Jacques M Bahi

Jean-François Couchot

Christophe Guyeux

Michel Salomon

Neural Networks and Chaos: Construction, Evaluation of Chaotic Networks, and Prediction of Chaos with MultiLayer Feedforward Network

Keywords: Construction, Evaluation of Chaotic Networks, and Prediction

come   L'archive ouverte pluridisciplinaire

Introduction

Several research works have proposed or used chaotic neural networks these last years. The complex dynamics of such networks lead to various potential application areas: associative memories [?] and digital security tools like hash functions [?], digital watermarking [?, ?], or cipher schemes [?]. In the former case, the background idea is to control chaotic dynamics in order to store patterns, with the key advantage of offering a large storage capacity. For the latter case, the use of chaotic dynamics is motivated by their unpredictability and random-like behaviors. Indeed, investigating new concepts is crucial for the computer security field, because new threats are constantly emerging. As an illustrative example, the former standard in hash functions, namely the SHA-1 algorithm, has been recently weakened after flaws were discovered.

Chaotic neural networks have been built with different approaches. In the context of associative memory, chaotic neurons like the nonlinear dynamic state neuron [?] frequently constitute the nodes of the network. These neurons have an inherent chaotic behavior, which is usually assessed through the computation of the Lyapunov exponent. An alternative approach is to consider a well-known neural network architecture: the MultiLayer Perceptron (MLP). These networks are suitable to model nonlinear relationships between data, due to their universal approximator capacity [?, ?]. Thus, this kind of networks can be trained to model a physical phenomenon known to be chaotic such as Chua's circuit [?]. Sometime a neural network, which is build by combining transfer functions and initial conditions that are both chaotic, is itself claimed to be chaotic [?].

What all of these chaotic neural networks have in common is that they are claimed to be chaotic despite a lack of any rigorous mathematical proof. The first contribution of this paper is to fill this gap, using a theoretical framework based on the Devaney's definition of chaos [?]. This mathematical theory of chaos provides both qualitative and quantitative tools to evaluate the complex behavior of a dynamical system: ergodicity, expansivity, and so on. More precisely, in this paper, which is an extension of a previous work [?], we establish the equivalence between chaotic iterations and a class of globally recurrent MLP. The second contribution is a study of the converse problem, indeed we investigate the ability of classical multilayer perceptrons to learn a particular family of discrete chaotic dynamical systems. This family is defined by a Boolean vector, an update function, and a sequence defining the component to update at each iteration. It has been previously established that such dynamical systems are chaotically iterated (as it is defined by Devaney) when the chosen function has a strongly connected iterations graph. In this document, we experiment several MLPs and try to learn some iterations of this kind. We show that non-chaotic iterations can be learned, whereas it is far more difficult for chaotic ones. That is to say, we have discovered at least one family of problems with a reasonable size, such that artificial neural networks should not be applied due to their inability to learn chaotic behaviors in this context.

The remainder of this research work is organized as follows. The next section is devoted to the basics of Devaney's chaos. Section 3 formally describes how to build a neural network that operates chaotically. Section 4 is devoted to the dual case of checking whether an existing neural network is chaotic or not. Topological properties of chaotic neural networks are discussed in Sect. 5. The Section 6.1 shows how to translate such iterations into an Artificial Neural Network (ANN), in order to evaluate the capability for this latter to learn chaotic behaviors. This ability is studied in Sect. 6.2, where various ANNs try to learn two sets of data: the first one is obtained by chaotic iterations while the second one results from a non-chaotic system. Prediction success rates are given and discussed for the two sets. The paper ends with a conclusion section where our contribution is summed up and intended future work is exposed.

Chaotic Iterations according to Devaney

In this section, the well-established notion of Devaney's mathematical chaos is firstly recalled. Preservation of the unpredictability of such dynamical system when implemented on a computer is obtained by using some discrete iterations called "asynchronous iterations", which are thus introduced. The result establishing the link between such iterations and Devaney's chaos is finally presented at the end of this section.

In what follows and for any function f , f n means the composition f •f •. . .•f (n times) and an iteration of a dynamical system is the step that consists in updating the global state x t at time t with respect to a function f s.t. x t+1 = f (x t ).

