , T ], ü + ω 2 β u = F t + µF n , t ∈ [T, T + T β ], u(T ) = u(T α ), u(T ) = 0, u(t) = u(T + T β ), u(t) = 0, t ∈ [T + T β , 2T ].
(5) A solution to system (5) is such that :

u(T ) = u(T α ) = U e + (U e -u 0 ) and u(2T ) = u(T + T β ) = U e -(u(T ) -U e ) = u 0 , u(2T ) = 0.
In other words the solution is 2T periodic. ii) Studying only the case of initial data with zero velocity may seem restrictive. In fact, introducing a nonzero initial velocity would not lead to important changes in the case of a linear stiffness matrix K. The case where unilateral contact and Coulomb friction is coupled with a nonlinear restoring force is much more intricate as can be seen in [START_REF] Léger | A fully nonlinear oscillator with contact and friction[END_REF] where the study of the equilibrium states is to be found. iii) Choosing u 0 < U e implies that the first phase of the motion is a sliding phase to the right. If u 0 ≥ U e then the first phase would be either motionless or sliding to the left but no other periodic solutions would be found.

Of course if u 0 < U e -d then R n (T α ) is greater than R ℓ therefore R n (T α ) / ∈ [R e , R ℓ
] so that the mass does not stay motionless in the time interval [T α , T ] and the motion is no longer governed by system [START_REF] Klarbring | Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction[END_REF]. Whether in the {R t , R n } plane or in the phase space {u t , ut }, the set of periodic trajectories fills completely the inside of the domain bounded by the periodic trajectory of largest amplitude represented on Figure 5 Proof. We are considering a period of the excitation T = T α + η with a given η strictly positive. Assume that after any trajectory involving k non periodic loops, the trajectory is at some outof-equilibrium position u k on the left side of the cone and take this position to be the initial data at a time t = 0, then there is a sliding phase to the right, then the trajectory jumps to the right side of the cone and there is a sliding phase to the left, then the trajectory comes back to a new out-of-equilibrium position u k+1 on the left side of the cone which makes a complete loop. More precisely, let τ 1 , τ 2 , τ 3 , τ 4 be successive times defined as:

-during a first sliding phase to the right, τ 1 is the time when the perturbation is set to ε 0 , -τ 2 is the time when the velocity of the sliding phase to the right goes through zero, so that τ 2 is the end of the sliding phase to the right, -during the sliding phase to the left, τ 3 is the time when the perturbation is set to zero, -τ 4 is the time when the velocity of the sliding phase to the left goes through zero.

                   ü + ω 2 α u = F t -µF n on (0, τ 1 ) ü + ω 2 α u = F t -µF n + ε 0 on (τ 1 , τ 2 ) ü + ω 2 β u = F t + µF n + ε 0 on (τ 2 , τ 3 ) ü + ω 2 β u = F t + µF n on (τ 3 , τ 4 ), (6)
using the continuity of the position u and of the velocity u at times τ 1 , τ 2 , τ 3 . From system (6) we calculate u k+1 := u(τ 4 ), where τ 4 is such that u(τ 4 ) = 0:

u k+1 = U e -(U e -U -d) 2 + d 2 + 2d(U e -U -d) cos(ω β η),
where U is an abstract initial position defined in the following way: 1) assume the perturbation is equal to ε 0 since the origin t = 0 instead of being set to ε 0 at time τ 1 , 2) then (U , 0) is the initial data which would lead to a trajectory which would coincide with the actual trajectory calculated in [τ 1 , τ 2 ],

3) as represented on Figure 6, this means that U is symmetrical of u(τ 2 ) with respect to U e . So that, (U e -u k+1

) 2 = (U e -U ) 2 -2d(U e -U -d)(1 -cos(ω β η)),
from which we obtain

u k+1 -U = 2d(U e -U -d)(1 -cos(ω β η)) (U e -U ) + (U e -u k+1
) .

If U ≥ U r then as u k+1 is larger than U , u k+1 obviously belongs to [U r , U e -d[. If not then

u k+1 -u k ≥ u k+1 -U ≥ 2d(U e -U r -d)(1 -cos(ω β η)) (U e -U d ) + (U e -U d ) ,
and setting

δ = 2d(U e -U r -d)(1 -cos(ω β η)) 2(U e -U d ) > 0 we have u k+1 -u k ≥ δ > 0.
This implies that u k is larger than U r as soon as kδ > U r -U d , that is at most after 2kT , therefore in finite time.

