François Bonnarel
email: francois.bonnarel@astro.unistra.fr

Patrick Dowler
email: patrick.dowler@nrc-cnrc.gc.ca

Keith Noddle

Douglas Tody
email: dtody@nrao.edu

IVOA Data Access Layer: Goals, Achievements and Current Trends

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Virtual Observatory can be defined as a set of standard protocols useful for interoperability of data, data archives, and applications. These standards facilitate the definition of shared datamodels for astronomical data of all kinds, shared transport formats and standard registration of on-line astronomical resources. Standard interfaces are also defined for applications and services. Data Access Layer protocols are an important part of the whole scheme. They standardize the way applications can query data services, enable data discovery, describe resources and define data access and retrieval methods via server side processing. In 2002 the IVOA created a dedicated working group (the Data Access Layer Working Group or DAL) to define and develop these universally accepted standards.

DAL protocols provide solutions to questions such as : "Describe in a standard way all datacubes containing the 21 cm line in an area of 1 deg radius around quasar 3C 273"."Retrieve the central part of one of this cube giving the spatial and spectral size of the sub-cube designated by a standard identifier"."Retrieve from all catalog services in the VO all the sources of all stellar objects around the center of Crab nebula"'.

The core work of the DAL WG is the definition of interface specifications. DAL defines standards allowing the development of interoperable services but without dictating implementation details. The DAL protocols define the way to query services and the response to be returned.

DAL protocols have strong relationships with other IVOA standards. Query operations and responses are based upon IVOA DataModel concepts. Usage of DAL standardized vocabulary is consistent with work done by the IVOA semantics working group. Standardized description and discovery of the services is done according to rules defined by the IVOA Registry protocols. Asynchronous mode support of the DAL services rely on functionalities defined by the IVOA Universal Worker Service (UWS). Eventually VO Applications are consumers of DAL services.

2. First generation of DAL specifications: ConeSearch, SIA1.0 One of the first successes of IVOA was the definition and recommendation of a TA-BLE transport protocol, VOTable based on a specific XML schema [START_REF] Ochsenbein | [END_REF]. Early DAL protocols were HTTP GET services with predefined input parameters and a specific output TABLE. ConeSearch ( Williams et al. (2011) ) standardized "region of interest" (ROI) constraints for catalog of sources, whilst SIA1.0 [START_REF] Tody | [END_REF] ) was devoted to discovery and retrieval of images and cubes. The Query phase used ROI constraints like ConeSearch, plus additional constraints. The VOTable response describes the images and cubes in a standard way. An implicit image data model (based upon the FITS standard) underlies this image description. Virtual data generation (cutouts and mosaics) are also possible as well as image retrieval. While quite successful and simple to implement and use (11785 ConeSearch services and 195 SIA1.0 services are available from the IVOA registry as of October 2015) this first generation of standards predates modern IVOA standards and is in the process of being replaced by newer protocols based upon updated service semantics and datamodels.

2nd generation protocols : TAP, ADQL, SSA and SLA

The Astronomical community makes large use of measurements stored in catalogs. Catalogs are searched, exchanged, cross-identified and plotted in various ways to make science. The relational model is well suited to describing attributes and relationships between the tables. Relational DBMS implement this model in various ways. The goal of the IVOA is to offer an interoperable common distant interface to services of tables and catalogs implemented in various relational DBMS. A first and unsuccessful attempt was made with Skynode using the VOQL query language. The TAP protocol (Dowler et al. (2011)) offers this functionality by exposing the database schema in a standardized way with TAP_SCHEMA and VOSI-tables description. It exposes the metadata of the tables provided by the service (table and column names, units, datatypes and extended types, IVOA Uniform Content Descriptor and Universal Types). Queries can be synchronous or asynchronous. Asynchronous queries rely on the IVOA UWS architecture ( Harrison & Rixon (2011)). TAP queries can be expressed in different languages in principle, but most of the TAP services exclusively use the mandatory Astronomical Data Query Language (ADQL - [START_REF] Ortiz | the VOQL-TEG Working Group, & the VOQL Working Group[END_REF]) which is an extension of SQL enriched by astronomy specific features for client requests. There are 35 TAP services in the IVOA registry at the current date of October 2015.

Spectra and "Spectral Energy Distribution" are specific kind of datasets very common in Astronomy. They occur each time the spatially integrated light emission of astronomical sources is spectrally analyzed and differentiated. The IVOA is driven by scientific imperatives to deliver standardized ways to exchange spectra. The Data model Working group delivered the Spectrum Data Model [START_REF] Mcdowell | IVOA Data Access Layer, t., & Data Model Working Groups[END_REF]), an abstract and exhaustive description of measurements and metadata within a spectrum which can be serialized in several ways. The Spectrum Data Model reuses some parts of the STC Data Model [START_REF] Rots | [END_REF] ) and of the Characterization Data Model (Louys et al. (2011b)). Simple Spectral Access protocol (SSA -Tody et al. ( 2012)) uses standard query parameters to constrain all axes of the data as well as more specific features of the spectrum model. The response is an organized VOTable where GROUPS are mapped from the Spectrum data model packages and where the Table elements are tagged as model attributes using the utype attribute. SSA protocol includes virtual data generation by server-side processing based on a "best match to query parameters" delivery strategy.

The Simple Line Access (SLA -Salgado et al. ( 2011) ) protocol allows retrieval of spectral lines as VOTABLE serializations of the Simple Spectral Line Data Model. This standard follows the same principles as SSA.

Recent or current developments of new protocols driven by new science needs

Recently the avalanche of terabytes of new astronomical data produced by current or planned survey projects in all wavelengths reinforced the need for developing protocols to discover, describe and access all kind of datasets more efficiently.

A TAP technology based upon a generic dataset observation-summary metadata model called ObsCore lead to the emergence of ObsTAP services (Louys et al. (2011a) ), allowing the discovery of observational datasets of all kinds: raw or calibrated images, cubes, spectra, event lists, etc...

In parallel the IVOA Committee of Science Priorities encouraged the Working group to provide more specific protocols adapted to multidimensional data as new large observational projects urgently need to provide open access and interoperability for cross-correlation of their data. In this context, DataLink (Dowler et al. (2015) ) defines various methods and technologies to link on-line additional resources useful for better usage to known or already discovered datasets. SIAV2.0 is a parameterised query (PQL) adapted to multidimensional data of the ObsCore 1.1 data model for simple data discovery. AccessData is driving server-side processing for extracting information from the datasets. Intended applications are cutout, filtering, re-sampling or re-gridding and combination of several datasets. IVOA DAL manages a large number of protocols implemented in many services. Many of the questions faced in developing these protocols are common questions, such as Query global syntax, management of valid and error queries, format of the responses, management of synchronous or asynchronous modes, availability or capability auto-description. The common patterns and features have been extracted into Data Access Layer interface (DALI - Dowler et al. (2014) ) specification for easy re-use and consistency.