Devaney's chaotic dynamical systems

Various domains such as physics, biology, or economy, contain systems that exhibit a chaotic behavior, a well-known example is the weather. These systems are in particular highly sensitive to initial conditions, a concept usually presented as the butterfly effect: small variations in the initial conditions possibly lead to widely different behaviors. Theoretically speaking, a system is sensitive if for each point x in the iteration space, one can find a point in each neighborhood of x having a significantly different future evolution. Conversely, a system seeded with the same initial conditions always has the same evolution. In other words, chaotic systems have a deterministic behavior defined through a physical or mathematical model and a high sensitivity to the initial conditions. Besides mathematically this kind of unpredictability is also referred to as deterministic chaos. For example, many weather forecast models exist, but they give only suitable predictions for about a week, because they are initialized with conditions that reflect only a partial knowledge of the current weather. Even if the differences are initially small, they are amplified in the course of time, and thus make difficult a long-term prediction. In fact, in a chaotic system, an approximation of the current state is a quite useless indicator for predicting future states.

From mathematical point of view, deterministic chaos has been thoroughly studied these last decades, with different research works that have provide various definitions of chaos. Among these definitions, the one given by Devaney [?] is well-established. This definition consists of three conditions: topological transitivity, density of periodic points, and sensitive point dependence on initial conditions.

Topological transitivity is checked when, for any point, any neighborhood of its future evolution eventually overlap with any other given region. This property implies that a dynamical system cannot be broken into simpler subsystems. Intuitively, its complexity does not allow any simplification.

However, chaos needs some regularity to "counteracts" the effects of transitivity. In the Devaney's formulation, a dense set of periodic points is the element of regularity that a chaotic dynamical system has to exhibit. We recall that a point x is a periodic point for f of period n ∈ N * if f n (x) = x. Then, the map f is regular on the topological space (X , τ ) if the set of periodic points for f is dense in X (for any x ∈ X , we can find at least one periodic point in any of its neighborhood). Thus, due to these two properties, two points close to each other can behave in a completely different manner, leading to unpredictability for the whole system.

Let us recall that f has sensitive dependence on initial conditions if there exists δ > 0 such that, for any x ∈ X and any neighborhood V of x, there exist y ∈ V and n > 0 such that d (f n (x), f n (y)) > δ. The value δ is called the constant of sensitivity of f .

Finally, the dynamical system that iterates f is chaotic according to Devaney on (X , τ ) if f is regular, topologically transitive, and has sensitive dependence to its initial conditions. In what follows, iterations are said to be chaotic (according to Devaney) when the corresponding dynamical system is chaotic, as it is defined in the Devaney's formulation.

Asynchronous Iterations

Let us firstly discuss about the domain of iteration. As far as we know, no result rules that the chaotic behavior of a dynamical system that has been theoretically proven on R remains valid on the floating-point numbers, which is the implementation domain. Thus, to avoid loss of chaos this work presents an alternative, that is to iterate Boolean maps: results that are theoretically obtained in that domain are preserved in implementations.

Let us denote by a; b the following interval of integers: {a, a + 1, . . . , b}, where a < b. In this section, a system under consideration iteratively modifies a collection of n components. Each component i ∈ 1; n takes its value x i among the domain B = {0, 1}. A configuration of the system at discrete time t is the vector x t = (x t 1 , . . . , x t n ) ∈ B n . The dynamics of the system is described according to a function f :

B n → B n such that f (x) = (f 1 (x), . . . , f n (x)).
Let be given a configuration x. In what follows N (i, x) = (x 1 , . . . , x i , . . . , x n ) is the configuration obtained by switching the i-th component of x (x i is indeed the negation of x i ). Intuitively, x and N (i, x) are neighbors. The discrete iterations of f are represented by the oriented graph of iterations Γ(f ). In such a graph, vertices are configurations of B n and there is an arc labeled i from x to N (i, x) if and only if f i (x) is N (i, x).

In the sequel, the strategy S = (S t ) t∈N is the sequence defining which component to update at time t and S t denotes its t-th term. This iteration scheme that only modifies one element at each iteration is usually referred as asynchronous iterations. More precisely, we have for any i,

1 ≤ i ≤ n,    x 0 ∈ B n x t+1 i = f i (x t ) if S t = i , x t i otherwise .
(1)

Next section shows the link between asynchronous iterations and Devaney's chaos.

On the link between asynchronous iterations and Devaney's Chaos

In this subsection we recall the link we have established between asynchronous iterations and Devaney's chaos. The theoretical framework is fully described in [?].