Proposition 3.4. Let T > T α , ε = ε 0 and K t -µW > 0, then the trajectories issued from any initial position u 0 / ∈ [U e -d, U e + d] are not periodic, and have a qualitative behavior which depends on T in the following way:

• i) for any T such that T α < T ≤ T α + T β /2, all sliding solutions tend to the largest amplitude periodic solution at infinity,

• ii) for any such that T > T α + T β /2 all sliding solutions reach one of the periodic solutions in finite time.

Proof. According to the values given in equation ( 4) and to remark (2-iii), proving Proposition 3.4 amounts to investigating the behavior of any trajectory starting with a zero velocity and an initial position u 0 such that

U d < u 0 < U e -d. (7)
Thanks to Lemma 3.3, we restrict the proof of Proposition 3.4 to trajectories starting from initial positions in the interval [U r , U e -d[ with a zero velocity. Setting u k = u(2kT ) it is easy to check that Lemma 3.1 implies that for all k, u k ∈ [U r , U e -d[, so that τ 1 = 0 in system [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF] and we can easily calculate u k+1 . We introduce the following notation: x k = U e -u k .

• If T ≥ T α + T β then u(τ 3 ) = 0 and the solution of system (6) gives:

x k+1 = d 2 + (x k -d) 2 -2d(x k -d), so that x k+1 = (d -(x k -d)) > 0. But as u k ∈ [U r , U e -d[ this implies that x k -d > 0 so that x k+1 -d = -(x k -d) < 0,
therefore, for any initial position in [U r , U e -d[ one of the periodic solutions is reached after 2T . Such a trajectory is represented on Figure 7 in the {R t , R n } plane.

• If T < T α + T β the solution of system (6) gives:

x k+1 = d 2 + (x k -d) 2 + 2d(x k -d) cos ω β (T -T α ). ( 8
)
The behaviour of the sequence {x k } defined by equation ( 8) depends on whether cos ω β (T -T α ) is positive or negative so that there are two cases: i) When T α < T ≤ T α + T β /2, then 0 ≤ cos ω β (T -T α ) < 1 and consequently

x k > d =⇒ x 2 k+1 -d 2 > 0 =⇒ x k+1 > d.
We deduce that ∀k x k > d so that d is a lower bound for the sequence {x k }. On the other hand equation ( 8) implies that

x 2 k+1 -x 2 k = 2d(d -x k )(1 -cos ω β (T -T α )) < 0, (9)
which means that the sequence {x k } is decreasing. Therefore the sequence {x k } converges, and passing to the limit in equation ( 9) we obtain that it converges towards d, which establishes point i) of Proposition 3.4. -ii) When T α + T β /2 < T < T α + T β , then cos ω β (T -T α ) < 0.

In this case there exists a subscript k * such that x k * ≤ d. In other words the trajectory converges in finite time to the periodic solution with initial condition (u k * , 0). Let us assume that this is not so, that is that for all k, x k > d. Then for the same reasons than in (i) equation ( 9) implies that the sequence {x k } converges towards d. But from equation (8) we have

x 2 k+1 -d 2 = (x k -d) 2 + 2d(x k -d) cos ω β (T -T α ) so that (x k -d) 2 + 2d(x k -d) cos ω β (T -T α ) > 0, ∀k or (x k -d) + 2d cos ω β (T -T α ) > 0, ∀k.
But this cannot be true for all k since the quantity 2d cos ω β (T -T α ) is given and strictly negative whereas x k converges to d. So that the assumption x k > d, ∀k is false and there exists k * such that x k * ≤ d.

We can see on Figure 8 two trajectories starting from the same initial data calculated with the numerical values given at equation ( 3): the first one, for T large enough, attains one of the periodic solutions in finite time, in fact after a very small number of sliding oscillations (Figure 8-a), the other one, for smaller T , converges to the largest amplitude periodic solution at infinity (Figure 8-b).

-1

1 1 2 -1 1 1 2 a -T > T α + T β /2 b -T α < T ≤ T α + T β /2

Periodic solutions for T = T α

This section focusses on point iii) of statement 2. As in section 3 the amplitude of the perturbation of the load is fixed at the value of the transition ε = ε 0 = 2µA Kt+µW . Here the position of the point U d corresponding to the limit of sliding solutions is important. It is easily shown that U d ≤ U r if and only if K t -3µW ≥ 0. The starting point is the following: Proposition 3.5. Assume T = T α and ε = ε 0 then any initial data (u 0 , u0 ) with u0 = 0 and

u 0 ∈ [max(U r , U d ), U e ] ( 10
)
leads to a periodic solution of period 2T .