Conclusion: Lessons learned, open questions

The use cases for accessing astronmical data are so disparate that it is infeasible to provide one single service protocol satisfying all the requirements and allowing all capabilities, hence the approach has been to define distinct but consistent protocols for the different classes of data. A complete solution will only be achieved when all current protocols are realised. In this context DALI helps to federate development of such combined services. However two branches of protocols have existed for several years and will still coexist and "collaborate" in the future. Relational-model protocols (TAP, ADQL, ObsTAP) allow the most complex queries and are well suited for generic discovery of astronomical datasets. However, they are complex to implement for data archives. At the same time object-model protocols historically called "simple" protocols (Cone Search, SIA version 1 and 2, SSA, SLA) are simpler and more flexible to implement and can be more user friendly. However they require a solid and well defined underlying data model for each class of astronomical data. The discovery and data access (e.g., AccessData) protocols currently being implemented will be integrated to provide capabilities ranging from discovery to advanced data access via virtual data generation. At present DataLink already allows a flexible set of combinations between protocols of the two branches (eg, AccessData functionalities can be attached to either ObsTAP services or to "Simple" access services using any of these techniques). The Parameter Query Language evolution remains an open question because it is difficult to balance user-friendly readability of the syntax with the homogeneity of the input query content and the VOTable query response. At the response level, utypes have proven to be very efficient to link various level elements in VOTable responses with VO DataModels attributes and class definitions and are plainly efficient for metadata description and discovery tasks. They are less efficient for full VO serialisation of datasets in VOTable, which is the reason why the DataModel working group proposed the "VO-DML mapping to VOTable" for full inclusion of the DM structures into the responses. Again it's still an open question to know how these innovations will benefit future versions of the DAL protocols.

Acknowledgments. The authors are the four successive DAL chairs in reverse chronological order. They want to thank specifically the four successive DAL working group vice-chairs: M.Molinaro, M.Fitzpatrick, J.Salgado and M.Dolensky and further more all the colleagues participating to the DAL WG as well as members of other IVOA WG. F.Bonnarel acknowledges support from the Astronomy ESFRI and Research Infrastructure Cluster -ASTERICS project, funded by the European Commission under the Horizon 2020 Programme (GA 653477).


Walter Riofrio

Sensorial Information and the Roots of Cognition in Evolution

Trying to understand when and how the cognitive phenomena arise in evolution continues being a hard problem of confronting. Another important issue is referring to find the explanatory framework which integrates the different observations and data in neuroscience investigations. These situations lead us to ask a question in regards to whether there are underlying general principles to the so-called neural code: What might the relationship be that is established between the neuronal responses and the sensory signs or signals that these supposedly represent? In fact, one important task is to find the ways in which we will be able to arrive at those explanations that will enable us to understand how living beings put together perceived sensory information as well as how they represent it. In this brief paper, we address some conceptual questions involved in the relationships between sensorial information perceived by brains and its representational capacities. Specially, we propose that it is possible to defend the physical origin of Information. Therefore, we will study the implications to use the information notion as an ascribed concept and its relationships with using this same notion like denoting something in the reality.

Introduction

When we say the brain has the capacity to make some kind of representations, with these we are saying that, the brain produces certain information processes types from the data or signals from the exterior. And following, this processes "some how" produce an integration in such a way that it is generated a coherent and coordinated answer as a results of the neuronal activities.

This representation could be an image, a sound, a flavour, an idea. In other words, we are in front of the so called "binding problem" (Kahneman, Treisman and Gibbs 1992;Ungerleider and Haxby 1994;von der Malsburg 2001von der Malsburg , 2002)).

In this way, one basic problem is to explain the relationship between the actions of interconnected set of cells specialized in receive and transmit ⋅ Walter Riofrio is Associate Professor of Theoretical and Evolutionary Biology at the Neuroscience and Behavior Division, Universidad Peruana Cayetano Heredia, Lima-Peru. This version is the first draft of January 2007.

information with the production of some kind of answers mediated by the representational capacity of this cells aggregations.

We found in the brain, the existence of different centres with the task to process certain types of signals (for example, visuals at the moment to observe a table or a book) and we realize that there are many neuronal connections in synchrony (Milner 1974;von der Malsburg 1995;Varela 1995;Kreiter and Singer 1996).

At one moment, it presents a tune between synchronic neural fires in one part of the brain with other brain portions that are activated in synchronic ways.

Until now, we do not know how to find a design -in laboratories and in computational simulations-that would resolve the implications derived to integrally the spatial and temporal coordination of all the processed signals which produce a coherent resultant, as the obstacle identification for us.

We approaches to this problem, with the proposal that the "low level" mental phenomena are not totally explained by virtue of mechanisms, and computations exclusively. In other words, we claim that it is needed to deepening on the information notion, and to connect it with the emergence notion, with the capacity of neurons structuring and with future scaling-developments in evolutionary terms.

Our working hypothesis proposes installing the questions in the most basic mental phenomena. With the final goal, to track until last consequences the origin of these capacities greatly extended in the living realm.

Our research proposal has the central objective to construct a theoretical approach (conceptually and with the possibility to design reliable algorithms), which will open the door to an overall comprehension of the successive steps, for the emergence of mental levels.

Integration of Sensorial Information

an ordinary situation: a person, walking through a park, begins to clap his hands; suddenly, the birds on the grass around him take off in flight. We can ask ourselves, "How did the birds perceive these claps?" Further, if there were another animal in the area, such as a cricket a mouse, how would it perceive them? These situations lead us to ask a question in regards to whether there are underlying general principles to the so-called neural code: What might the relationship be that is established between the neuronal responses and the sensory signs or signals that these supposedly represent?

It is important to understand that the terms "stimulus" and "response" include within them an extremely large group of components, such as the different ways for receiving a determined type of sensory information living beings possess or the various and multiple forms in which this specific type of sensory information flows through a determined group of reception centers (smell, taste, sight, etc.) for that information. A typical case is what happens in the different dynamics that are observed in the predator-prey interaction, for example.