We introduce the function F f that is defined for any given application f :

B n → B n by F f : 1; n × B n → B n , s.t. F f (s, x) j = f j (x) if j = s , x j otherwise . (2)
With such a notation, asynchronously obtained configurations are defined for times t = 0, 1, 2, . . . by: x 0 ∈ B n and

x t+1 = F f (S t , x t ) . (3)
Finally, iterations defined in Eq. ( 3) can be described by the following system:

   X 0 = ((S t ) t∈N , x 0 ) ∈ 1; n N × B n X k+1 = G f (X k ) where G f ((S t ) t∈N , x) = σ((S t ) t∈N ), F f (S 0 , x) , (4)
where σ is the so-called shift function that removes the first term of the strategy (i.e., S 0 ). This definition allows to link asynchronous iterations with classical iterations of a dynamical system. Note that it can be extended by considering subsets for S t .

To study topological properties of these iterations, we are then left to introduce a distance d between two points (S, x) and ( Š, x) in

X = 1; n N × B n . Let ∆(x, y) = 0 if x = y, and ∆(x, y) = 1 else, be a distance on B. The distance d is defined by d((S, x); ( Š, x)) = d e (x, x) + d s (S, Š) , (5)
where

d e (x, x) = n j=1 ∆(x j , xj ) ∈ 0; n (6)
and

d s (S, Š) = 9 2n ∞ t=0 |S t -Št | 10 t+1 ∈ [0; 1] . ( 7
)
This distance is defined to reflect the following information. Firstly, the more two systems have different components, the larger the distance between them. Secondly, two systems with similar components and strategies, which have the same starting terms, must induce only a small distance. The proposed distance fulfills these requirements: on the one hand its floor value reflects the difference between the cells, on the other hand its fractional part measures the difference between the strategies.

The relation between Γ(f ) and G f is obvious: there exists a path from x to x in Γ(f ) if and only if there exists a strategy s such that iterations of G f from the initial point (s, x) reach the configuration x . Using this link, Guyeux [?] has proven that,

Theorem 1 Let f : B n → B n . Iterations of G f are chaotic according to De- vaney if and only if Γ(f ) is strongly connected.
Checking if a graph is strongly connected is not difficult (by the Tarjan's algorithm for instance). Let be given a strategy S and a function f such that Γ(f ) is strongly connected. In that case, iterations of the function G f as defined in Eq. ( 4) are chaotic according to Devaney.

Let us then define two functions f 0 and f 1 both in B n → B n that are used all along this paper. The former is the vectorial negation, i.e., f 0 (x 1 , . . . , x n ) = (x 1 , . . . , x n ). The latter is f 1 (x 1 , . . . , x n ) = (x 1 , x 1 , x 2 , . . . , x n-1 ). It is not hard to check that Γ(f 0 ) and Γ(f 1 ) are both strongly connected, then iterations of G f0 and of G f1 are chaotic according to Devaney.

With this material, we are now able to build a first chaotic neural network, as defined in the Devaney's formulation.

A chaotic neural network in the sense of Devaney

Let us build a multilayer perceptron neural network modeling F f0 : 1; n × B n → B n associated to the vectorial negation. More precisely, for all inputs (s, x) ∈ 1; n × B n , the output layer will produce F f0 (s, x). It is then possible to link the output layer and the input one, in order to model the dependence between two successive iterations. As a result we obtain a global recurrent neural network that behaves as follows (see Fig. 1).

• The network is initialized with the input vector S 0 , x 0 ∈ 1; n × B n and computes the output vector x 1 = F f0 S 0 , x 0 . This last vector is published as an output one of the chaotic neural network and is sent back to the input layer through the feedback links.

• When the network is activated at the t th iteration, the state of the system x t ∈ B n received from the output layer and the initial term of the sequence (S t ) t∈N (i.e., S 0 ∈ 1; n ) are used to compute the new output vector. This new vector, which represents the new state of the dynamical system, satisfies:

x t+1 = F f0 (S 0 , x t ) ∈ B n . ( 8
)
The behavior of the neural network is such that when the initial state is x 0 ∈ B n and a sequence (S t ) t∈N is given as outside input, then the sequence of successive published output vectors (x t ) t∈N * is exactly the one produced by the chaotic iterations formally described in Eq. ( 4). It means that mathematically

Output layer

Hidden layers , and therefore that they are equivalent reformulations of the iterations of G f0 in X . Finally, since the proposed neural network is built to model the behavior of G f0 , whose iterations are chaotic according to the Devaney's definition of chaos, we can conclude that the network is also chaotic in this sense.