Proof. If u 0 is chosen in [U r , U e ] then the motion shall be governed by a system similar to system (5), except that as T = T α the motionless phase during [T α , T ] in [START_REF] Klarbring | Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction[END_REF] does not exist so that the trajectory is the solution to:

           ü + ω 2 α u = F t + ε -µF n , t ∈ [0, T α ], u(0) = u 0 , u(0) = 0, ü + ω 2 β u = F t + µF n , t ∈ [T α , T α + T β ], u(T α ) = u(T α ), u(T α ) = 0, u(t) = u(T α + T β ), u(t) = 0, t ∈ [T α + T β , 2T ].
(11) A solution to system (11) is such that :

u(T α ) = U e + U e -u 0 and u(2T ) = u(T α + T β ) = U e -(u(T ) -U e ) = u 0 with u(2T ) = 0.
In other words the solution is 2T periodic. The set of initial conditions that give a periodic solution is larger than when T > T α simply because here when

T = T α the normal reaction R n (T α ) does not belong to [R e , R ℓ ] if u 0 ∈ [U r , U e -d[
so that there is no motionless phase. Comparing this set with the one obtained for the case T > T α we observe that again it fills completely a domain, either in the {R t , R n } plane or in the phase space, which is bounded by the maximal amplitude periodic solution, but this set is now strictly larger than for T > T α . It is represented on Figure 9, where we stress the fact that the scales on both axis of Figures 9-a,b are the same as those used for Figure 5-a,b. Remark 3. Following Remark 1, the interval given by equation ( 10) is not empty if and only if K t -3µW > 0. If K t -3µW ≤ 0, there exist no non periodic sliding solutions. All the sliding trajectories are periodic, and any other trajectory involves jumps. [START_REF] Jean | The nonsmooth contact dynamics method[END_REF] When T is smaller than T α Proposition 4.1. Assume T < T α and ε = ε 0 , then any initial data such that u 0 ∈ ]U d , U e [ and u0 = 0 leads to a trajectory which converges to the single equilibrium.

An elementary corollary of Proposition 4.1 is that, due to the well-posedness of the Cauchy problem, no periodic solution exists (except the trivial one which is the single equilibrium point).

The proof of Proposition 4.1 is carried out in several steps, it will establish that all the trajectories of sliding motion converge to the equilibrium, but depending on the period and on the initial data the convergence holds in finite time or at infinity.

Step 1. Let T < T α /2. Then there exists an initial data u 0 such that the trajectory arrives exactly at the equilibrium point U e with a zero velocity after a finite number of oscillations of the external force and involves only sliding phases on the left side of the cone.

To establish this existence result we are going to explicit the trajectory by a direct construction. As long as the reaction remains on the left side of the cone, the position u(t) is smaller than U e and the velocity u(t) is strictly positive, the successive phases of the motion are given by:

                 i = 0, ...n -1, t ∈ ]2iT, (2i + 1)T ] : u(t) = U e + (u o -U e ) cos ω α t - ε 0 α 2 2i j=1 (-1) j cos ω α (t -jT ), t ∈ ](2i + 1)T, (2i + 2)T ] : u(t) = U e - ε 0 α 2 + (u o -U e ) cos ω α t - ε 0 α 2 2i+1 j=1 (-1) j cos ω α (t -jT ),
(12) where the number n of oscillations is associated with the period T of the load by:

T α 2n + 1 ≤ T < T α 2n -1 . ( 13
)
Consequently for any given half-period T in ]0, T α /2], we can calculate from equation ( 12) an intial data (u 0 , 0) such that the trajectory leads to the equilibrium U e with zero velocity after n oscillations (obviously a trajectory can reach U e with a zero velocity during the interval [0, T ] only in the trivial case u 0 = U e ). An example of such a trajectory is represented on Figure 10 for n = 7. Let us give the fully explicit corresponding calculations in the case T = T α /4, i.e. n = 2 from equation( 13). This implies that the velocity can be zero for the first time at some t ∈ ]3T, 4T ] given by tan

ω α t = sin ω α T -sin 2ω α T + sin 3ω α T cos ω α T -cos 2ω α T + 3 cos ω α T + α 2 ε 0 (u 0 -U e )
and time t will be the final point of the trajectory if u( t) = U e , which gives the initial data for such a trajectory

ũ0 = U e - ε 0 α 2 cos 2ω α T (2 cos ω α T -1) + cos 2 2ω α T (2 cos ω α T -1) 2 -4 cos ω α T (cos ω α T -1) 1/2
.