An interesting discussion of these topics is one we find in a recent work on the behavior of bats as they are hunting prey (Ghose et al. 2006). It seems that on a wide scale evolutionary basis the constant bearing (CB) strategy is the one most often used. This strategy is the optimal way a predator to capture its prey or a moving object, following a straight line or a predictable trajectory (Olberg et al. 2000;Shaffer et al. 2004).

in the case of bats, the authors propose that the most efficient strategy of this type of predator-prey interaction is not the CB but the CATD strategy (constant absolute target direction). This strategy predicts a bearing that would be defined by the absolute direction between the predator and the prey. In their studies, the researchers find that the bat capture its prey by adjusting the direction of its flight path and its speed of pursuit. And so, it would consistently maintain the absolute direction to its target throughout the pursuit: "…When the bat converges to (and maintains) the optimal bearing, the absolute direction to the target does not change. The CATD strategy produces a trajectory which, from the viewpoint of the target, makes the pursuer ''appear'' stationary against a distant background, and vice-versa." (Ghose et al. 2006, pp. 0006).

In fact, one important task is to find the ways in which we will be able to arrive at those explanations that will enable us to understand how living beings put together perceived sensory information as well as how they represent it.

Emergence of Information

Focusing our attention for a few moments on a controversy that is produced among the consequences that are derived from the laws of quantum physics and the physics of black holes, we have, on one hand, the former, which states that information (in a similar manner to energy) "cannot be created or destroyed." On the other hand, in a black hole, "information escape." This is known as the "black hole information paradox " (Danielsson and Schiffer 1993;Susskind 2003).

It is possible for us to have the following situations: first, we ask for the information that is contained in a determined object, for example, a CD Secondly, we wonder about the information that would be needed to in a random sort of way somewhere in the universe all the necessary phenomena of physics in order for the parts of a watch to "spontaneously assemble". And, the third situation is one of the possibility of measuring the information content that is necessary for an ant colony to explore the required strategies for detecting, finding, and transporting a source of sugar to their nest that lies some distance away from said source.

We have to ask ourselves if, in these cases, whether the term "information" is being used in the same way. If what we are talking about is the way in which can formalize the measurement of the information content in these situations, then the answer is probably yes.

Nevertheless, it seems that for the case of the ants there are some extras to the notion of information that are not found in the other situations. We can affirm that among the phenomena produced in the ant colony, it must have been possible for them to transmit certain among other so as to be able to produce the coordinated behaviors as a whole that we observed. This is something that we cannot state with certainty in the other two cases. So, it seems that we are faced with two uses of the notion of information; the first use has something to do with the way in which "we ascribe" a certain information content that is susceptible to being treated in formal terms by some suitable information theory for that particular case.

The second case has more to do with the "production of information" that is used by entities, which are adaptive, dynamic systems, and that, likewise, may be susceptible to being treated in formal terms. What difference have we found between these two cases?

A first approximation is to propose that this difference among the uses of the notion of information (in the cases we set out above) is not casual. Instead, it could be revealing a problem that ultimately makes us question the composition and the interaction of the most basic things of the world (Godfrey-Smith 1998;Rouvray and Bonchev 2005).

It might be a problem that establishes the difference between using a notion through ascription (imposed by us for the purpose of deepening the understanding of determined phenomenology) and using this same notion as indicating its existence in physical terms for certain entities of the world.

The second use could be making us understand that the notion of in the world; it is a capacity, a property, or a characteristic of certain things in the world. In other words, the term "information" has a physical reference in reality.

The first use could indicate that some notions (for example, the notion of information) might be susceptible to being largely abstracted and, therefore, used on "a wide-scale basis" to almost anything in the universe.

Hence, the first use (the "wide-scale approach") has its theoretical and applicable legitimacy since we have discovered that it is real. An important consequence that is derived from our earlier reasoning, therefore, is that it might be possible to propose that there was a time when information appeared in the universe, that information had an origin.

The central question within this analysis refers to the causes that have occurred during evolution for the appearance of the production of interpretations (meanings) in the brains of these animals from the sensory information they receive. It would be interesting if we could discover that

biological information (understood in a wider sense) already carries a certain "meaning" (a basic semantic) and is perhaps associated with the molecular processes that generate the biological functions within a network of processes. Thus, it would not seem so illogical to think that the search for the origins of cognition will take us directly to a deepening of the notions of biological information and biological functions as well as to the possible relationships that take place between them.


Walter Riofrio

Dynamic changes in Prebiotic Systems

Keywords: cohesion, complex systems, emergence, evolvability, far from thermodynamic equilibrium, open-ended evolution

Introduction

Understanding the origin of living systems is grasping the most basic capacities that are used in the construction of the organization of these systems [START_REF] Shapiro | Origins: A Skeptic's Guide to the Creation of Life on Earth[END_REF][START_REF] Fox | Molecular Evolution and the Origin of Life[END_REF][START_REF] Lurquin | Origins of Life and the Universe[END_REF].

The prebiotic world can be conceived of as a group of systems that generated the move from the inanimate to the animate. In more precise terms, the prebiotic world must have been constituted by a series of more or less successive steps and each one of them, was represented by a specific type of dynamic system. In each successive stage, the respective dynamic system must have contained those capacities/ improvements possessed by the systems from previous steps and, in It is possible to think that in the prebiotic epoch the conditions should have existed that enabled the development of the rudiments of a dynamic organization containing the essentials for the subsequent emergence of life, a specific type of dynamic that would be associated with the of the different prebiotic systems.

Variability in prebiotic systems

know that two variability types exist in biological systems: one that is produced by genetic changes (mutations) and is therefore inheritable and the second, which is produced by changes in the environment and associated with the system's capacity to maintain its cohesion [START_REF]Phenotypic plasticity in the interactions and evolution of species[END_REF][START_REF] Hoffmann | Heritable variation and evolution under favourable and unfavourable conditions[END_REF]. It is associated with multiple strategies that produce homeostatic phenomena [START_REF] Marder | Variability, compensation and homeostasis in neuron and network function[END_REF]. Although this second variability type is not "inheritable", in the sense that we are accustomed to thinking of in biology, it is still possible to think about an open-ended evolution that focuses on increasing the complexity of the systems.

More important than preserving some of components that are reliably transmitted to the successive generations, it seems reasonable to claim that during the prebiotic epoch, it was crucial to maintain the cohesion of its organization at all costs. The dynamic of its organization is revealed in the fundamental properties that make its integrity and identity explicit. Therefore, maintaining the conditions that enable the stabilization of these propertiesmaintaining them robustly -is that which must have been preserved in the different systems of the prebiotic world.