1 0 2 0 2 x x x x 1 +1 2 +1 x x t t t t 1 -1 -1 -1 -1 -1 bias / threshold
The previous construction scheme is not restricted to function f 0 . It can be extended to any function f such that G f is a chaotic map by training the network to model F f : 1; n × B n → B n . Due to Theorem 1, we can find alternative functions f for f 0 through a simple check of their graph of iterations Γ(f ). For example, we can build another chaotic neural network by using f 1 instead of f 0 .

Checking whether a neural network is chaotic or not

We focus now on the case where a neural network is already available, and for which we want to know if it is chaotic. Typically, in many research papers neural network are usually claimed to be chaotic without any convincing mathematical proof. We propose an approach to overcome this drawback for a particular category of multilayer perceptrons defined below, and for the Devaney's formulation of chaos. In spite of this restriction, we think that this approach can be extended to a large variety of neural networks. We consider a multilayer perceptron of the following form: inputs are n binary digits and one integer value, while outputs are n bits. Moreover, each binary output is connected with a feedback connection to an input one.

• During initialization, the network is seeded with n bits denoted x 0 1 , . . . , x 0 n and an integer value S 0 that belongs to 1; n .

• At iteration t, the last output vector (x t 1 , . . . , x t n ) defines the n bits used to compute the new output one x t+1 1 , . . . , x t+1 n . While the remaining input receives a new integer value S t ∈ 1; n , which is provided by the outside world.

The topological behavior of these particular neural networks can be proven to be chaotic through the following process. Firstly, we denote by F : 1; n ×B n → B n the function that maps the value (s, (x 1 , . . . , x n )) ∈ 1; n ×B n into the value (y 1 , . . . , y n ) ∈ B n , where (y 1 , . . . , y n ) is the response of the neural network after the initialization of its input layer with (s, (x 1 , . . . , x n )). Secondly, we define f :

B n → B n such that f (x 1 , x 2 , . . . , x n ) is equal to (F (1, (x 1 , x 2 , . . . , x n )) , . . . , F (n, (x 1 , x 2 , . . . , x n ))) . (9)
Thus, for any j, 1 ≤ j ≤ n, we have

f j (x 1 , x 2 , . . . , x n ) = F (j, (x 1 , x 2 , . . . , x n )).
If this recurrent neural network is seeded with x 0 1 , . . . , x 0 n and S ∈ 1; n N , it produces exactly the same output vectors than the chaotic iterations of F f with initial condition S, (x 0 1 , . . . , x 0 n ) ∈ 1; n N × B n . Theoretically speaking, such iterations of F f are thus a formal model of these kind of recurrent neural networks. In the rest of this paper, we will call such multilayer perceptrons "CI-MLP(f )", which stands for "Chaotic Iterations based MultiLayer Perceptron".

Checking if CI-MLP(f ) behaves chaotically according to Devaney's definition of chaos is simple: we need just to verify if the associated graph of iterations Γ(f ) is strongly connected or not. As an incidental consequence, we finally obtain an equivalence between chaotic iterations and CI-MLP(f ). Therefore, we can obviously study such multilayer perceptrons with mathematical tools like topology to establish, for example, their convergence or, contrarily, their unpredictable behavior. An example of such a study is given in the next section.

Topological properties of chaotic neural networks

Let us first recall two fundamental definitions from the mathematical theory of chaos.

Definition 1 A function f is said to be expansive if ∃ε > 0, ∀x = y, ∃n ∈ N such that d (f n (x), f n (y)) ≥ ε.
In other words, a small error on any initial condition is always amplified until ε, which denotes the constant of expansivity of f .

Definition 2 A discrete dynamical system is said to be topologically mixing if and only if, for any pair of disjoint open sets U ,V = ∅, we can find some n 0 ∈ N such that for any n, n ≥ n 0 , we have

f n (U ) ∩ V = ∅.
Topologically mixing means that the dynamical system evolves in time such that any given region of its topological space might overlap with any other region.

It has been proven in Ref.

[?] that chaotic iterations are expansive and topologically mixing when f is the vectorial negation f 0 . Consequently, these properties are inherited by the CI-MLP(f 0 ) recurrent neural network previously presented, which induce a greater unpredictability. Any difference on the initial value of the input layer is in particular magnified up to be equal to the expansivity constant.