From now onwards, we denote by ũ0,α such an initial position to stress the fact that it corresponds to sliding phases to the right (and accordingly ũ0,β will denote an equivalent initial position of sliding phases to the left, when necessary). The corresponding trajectory is represented Figure 11-a.

Step 2. Let T be a given half-period of the force in ]0, T α /2], and (ũ 0,α , 0) the initial data leading to a trajectory which reaches the equilibrium U e in finite time for this period. Then, any initial data (u 0 , 0) with u 0 ∈ ]ũ 0,α , U e [ leads to a trajectory which converges to U e at infinity. Proof. The proof of Step 2 is based upon the following result:

Lemma 4.2. Let T ∈ ]0, T α /2
] and an initial data (u 0 , 0) such that U e -u 0 is strictly positive and satisfies the following inequality

U e -u 0 < 1 + tan 2 ω α T 2 α 2 ε 0 cos ω α T , (14)
then the trajectory involves only sliding phases to the right which pile up at the left of U e .

proof. Let us first observe that inequality (14) implies that starting from a point u 0 at t = 0, the velocity will be equal to zero at some time t 1 in ]T, 2T [. So that the motion is governed by

     t ∈ [0, T ] : u(t) = U e + (u 0 -U e ) cos ω α t, t ∈ [T, t 1 ] : u(t) = U e - ε 0 α 2 + (u 0 -U e ) cos ω α t + ε 0 α 2 cos ω α (t -T ), t ∈ [t 1 , 2T ] : u(t) ≡ u(t 1 ). (15)
Of course we have u 1 := u(t 1 ) > u 0 . Let then u k-1 := u(t k-1 ) be a point which is at equilibrium when the perturbation is equal to zero and from which a motion starts sliding to the right when the perturbation is set to ε 0 , and take (u k-1 , 0) as initial data. Inequality (14) implies that the trajectory involves only equation (15) so that we obtain the time t k and the position u k := u(t k ) when the velocity is equal to zero again:

       tan ω α t k = sin ω α T cos ω α T + α 2 ε 0 (u k -U e ) , u k = U e - α 2 ε 0 + ε 2 0 α 4 + (u k-1 -U e ) 2 + 2ε 0 α 2 (u k-1 -U e ) cos ω α T 1/2 . ( 16
)
We have then defined a sequence {u k } from equation ( 16). The sequence {u k } is shown to be such that u k+1 > u k , and u k+1 < U e and we easily check that it converges to U e , which completes the proof of Lemma 4.2.

Step 2 follows as a simple corollary. An example of such a case is represented on Figure 11-b, which is interesting to be compared to 11-a. The meaning of this comparison is that generically the convergence to equilibrium holds at infinity, but there exists particular values of the initial data for which the convergence holds in finite time, only two oscillations in the example of Figure 11-a.

Step 3. Let T be a given half-period of the external force, with T < T α /2 and (ũ 0,α , 0) the initial data leading to a trajectory which reaches the equilibrium U e in finite time for this period. Then, any initial data (u 0 , 0) with u 0 ∈ ]U d , ũ0,α [ leads to a trajectory which jumps to the other side of the cone and converges to U e .

Proof. We recall that the special position U d has been defined in equation ( 4) as the position such that all trajectories issued from (u 0 , 0) with u 0 < U d loose contact. The sketch of the proof is then the following: at first, we observe that as soon as u 0 ∈ ]U d , ũ0,α [, the trajectory sliding to the right goes beyond the equilibrium U e and, when its passes through zero, jumps to the other side of the cone. Secondly, as mentionned in Step 1, a position ũ0,β such that a trajectory starting from (ũ 0,β , 0) involves only sliding phases to the left and reaches the equilibrium in finite time, does exists if T < T β /2. Thirdly we show that any trajectory starting from (u 0 , 0) with u 0 ∈ ]U d , ũ0,α [ either enters the interval [ũ 0,α , ũ0,β ] in finite time if T < T β /2, or enters the interval [ũ 0,α , U e [ in finite time if not. From step 2, this implies the generic convergence at infinity to U e . Figure 12 represents an example of this convergence. Step 4. The range T ∈ ]T α /2, T α [ must be shared into two parts. We shall present the proof only in one case.

• (T α + T β )/2 < T < T α : trajectory always oscillates around U e and converges to U e at infinity; moreover the closer T is to T α , the slower the convergence is.