To characterize the identity of the dynamic organization of these systems is to identify the most fundamental properties that define them. Once these properties are located, it is a question of widening the notion of evolution by natural

selection towards an open-ended evolution. Therefore, instead of occupying ourselves with having to follow up on the inheritable variability, we are now interested in the variability that preserves the cohesion of the fundamental properties in contexts of increasing the complexity of the networks of processes allow the expression of these properties.

If we apply this approach to the variations in the prebiotic universe, then we will inevitably arrive at the conclusion that the fundamental causes for variability in those systems were changes in its environment. In other words, since there was nothing like "genetic materials" at the beginnings of the prebiotic world, the system evolved in the face of problems generated by the environment through different ways of preserving the basic which characterized them. So, the dynamic systems of the prebiotic world must have developed -taking into consideration that they survived these adverse conditions -strategies that enabled them to establish nexuses between the external environmental changes and the internal developments that perpetuated the dynamic cohesion of their organization.

We now realize that there are a variety of processes in the plasmatic membranes of cells that enables them, within certain limits, to suitably manage their conditions and internal requirements. Including isolation from its environment, it has a capacity that actively selects certain matter-energy components from its exterior and carries them to its interior.

We may hypothesize that the extreme complexity and sophistication of these processes correspond to the final phases of strategies developed very early by evolutionary processes in which improvements were gradually produced. Moreover, their presence is important for resolving problems that seriously place the cellular integrity at risk if they are not worked out, problems such as the osmotic balance, the electrochemical transportation, and the contribution of compounds to internal processes.

The above would reveal to us the earliest ancestor of these actions. It would be pointing out that its importance and age lie in the intimate relationship they have with the spontaneous origin of a dynamic organization of a system that will guide us -much, much later -towards the world of the living.

Evolvabilty and Inheritance

In this way, we may hypothesize that from their most primitive origins, the true ancestors of the living systems had the capacity to interact with and to express an active behavior towards their environment. If we are in right track, then, the successive systems in the prebiotic universe had as one of their most important dynamic structures a protoplasmic membrane that formed a fundamental part of their organizational composition. This structure would have been formed of molecules interacting in processes they would have simulated at very basic levels, those we can still see in the plasmatic membranes of the current cells.

Apart of to be a physical barrier, this protoplasmic membrane permits certain properties of chemistry to have a focal point of action to generate problematic circumstances which in turn produce the conditioning situations for exploring different solution strategies. And is this kind of protoplasmic membrane what the systems in prebiotic epochs with the capacity of being evolvable from its very origins.

We talk about evolvability when we are able to observe a phenotypic variation that is susceptible to being a target of natural selection or, in other words, when an organism has the capacity to generate a phenotypic variation that is inheritable [START_REF] Altenberg | The Evolution of Evolvability in Genetic Programming[END_REF][START_REF] Conrad | The geometry of evolution[END_REF][START_REF] Dawkins | The Evolution of Evolvability[END_REF][START_REF] Kirschner | Evolvability[END_REF][START_REF] Wagner | Complex adaptations and the evolution of evolvability[END_REF]. Nevertheless, when we are thinking upon the origins of the prebiotic world, we cannot establish these types of associations. It is nearly impossible to think that in those remote epochs there existed molecular entities so complex and sophisticated as nucleic acids or proteins. Hence, it is not possible to talk about evolution in terms of natural selection during those times; what we are proposing, then, is that it is possible to talk about evolvability in the prebiotic world.

Once these systems are confronted by a specific problem that the environment generates, the different possible solutions (strategies) produced in the system's protoplasmic membranes as a product of the reproduction of these systems are nothing more than the maintenance of the integrity of their dynamic organization. This sort of prebiotic "inheritance" must have been associated with a certain evolvability capacity rather than with a Darwinian type of evolution.

In other words, every time the systems reproduced we could see that inside the processes, the components could be changed, but as a whole, the system's descendents attempted to keep their most fundamental properties intact and compatible with its survival; that is, in conditions compatible with the far from equilibrium state.

To end this work, it seems feasible to propose the idea that the basis of hereditary similarity in "prebiotic evolution" depended on the way cell-like systems were embedded in their environments, rather than on internal, gene-like mechanisms. We also suggest evolution is happening at small-time scales -so it

  addition, it had the chance to explore new strategies or to improve on what was already there.

is continuous -due to that the membrane makes the cell "evolvable". Also, we consider interesting to continue the exploration of the possible consequences derived of the previous statements and proposals sketched in this article, with the ultimate goal of to widen current comprehension of evolution.


David J Wagg
email: david.wagg@bristol.ac.uk

Multiple Non-Smooth Events in Multi-Degree-of-Freedom Vibro-Impact Systems

Keywords: chatter, multiple constraint, sliding, sticking, vibro-impact

The behaviour of a multi-degree-of-freedom vibro-impact system is studied using a 2 degree-of-freedom impact oscillator as a motivating example. A multi-modal model is used to simulate the behaviour of the system, and examine the complex dynamics which occurs when both degrees of freedom are subjected to a motion limiting constraint. In particular, the chattering and sticking behaviour which occurs for low forcing frequencies is discussed. In this region, a variety of non-smooth events can occur, including newly studied phenomena such as sliding bifurcations. In this paper, the multiple non-smooth events which can occur in the 2 degree-of-freedom system are categorised, and demonstrated using numerical simulations.

Introduction

This paper deals with the dynamics of multi-degree-of-freedom impact oscillators subject to multiple motion limiting constraints. These systems consist of a set of coupled masses, where the motion of each of the masses is restricted by a series of motion limiting constraints. This paper will consider a 2 degreeof-freedom system with constraints placed a different distances from each mass [START_REF] Wagg | Dynamics of a two degree of freedom vibro-impact system with multiple motion limiting constraints[END_REF]. Several authors have considered two constraints placed an equal distance either side of an oscillating mass, e.g. [START_REF] Shaw | The onset of chaos in a two-degree of freedom impacting system[END_REF][START_REF] Hogan | Graph theory and piecewise smooth dynamical systems of arbitrary dimension[END_REF][START_REF] Lenci | Regular nonlinear dynamics and bifurcations of an impacting system under general periodic excitation[END_REF].