Let us then focus on the consequences for a neural network to be chaotic according to Devaney's definition. Intuitively, the topological transitivity property implies indecomposability, which is formally defined as follows:

Definition 3 A dynamical system (X , f ) is not decomposable if it is not the union of two closed sets A, B ⊂ such that f (A) ⊂ A, f (B) ⊂ B.
Hence, reducing the set of outputs generated by CI-MLP(f ), in order to simplify its complexity, is impossible if Γ(f ) is strongly connected. Moreover, under this hypothesis CI-MLPs(f ) are strongly transitive:

Definition 4 A dynamical system (X , f ) is strongly transitive if ∀x, y ∈ X , ∀r > 0, ∃z ∈ X , d(z, x) ≤ r ⇒ ∃n ∈ N * , f n (z) = y.
According to this definition, for all pairs of points (x, y) in the phase space, a point z can be found in the neighborhood of x such that one of its iterates f n (z) is y. Indeed, this result has been established during the proof of the transitivity presented in Ref. [?]. Among other things, the strong transitivity leads to the fact that without the knowledge of the initial input layer, all outputs are possible. Additionally, no point of the output space can be discarded when studying CI-MLPs: this space is intrinsically complicated and it cannot be decomposed or simplified.

Furthermore, these recurrent neural networks exhibit the instability property:

Definition 5 A dynamical system (X , f ) is unstable if for all x ∈ X , the orbit γ x : n ∈ N -→ f n (x) is unstable, that means: ∃ε > 0, ∀δ > 0, ∃y ∈ X , ∃n ∈ N, such that d(x, y) < δ and d (γ x (n), γ y (n)) ≥ ε.
This property, which is implied by the sensitive point dependence on initial conditions, leads to the fact that in all neighborhoods of any point x, there are points that can be apart by ε in the future through iterations of the CI-MLP(f ). Thus, we can claim that the behavior of these MLPs is unstable when Γ(f ) is strongly connected.

Let us now consider a compact metric space (M, d) and f : M → M a continuous map. For each natural number n, a new metric d n is defined on M by

d n (x, y) = max{d(f i (x), f i (y)) : 0 ≤ i < n} . ( 10
)
Given any ε > 0 and n 1, two points of M are ε-close with respect to this metric if their first n iterates are ε-close.

Then we have the following result [?],

Theorem 2 (X , d) is compact and the topological entropy of (X , G f0 ) is infinite.

Figure 2 is a summary of addressed neural networks and chaos problems. In Section 3 we have explained how to construct a truly chaotic neural networks, A for instance. Section 4 has shown how to check whether a given MLP A or C is chaotic or not in the sense of Devaney, and how to study its topological behavior. The last thing to investigate, when comparing neural networks and Devaney's chaos, is to determine whether an artificial neural network C is able to learn or predict some chaotic behaviors of B, as it is defined in the Devaney's formulation (when they are not specifically constructed for this purpose). This statement is studied in the next section.

Representing Chaotic Iterations for Neural Networks

The problem of deciding whether classical feedforward ANNs are suitable to approximate topological chaotic iterations may then be reduced to evaluate such neural networks on iterations of functions with Strongly Connected Component (SCC) graph of iterations. To compare with non-chaotic iterations, the experiments detailed in the following sections are carried out using both kinds of function (chaotic and non-chaotic). Let us emphasize on the difference between this kind of neural networks and the Chaotic Iterations based multilayer peceptron.

We are then left to compute two disjoint function sets that contain either functions with topological chaos properties or not, depending on the strong connectivity of their iterations graph. This can be achieved for instance by removing a set of edges from the iteration graph Γ(f 0 ) of the vectorial negation function f 0 . One can deduce whether a function verifies the topological chaos property or not by checking the strong connectivity of the resulting graph of iterations.

For instance let us consider the functions f and g from B 4 to B 4 respectively defined by the following lists: [0, 0,2,3,13,13,6,3,8,9,10,11,8,13,14,15] and [11,14,13,14,11,10,1,8,7,6,5,4,3,2,1,0] .

In other words, the image of 0011 by g is 1110: it is obtained as the binary value of the fourth element in the second list (namely 14). It is not hard to verify that Γ(f ) is not SCC (e.g., f (1111) is 1111) whereas Γ(g) is. The remaining of this section shows how to translate iterations of such functions into a model amenable to be learned by an ANN. Formally, input and output vectors are pairs ((S t ) t∈N , x) and σ((S t ) t∈N ), F f (S 0 , x) as defined in Eq. (4).