The proof of this point stems from the fact that for T larger than (T α + T β )/2 the motion stays at rest during the time interval [(T α +T β )/2, 2T ] up to the time when the perturbation is set to ε 0 again. This implies that each loop starts from an equilibrium position with a zero velocity, and this situation does not depend on the amplitude so that no phase difference accumulates from one loop to the next and only the value T α -T determines the rate of convergence towards U e . The trajectory is represented on Figure 13-a. This can be written explicitely.

Proof. Assume the initial data is (u 0 , 0) and the trajectory starts a sliding phase to the right at time t=0 when the perturbation is set to ε 0 . Then the trajectory is built piecewisely as the solution to the following system:

                             v1 + ω 2 α v 1 = F t + ε 0 -µF n , t ∈]0, T [, v 1 (0) = u 0 , v1 (0) = 0, v2 + ω 2 α v 2 = F t -µF n , t ∈]T, t[, v 2 (T ) = v 1 (T ), v2 (T ) = v1 (T ), where t such that v2 ( t) = 0. v3 + ω 2 β v 3 = F t + µF n , t ∈] t, t + T β [, v 3 ( t) = v 2 ( t), v3 ( t) = 0, ( 17
)
where the part v 3 involves only sliding to the left while the external force remains equal to zero. The end of this phase is a rest position up to the time 2T when the perturbation is set to ε 0 again. Through classical elementary calculations we get v 2 ( t) as a function of u 0 , and the arrival position of the loop:

v 3 ( t + T β ) := u 1 = U e -(v 2 ( t) -U e ). ( 18
)
Of course these calculations can be changed into those of a loop starting at (u k , 0) and arriving at (u k+1 , 0). For all k, let us introduce d k := U e -u k . Then equation (18) gives

d k+1 = d 2 k + ε 2 0 ω 4 α - 2ε 0 ω 2 α d k cos ω α T - ε 0 ω 2 α , (19)
Since (T α + T β )/2 < T < T α we have that π 2 < ω α T < π, so that equation (19) implies d k+1 < d k . The sequence {d k } has a lower bound which is zero, and passing to the limit in equation (18), we obtain the convergence of the trajectory to U e at infinity.

• T α /2 < T < (T α + T β )/2: the trajectory is less smooth, but still converges to U e .

The difference with the case (T α + T β )/2 < T < T α is that we now have u(2T ) = 0 which implies that instead of having loops which all start at some data (u k , 0), so that all the loops were solution of the same system with formally the same initial data, there is now a phase difference which accumulates at each loop between the trajectory and the loading. We can nevertheless, through calculations slightly more complicated that the previous ones, prove that the trajectory converges to U e . The rate of convergence can be non uniform, but is more or less stronger depending on whether T is close to T α /2 or to (T α + T β )/2. An example of this last case is represented on Figure 13-b.

Concluding remarks: towards a more general excitation

This short concluding section can be taken on the one hand as an introduction to the genericity of the results of the investigation, and on the other hand as a useful complement to the qualitative analysis presented in [START_REF] Léger | Qualitative analysis of a forced nonsmooth oscillator with contact and friction[END_REF]. The loadings F t (t) and F n (t) are still taken of the form:

F t (t) = F t + P t (t) and F n (t) = F n + P n (t),
where P t (t) and P n (t) are respectively a tangential perturbation and a normal one, and we still restrict our attention to the case of a tangential perturbation, but we now choose P t (t) = ε sin(γt). We do not give theoretical results, but only numerical ones obtained by symbolic or numerical calculations.

The transition

We first present the two main qualitative features:

• The exact value of the amplitude for the transition is not affected. As it was done on Figure 3 the set of equilibrium solutions can still be represented in the (R t , R n ) plane. We deduce that when ε = (µA)/K t the set {R n }(t) reduces to a single point, so that the set of normal component of the reactions at equilibrium is a non zero measure interval when ε < (µA)/K t and an empty set when ε > (µA)/K t . For example in the case of the numerical values given at equation (3) we obtain (µA)/K t = 0.75, which would also be the value obtained with a rectangular wave tangential force oscillating between -ε and +ε.

• The qualitative meaning of the transition is not affected. The transition shares the part of the {period -amplitude} plane where the trajectories do not loose contact into two parts, the lower one where non periodic solutions exist but infinitely many equilibrium states do, and the upper one, where periodic solutions exist but equilibrium states no longer do.