Two degree-of-freedom impact oscillator systems have been studied in relation to a range of applications [START_REF] Masri | Theory of the dynamic vibration neutraliser with motion-limiting stops[END_REF][START_REF] Chatterjee | On impact dampers for non-linear vibrating systems[END_REF][START_REF] Neilson | Chaotic motion of a rotor system with a bearing clearance[END_REF]. More general multi-degree-of-freedom impact systems have also been considered by several other authors [START_REF] Theodossiades | Periodic and chaotic dynamics of motor-driven gear-pair system with backlash[END_REF][START_REF] Cusumano | Period-infinity periodic motions, chaos and spatial coherence in a 10 degree of freedom impact oscillator[END_REF][START_REF] Nigm | Effect of an impact damper on a multi-degree of freedom system[END_REF][START_REF] Pfeiffer | Multibody Dynamics with Unilateral Contacts[END_REF][START_REF] Lamarque | Modal analysis of mechanical systems with impact non-linearities: Limitations to a modal superposition[END_REF]. Of particular interest have been periodic impacting orbits which occur in multi-degree-of-freedom impact systems. Systems with a single impact stop have been studied [START_REF] Natsiavas | Dynamics of multiple-degree-of-freedom oscillators with colliding components[END_REF][START_REF] Pun | Forced vibration of a multidegree impact oscillator[END_REF], and the method for finding period (1, n) solutions developed for single degree-of-freedom impact oscillators by [START_REF] Shaw | A periodically forced piecewise linear oscillator[END_REF] has been extended to multi-degree-of-freedom impact oscillators [START_REF] Natsiavas | Dynamics of multiple-degree-of-freedom oscillators with colliding components[END_REF]. Studies into bifurcations in this type of system have also been carried out, for example [START_REF] Luo | Hopf bifurcations and chaos of a two-degree-of-freedom vibro-impact system in two strong resonance cases[END_REF].

Chatter and sticking in single degree-of-freedom impact oscillators has been studied in detail [START_REF] Budd | Chattering and related behaviour in impact oscillators[END_REF][START_REF] Nordmark | On Chattering Bifurcations in 1DOF Impact Oscillator Models[END_REF] and also noted in 2 degree-of-freedom systems [START_REF] Wagg | Chatter, sticking and chaotic impacting motion in a two-degree of freedom impact oscillator[END_REF]. The behaviour of periodic sticking motions in both single and multi-degree-of-freedom systems has also been studied [START_REF] Toulemonde | Sticking motions of impact oscillators[END_REF]. The sticking phenomena observed here is analogous to sliding in other systems [START_REF] Bernardo | Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations[END_REF], as discussed in [START_REF] Wagg | Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator[END_REF]. In this paper, we consider the same 2 degree-of-freedom example studied in [START_REF] Wagg | Dynamics of a two degree of freedom vibro-impact system with multiple motion limiting constraints[END_REF], but examine the multi-dimensional non-smooth events which can occur in the chatter and sticking region.

Mathematical Model

A schematic model of the generalized N degree-of-freedom coupled linear oscillator system with N lumped masses is shown schematically in Figure 1(a). In general, the equations of motion for the coupled masses can be expressed as

m i ẍi + c i ( ẋi -ẋi-1 ) + c i+1 ( ẋi -ẋi+1 ) + k i (x i -x i-1 ) + k i+1 (x i -x i+1 ) = f i (t), (1)
for i = 1, 2 . . . , N -1 and

m N ẍN + c N ( ẋN -ẋN-1 ) + k N (x N -x N -1 ) = f N (t) (2)
for i = N , where x i and f i (t) represent the displacement and forcing of mass m i , and an overdot is used to represent differentiation with respect to time t. These expressions govern the motion while all the displacements x i are less than some fixed set of values s i corresponding to the position of the motion constraints.

The equations of motion for the coupled masses can be expressed in matrix form as

[M]ẍ + [C]ẋ + [K ]x = f(t), (x i -s i ) ≶ 0 ∀s i ≷ 0 (3)
where [M], [C], [K ] are the mass, damping and stiffness matrices respectively, x = {x 1 , x 2 , . . . , x N } T is the displacement vector and f(t) = { f 1 , f 2 , . . . , f N } T the external forcing vector. The coupling between masses occurs via the matrices [C] and [K ], which are non-diagonal. The mass matrix [M] is a diagonal matrix. Equation (3) has the dual condition for free flight that (x is i ) < 0 for s i > 0 and (x is i ) > 0 for s i < 0.

It is assumed that the damping matrix [C] is linearly proportional to the stiffness matrix [K ], such that Equation (3) can be decoupled in the standard way [START_REF] Meirovitch | Analytical Methods in Vibration[END_REF]. In this work, the case where m j = m, c j = c, k j = k for j = 1, 2, . . . , N , is considered. Then Equation (3) can be written in the form

[I ]ẍ + c m [E]ẋ + k m [E]x = 1 m f(t), (x i -s i ) ≶ 0 ∀s i ≷ 0 (4)
where [E] is the N × N coupling matrix

[E] = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2 -1 0 . . . 0 -1 2 -1 . . . 0 . . . . . . . . . . . . . . . 0 . . . -1 2 -1 0 . . . 0 -1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ( 5
)
and [I ] is the identity matrix.

The natural frequencies are given by ω nj = λ j k/m for j = 1, 2, . . . , N where λ j are the eigenvalues of matrix [E], and the corresponding normalized eigenvectors ξ j can be used to construct a orthogonal modal matrix [ ] = [{ξ 1 }, {ξ 2 }, . . . , {ξ N }]. Equation (4) can then be transformed into a modal form by defining modal coordinates x = [ ]q where q = {q 1 , q 2 , . . . q N } T , such that

[I ] q + c m [ ] q + k m [ ]q = 1 m [ ] T f(t) (6)
where

[ ] = [ ] T [E][ ] is the diagonal matrix of the eigenvalues, λ j , j = 1, 2, . . . N .
In this modal formulation the vector ψ i = { i1 , i2 , . . . , i N } T , is defined such that an impact occurs when ψ T i q = x i . Hence, Equation ( 6) is valid only for (ψ T i qs i ) ≶ 0 ∀s i ≷ 0, which is equivalent to the condition that (x is i ) ≶ 0 ∀s i ≷ 0 for the ith impacting mass.

The system is considered to be subject to harmonic forcing of the form f(t) = A cos( t), where A = {A 1 , A 2 , . . . , A N } T , then Equation ( 6) can be simplified such that for each mode

q j + 2ζ j ω nj q j + ω 2 nj q = f j m cos( t), j = 1, 2, . . . , N (7)
where

f = [ ] T A, f = { f 1 , f 2 , . . . , f N } T and ζ j = (c/2) λ j /
km is the modal damping coefficient. Equation [START_REF] Neilson | Chaotic motion of a rotor system with a bearing clearance[END_REF] has the well known exact solution for under-damped oscillations 0 < ζ j < 1 [START_REF] Meirovitch | Analytical Methods in Vibration[END_REF] such that for each mode an exact solution can be obtained, and from this the total displacements for x [START_REF] Wagg | Dynamics of a two degree of freedom vibro-impact system with multiple motion limiting constraints[END_REF].

MODELLING IMPACTS FOR SYSTEMS WITH MULTIPLE CONSTRAINTS

An impact occurs for the ith mass when x i = s i , while for j = i:(x js j ) ≶ 0 ∀s j ≷ 0. To model the impact, an instantaneous coefficient of restitution rule is used [START_REF] Thompson | Nonlinear Dynamics and Chaos[END_REF] ẋi (t

+ ) = -r ẋi (t -) x i = s i (8)
where, t -is the time just before impact, t + is the time just after impact and r is the coefficient of restitution with a value in the range r ∈ [0, 1].

For systems with multiple constraints the matrix form the coefficient of restitution rule is

ẋ(t + ) = [R]ẋ(t -) (x i -s i ) = 0 (9)
where [R] is the N × N diagonal coefficient of restitution matrix. For a system with N impacting masses [R] will have a different form depending on whether single, multiple or all the masses make contact during the impact process [START_REF] Wagg | Dynamics of a two degree of freedom vibro-impact system with multiple motion limiting constraints[END_REF].

In modal form the coefficient of restitution rule, Equation ( 9), for a single impact becomes

[ ] q(τ + ) = [R][ ] q(τ -), ψ T i q -s i = 0. ( 10
)
This leads to the relation for the modal velocities after impact

q(τ + ) = [ R] q(τ -), ψ T i q -s i = 0, (11)
where

[ R] = [ ] -1 [R][ ]
is the set of matrices which represents a linear transform of modal velocities just before impact to modal velocities just after impact for all the possible impact cases.

STICKING MOTION

Sticking motions occur when one or more of the masses is held motionless against the stop for a finite period of time. Sticking motions can occur in multi-degree-of-freedom impact oscillators after a complete chatter sequence has occurred [START_REF] Budd | Chattering and related behaviour in impact oscillators[END_REF][START_REF] Wagg | Chatter, sticking and chaotic impacting motion in a two-degree of freedom impact oscillator[END_REF]. A chatter sequence becomes complete when the time between two successive impacts, δt becomes small, while at the same time the forces on the mass hold it against the impact stop. So, in order to reach a sticking solution for a single mass (the pth say) chatter must be complete, i.e δt ≈ 0 and the force acting on the sticking mass must hold it against the constraint, which is equivalent to the condition F p s p > 0. This is similar to conditions for a relay system referred to as the reaching conditions [START_REF] Bernardo | Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations[END_REF]. There is one possible exception to these conditions, that is if a mass comes into contact with the stop with zero velocity and acceleration and simultaneously F p s p > 0 becomes true. This non-generic case is an example of a grazing-sliding bifurcation discussed in Section 3.3.

To find the force F p , we substitute x p = s p and ẋ p = 0 into the pth line of Equation ( 4). So for 1 ≤ p < N from Equation ( 1) with all m, c and k values equal

F p = c( ẋ p-1 + ẋ p+1 ) + k(x p-1 + x p+1 ) + f p (t) + 2ks p , ( 12
)
and for p = N from Equation (2),

F p = c ẋ p-1 + kx p-1 + f p (t) -ks p . ( 13
)
The end of sticking is defined as when F p changes sign. As a result, Equations ( 12) and ( 13) set equal to zero, can be used to define the exit boundary of the sticking region. So, the region of sticking trajectories can be defined as S, which is bounded on one side by the exit boundary ∂ S defined by F p = 0.

A 2 Degree-of-Freedom System Example

A 2 degree-of-freedom impact oscillator with multiple constraints is shown schematically in Figure 1(b). The following parameter values have been selected: masses 4), the equations of motion for two coupled masses can be expressed as

m 1 = m 2 = 1, stiffness k 1 = k 2 = 1, viscous damping c 1 = c 2 = 0.1, coefficient of restitution r = 0.7. From Equation (
ẍ1 + c m (2 ẋ1 -ẋ2 ) + k m (2x 1 -x 2 ) = A 1 m cos( t), ( 14
) ẍ2 + c m ( ẋ2 -ẋ1 ) + k m (x 2 -x 1 ) = A 2 m cos( t). ( 15
)
where x 1 represents the displacement of mass m 1 and x 2 the displacement of mass m 2 . When (x i -s i ) = 0 for i = 1, 2 an impact occurs and an instantaneous coefficient of restitution rule is applied via Equation ( 9). For this system the three [R] matrices are

[R 1 ] = -r 0 0 1 , [R 2 ] = 1 0 0 -r , [R 3 ] = -r 0 0 -r . ( 16
)
The eigenvalues of the 2 × 2 coupling matrix [E] are λ 1 = 0.382 and λ 2 = 2.618, and the corresponding normalised eigenvectors, ξ 1 = [0.526, 0.851] T and ξ 2 = [-0.851, 0.526] T , which give the mode shapes for the non-impacting system. Using the modal transform described in Section 2, we can express the modal equations of motion for this example as

q1 + 2ζ 1 ω n1 q1 + ω n1 q 1 = f 1 m cos( t), ( 17
) q2 + 2ζ 2 ω n2 q2 + ω n2 q 2 = f 2 m cos( t). ( 18
)
For this example there are two modal impact vectors, ψ 1 = [ 11 , 12 ] and ψ 2 = [ 21 , 22 ], such that ψ 1 q = s 1 and ψ 2 q = s 2 , where q = [q 1 , q 2 ] T . For the numerical simulations in this paper, we set the forcing amplitudes as A 2 = 0 and A 1 = 0.5 and take initial conditions q 1 (t 0 ) = q 2 (t 0 ) = q1 (t 0 ) = q1 (t 0 ) = t 0 = 0.