When ε is small enough

In this case the set {R n }(t) is a nonzero measure interval. The existence of an infinity of equilibrium solutions is then a straightforward consequence of Lemma (3.1) no matter the perturbation, whether of rectangular or sinusoidal wave shape. But it is interesting, both from the point of view of a constructive proof of the existence of equilibrium states and of specific notions of stability, to look at the trajectories starting from any initial data out of equilibrium. We still observe that all the trajectories go to equilibrium and an equilibrium state is always reached in finite time. Numerical simulations have been performed for different values of the frequency γ. The qualitative behavior represented on Figure 14 is very close to that of the case where the perturbating load was a rectangular wave, as studied in [START_REF] Léger | Qualitative analysis of a forced nonsmooth oscillator with contact and friction[END_REF]. perturbation was a rectangular wave, as can be seen on Figures 15-a,b. Nevertheless, some differences appear since the trajectory represented on Figure 16-a is slightly more complicated than what had been obtained before at the same points of the {period -amplitude} plane. Moreover, the complexity seems to be increasing for ε sufficiently large and large periods, which had never been observed with the rectangular wave. An exemple is given on Figure 16-b.

The behaviour on the transition

Here it seems that there is a qualitative difference with the case of the rectangular waves. The main result of this numerical investigation is the following:

• Whatever the frequency of the oscillating perturbation, the trajectory tends to the single equilibrium at infinity. Two examples of trajectories are represented on Figure 17. Figure 17: Evolution in the phase space (u t , ut ) when ε = 0.75 value of T on the transition line, we obtain only these two different types of trajectories. In the case of the rectangular wave these trajectories represent the qualitative dynamics only when T < T α . When T > T α , a rectangular wave leads to trajectories which no longer converge to the single equilibrium. They are either periodic or converge towards a periodic trajectory. This result of non existence of periodic solutions on the transition line in the case of a sinusoidal excitation must be handled with care since it only follows from numerical experiments for the moment, but it is an important qualitative point, which suggests further works in these lines.

Figure 2 :

 2 Figure 2: Equilibrium solutions in the (R t , R n ) plane.

Figure 3 :Figure 4 :

 34 Figure 3: The set of equilibria under oscillating loading. dotted line : when the tangential perturbation is equal to zero, full line : when the tangential perturbation is equal to ε.

  R ℓ ] and Lemma 3.1 implies that the mass shall remain motionless for t ∈ [T α , T ]. When the load changes it starts sliding to the left and once again Lemma 3.1 implies that the mass shall remain motionless for t ∈ [T + T β , 2T ]. So that for u 0 ∈ [U e -d, U e ] the motion is governed by the following system:

Remark 2 .

 2 i) On all the figures the use of the numerical values defined at equation (3) imply that the interval [U e -d, U e + d] is equal to [0.32, 1.28] .

  . Proposition 3.4 will prove how trajectories starting out of interval [U e -d, U e + d] behave and will also show that there are no other periodic solutions than those given by proposition 3.2. Let us start by showing that all sliding trajectories enter the interval [U r , U e -d[ in finite time.

Figure 5 :

 5 Figure 5: The maximal size periodic solution for ε = ε 0 and T > T α .

Figure 6 :

 6 Figure 6: One loop of the orbit for T > T α , with some particular values useful for the calculations

Figure 7 :

 7 Figure 7: A trajectory starting from initial positions in the interval [U r , U e -d[ with a zero velocity for T ≥ T α + T β .

Figure 8 :

 8 Figure 8: Two nonperiodic trajectories for ε = ε 0 and T > T α .

Figure 9 :

 9 Figure 9: The maximal amplitude periodic solution for ε = ε 0 and T = T α with U d < U r .

Figure 10 :

 10 Figure 10: An example of trajectory represented in the phase space converging to the single equilibrium in finite time.

  Figure 11: One trajectory, here represented in the phase space, converges to the single equilibrium in finite time while the other converges to the equilibrium at infinity.

0 1 1 Figure 12 :

 112 Figure 12: The trajectory enters the interval [ũ 0,α , ũ0,β ] in finite time and converges towards U e at infinity.

2 Figure 13 :

 213 Figure 13: Two trajectories represented in the phase space for Tα 2 < T < T α .

5 ε 2 Figure 14 : 7 a 2 Figure 15 :

 52147215 Figure 14: Evolution in the phase space (u t , ut ) when ε = 0.6

a 1 Figure 16 :

 116 Figure16: Evolution in the phase space (u t , ut ) for large periods