SOLUTIONS FOR STICKING

In the case when x 1 = s 1 and ẋ1 = 0 the reduced equation of motion with

A 2 = 0, is ẍ2 + c m ẋ2 + k m (x 2 -s 1 ) = 0, (19)
and the force which holds the mass against the stop during sticking is given by

F 2 = c ẋ2 + k(x 2 -2s 1 ) + A 1 cos( t). ( 20
)
Equation ( 19) has the exact solution

x 2 = e -ζ ωn (t-t s ) (C 1 cos( ωd (t -t s )) + C 2 sin( ωd (t -t s ))) + s 1 , (21)
where ωn = √ k/m, ζ = c/2m ωn and ωd = ωn 1 -ζ 2 . At the start of the sticking period t s = t and the constants C 1 and C 2 can be found using the initial conditions x 1 (t s ) = s 1 and ẋ1 (t s ) = 0.

In the case when x 2 = s 2 and ẋ2 = 0, the reduced equation of motion is given by

ẍ1 + 2 c m ẋ1 + k m (2x 1 -s 2 ) = A 1 m cos( t). ( 22
)
The force which holds the mass against the stop during sticking is given by

F 1 = c ẋ1 + k(x 1 -s 2 ). (23)
Equation ( 22) has the exact solution

x 1 = e -2 ζ ωn (t-t s ) (C 1 cos(2ω * d (t -t s )) + C 2 sin(2ω * d (t -t s ))) + C 3 cos( t -φ * ) -s 2 /2, ( 24
)
where ωn = √ k/m, ζ = c/2m ωn and ω * d = ωn 0.5 -ζ 2 and t 0 is taken at the start of the sticking period. Full details of the derivation of these sticking solutions can be found in [START_REF] Wagg | Dynamics of a two degree of freedom vibro-impact system with multiple motion limiting constraints[END_REF].

PERIODIC STICKING MOTION

A numerically computed example of sticking motion is shown in Figure 2(a) with stop distance values s 1 = -0.3 and s 2 = 0.1, forcing amplitude A 1 = 0.5, and forcing frequency = 0.2. The figure shows the displacement of both masses for this set of parameter values. The motion is period infinity steadystate motion and each mass has a complete chatter sequence and sticking period during one excitation period. Complete chatter motions with sticking are referred to as period infinity periodic motions because an infinite number of instantaneous impacts occur in one period [START_REF] Budd | Chattering and related behaviour in impact oscillators[END_REF]. In Figure 2(b) we show a close up of the chatter sequence computed for mass 2. The chatter peaks diminish approximately exponentially and can be studied via mappings [START_REF] Budd | Chattering and related behaviour in impact oscillators[END_REF][START_REF] Nordmark | On Chattering Bifurcations in 1DOF Impact Oscillator Models[END_REF]. The simulation was computed by using the exact solutions to the modal equations for both free flight and sticking for a fixed time step of 0.01. The time of impacts were found using linear interpolation with a tolerance of 10 -9 . More details on the computations for this example are given in [START_REF] Wagg | Dynamics of a two degree of freedom vibro-impact system with multiple motion limiting constraints[END_REF], for a general survey of numerical methods for mechanical systems with impacts see [START_REF] Janin | Comparison of several numerical methods of mechanical systems with impacts[END_REF].

There is a special class of orbits which have a complete chatter sequence, but then lift off before a finite sticking time occurs -referred to as border orbits [START_REF] Nordmark | On Chattering Bifurcations in 1DOF Impact Oscillator Models[END_REF]. In vibro-impact systems these border orbits represent the boundary between the sticking and non-sticking regimes as a system parameter is varied. This can be seen in Figure 3 where examples of bifurcation diagrams for the 2 degree-of-freedom system with unequal constraints of different sign, computed for a forcing amplitude A 1 = 0.5 are shown. In Figure 3(a) the impact velocity, v 1 = ẋ1 of mass 1 is shown against forcing frequency, in the range 0.25-0.7, and in (b) the impact velocity, v 2 = ẋ2 is shown against the same frequency range. We note that because s 1 < 0 the impact velocities for mass 1 are all less than zero, and likewise as s 2 > 0 the impact velocities for mass 2 are all greater than zero.

The region of sticking motions all exist at forcing frequencies, < 0.5, which can be seen from the chatter impact velocities successively decreasing toward zero. At low frequency, < 0.5 periodic sticking motions preceded by complete chatter exist. Then as is increased past the sticking region, chatter becomes incomplete -and the system passes through a border orbit. In fact, careful observation shows that for mass 1, chatter remains incomplete for 0.3 0.35 and 0.37 0.4, such that additional border orbits exist close to these parameter values. In Section 2.2 the relationship F p = 0 was used to define the boundary in phase space where sticking ends. For example, for the system shown in Figure 2, the case when x 2 = s 2 and ẋ2 = 0, the system trajectories during sticking will be restricted to the x 1 , ẋ1 space. By setting Equation ( 23) to zero we define the relationship for the end of sticking as ẋ1 = -(k/c)x 1 + (k/c)s 2 = ẋ1 = -10x 1 + 1, which defines the exit boundary of the sticking region S which is denoted ∂ S. For sticking to exist the condition F p s p > 0 must apply, which in this case is the region on the positive side of the ∂ S. Note also that ∂ S includes the point (0.1, 0) which corresponds to the (x 2 , ẋ2 ) values during sticking. However, when mass 2 sticks, setting Equation [START_REF] Toulemonde | Sticking motions of impact oscillators[END_REF] to zero results in a relationship for the end of sticking given by ẋ2 = -(k/c)(x 2 -2s 1 ) -A 1 cos( t) = ẋ2 = -10(x 2 + 0.6) -0.5 cos( t). This exit boundary for the sticking region is now dependent of the forcing frequency .

In the work on sliding orbits by [START_REF] Bernardo | Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations[END_REF], S was defined using Utkin's equivalent control method. However, in this case only a subset of the system states are restricted to S, with the result that we cannot define S simply in terms of the system parameters alone, we must include some of the system states.

MULTIPLE NON-SMOOTH EVENTS

Sliding bifurcations have been studied in detail for relay systems [START_REF] Bernardo | Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations[END_REF] and friction oscillators [START_REF] Bernardo | Sliding bifurcations: A novel mechanism for the onset of chaos in dry friction oscillators[END_REF]. The occurrence of the multi-sliding bifurcation in vibro-impact systems was first highlighted by [START_REF] Toulemonde | Sticking motions of impact oscillators[END_REF] (referred to as a rising bifurcation) and has been studied for 2 degree-of-freedom systems by [START_REF] Wagg | Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator[END_REF]. Multi-sliding occurs when a sliding orbit in the region S becomes tangent (in a similar way to grazing) to the exit boundary ∂ S as a system parameter is varied. In the system studied here ∂ S is defined by the condition F p = 0, and so we can characterize a multi-sliding event using F p . An example is shown in Figure 4, where the force F 2 is shown just before, Figure 4(a) and just after a multi-sliding bifurcation, Figure 4(b). In both Figure 4(a) and (b), the solid line represents the force on the mass during the sticking phase. In this example, a negative force is required to keep mass 1 stuck to the motion constraint. So when the force signal intersects the zero axis, the mass lifts off from the constraint. In the special case when the force becomes tangent to the zero axis, a multi-sliding bifurcation occurs -Figure 4(a). In Figure 4(b), the situation just after multi-sliding is shown, where the sticking orbit is now divided into two segments. In between the segments a chatter sequence exists [START_REF] Wagg | Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator[END_REF].

In addition to multi-sliding, other sliding bifurcations can occur in vibro-impact systems. For example, for the special case of r = 0, the system will have a grazing-sliding bifurcation each time a grazing event occurs -exceptions to this are discussed in [START_REF] Wagg | Periodic sticking motion in a two-degree of freedom impact oscillator[END_REF]. This is similar to the grazing-sliding in the friction oscillator example studied by [START_REF] Bernardo | Sliding bifurcations: A novel mechanism for the onset of chaos in dry friction oscillators[END_REF]. The non-generic example mentioned in Section 2.2 of a mass coming into contact with the constraint with zero velocity and as the force changed sign, is also an example of grazing-sliding.

In general, sliding bifurcations occur to sliding orbits from an individual mass in the MDOF system -including all the examples we have shown thus far. However, because there are two masses in the system, there is also the possibility of non-smooth events occurring simultaneously to both masses. The three non-smooth events we are considering in these examples are impact, grazing and sticking. It is therefore possible to list the possible combinations of multiple non-smooth events in a two mass system (Table 1).

An example is shown in Figure 5(a) where mass 1 has a complete chatter and sticking sequence, and mass 2 has two impacts during the sticking phase of mass 1 (SI), where both motion constraints are set at s 1 = s 2 = 0.3. This is distinct from a double impact (DI) which is the case where x 1 = s 1 and x 2 = s 2 simultaneously without chatter or sticking -a rare event. However, because our model of the impact assumes that an infinite number of impacts occur during sticking, the points in Figure 5(a) could possibly be viewed as double impacts (DI) -i.e. both mass 1 and 2 are in contact with the motion constraints simultaneously. Despite this, it is preferable if a distinction between these two cases is made.

A second example is shown in Figure 5(b), where the motion constraint for mass 2 has been moved to s 2 = 0.45, and the coefficient of restitution is close to zero (r = 0.001). In this case, mass 1 has a very short chatter sequence, and a long sticking phase. At t ≈ 920, mass 2 grazes with s 2 = 0.45 while mass 1 is stuck (GS) -not the same as the grazing-sliding bifurcation -subsequent dynamics involve dual sticking of the masses (DS).

Finally, in Figure 5(c), an example is shown which is close to dual sticking (DS), at very low forcing frequency of = 0.02, with r = 0.7. In this case mass 1 is stuck to s 1 , while mass 2 has a very long chatter sequence, and also appears to be stuck (at this scale), but in fact is still chattering at the end of the sticking phase of mass 1 -a reduction of and/or r leads to the dual sticking case (DS).

Conclusions

In this paper, we have considered the dynamics of multi-degree-of-freedom impact oscillators with multiple constraints using a 2 degree-of-freedom example to illustrate the dynamical complexities of these systems. We have considered the mathematical modelling of these multiple constraint systems using a modal formulation, including a modal form of the coefficient of restitution rule to model single, multiple and simultaneous impact events. The focus of this paper has been the chatter and sticking region for a 2 degree-of-freedom example. In particular, we have discussed two types of sliding bifurcationmulti-sliding and grazing-sliding -which can occur in these systems.

The fact that the system considered has two masses means that non-smooth events such as impact, grazing and sticking can occur simultaneously for certain ranges of parameter values. The possible combination of these events has been discussed, and numerical simulations of three examples; impact and sticking, sticking and grazing and the dual sticking case have been presented.

Figure 1 .

 1 Figure 1. Schematic representation of an impact oscillator with multiple motion limiting constraints (a) N degree-of-freedom (b) a 2 degree-of-freedom.

Figure 2 .

 2 Figure 2. Numerically computed displacement time series of a 2 degree-of-freedom impact oscillator with constraints s 1 = -0.3 and s 2 = 0.1; Solid line mass 1; dashed line mass 2. (a) Showing chatter and sticking motion. (b) Close up of the chatter region for mass 2.

Figure 3 .

 3 Figure 3. Numerically computed 2 degree-of-freedom impact oscillator bifurcation diagram with impact stops s 1 = -0.3, s 2 = 0.1. Parameter values m 1 = m 2 = 1, k 1 = k 2 = 1, c 1 = c 2 = 0.1, r = 0.7, forcing A 2 = 0.0, A 1 = 0.5. (a) Impact velocity v 1 vs. forcing frequency . (b) Impact velocity v 2 vs. forcing frequency .

Figure 4 .

 4 Figure 4. Multi-sliding bifurcation. Force F 2 during a sticking phase of mass one. (a) Just before multi-sliding = 0.256, (b) just after multi-sliding = 0.2561.

Figure 5 .

 5 Figure 5. Numerically computed displacement time series of a 2 degree-of-freedom impact oscillator with constraints s 1 = 0.3. Solid line mass 1, dashed line mass 2. (a) Additional impacts during sticking = 0.2561, s 2 = 0.3. (b) Grazing on mass 2, while sticking on mass 1 for r = 0.001, = 0.2995 s 2 = 0.45. (c) Just before dual sticking r = 0.7, = 0.02, s 2 = 0.3.

Table 1 .

 1 Possible multiple non-smooth events

	Event
	Dual grazing	DG
	Dual sticking	DS
	Double impact	DI
	Grazing-sticking	GS
	Grazing-impact	GI
	Sticking-impact	SI

Acknowledgements

The author would like to acknowledge the support of the EPSRC via an advanced research fellowship. The author would also like to thank Chris Budd, Mario di Bernardo and Arne Nordmark for helpful discussions regarding this work.