Firstly, let us focus on how to memorize configurations. Two distinct translations are proposed. In the first case, we take one input in B per component; in the second case, configurations are memorized as natural numbers. A coarse attempt to memorize configuration as natural number could consist in labeling each configuration with its translation into decimal numeral system. However, such a representation induces too many changes between a configuration labeled by a power of two and its direct previous configuration: for instance, 16 (10000) and 15 (01111) are close in a decimal ordering, but their Hamming distance is 5. This is why Gray codes [?] have been preferred.

Secondly, let us detail how to deal with strategies. Obviously, it is not possible to translate in a finite way an infinite strategy, even if both (S t ) t∈N and σ((S t ) t∈N ) belong to {1, . . . , n} N . Input strategies are then reduced to have a length of size l ∈ 2, k , where k is a parameter of the evaluation. Notice that l is greater than or equal to 2 since we do not want the shift σ function to return an empty strategy. Strategies are memorized as natural numbers expressed in base n + 1. At each iteration, either none or one component is modified (among the n components) leading to a radix with n + 1 entries. Finally, we give an other input, namely m ∈ 1, l -1 , which is the number of successive iterations that are applied starting from x. Outputs are translated with the same rules.

To address the complexity issue of the problem, let us compute the size of the data set an ANN has to deal with. Each input vector of an input-output pair is composed of a configuration x, an excerpt S of the strategy to iterate of size l ∈ 2, k , and a number m ∈ 1, l -1 of iterations that are executed.

Firstly, there are 2 n configurations x, with n l strategies of size l for each of them. Secondly, for a given configuration there are ω = 1 × n 2 + 2 × n 3 + . . . + (k -1) × n k ways of writing the pair (m, S). Furthermore, it is not hard to establish that

(n -1) × ω = (k -1) × n k+1 - k i=2 n i then ω = (k -1) × n k+1 n -1 - n k+1 -n 2 (n -1) 2 .
And then, finally, the number of input-output pairs for our ANNs is

2 n × (k -1) × n k+1 n -1 - n k+1 -n 2 (n -1) 2 .
For instance, for 4 binary components and a strategy of at most 3 terms we obtain 2304 input-output pairs.

Experiments

To study if chaotic iterations can be predicted, we choose to train the multilayer perceptron. As stated before, this kind of network is in particular well-known for its universal approximation property [?, ?]. Furthermore, MLPs have been already considered for chaotic time series prediction. For example, in [?] the authors have shown that a feedforward MLP with two hidden layers, and trained with Bayesian Regulation back-propagation, can learn successfully the dynamics of Chua's circuit.

In these experiments we consider MLPs having one hidden layer of sigmoidal neurons and output neurons with a linear activation function. They are trained using the Limited-memory Broyden-Fletcher-Goldfarb-Shanno quasi-newton algorithm in combination with the Wolfe linear search. The training process is performed until a maximum number of epochs is reached. To prevent overfitting and to estimate the generalization performance we use holdout validation by splitting the data set into learning, validation, and test subsets. These subsets are obtained through random selection such that their respective size represents 65%, 10%, and 25% of the whole data set.

Several neural networks are trained for both iterations coding schemes. In both cases iterations have the following layout: configurations of four components and strategies with at most three terms. Thus, for the first coding scheme a data set pair is composed of 6 inputs and 5 outputs, while for the second one it is respectively 3 inputs and 2 outputs. As noticed at the end of the previous section, this leads to data sets that consist of 2304 pairs. The networks differ in the size of the hidden layer and the maximum number of training epochs. We remember that to evaluate the ability of neural networks to predict a chaotic behavior for each coding scheme, the trainings of two data sets, one of them describing chaotic iterations, are compared.

Thereafter we give, for the different learning setups and data sets, the mean prediction success rate obtained for each output. Such a rate represents the percentage of input-output pairs belonging to the test subset for which the corresponding output value was correctly predicted. These values are computed considering 10 trainings with random subsets construction, weights and biases initialization. Firstly, neural networks having 10 and 25 hidden neurons are trained, with a maximum number of epochs that takes its value in {125, 250, 500} (see Tables 1 and2). Secondly, we refine the second coding scheme by splitting the output vector such that each output is learned by a specific neural network (Table 3). In this last case, we increase the size of the hidden layer up to 40 neurons and we consider larger number of epochs.

Table 1 presents the rates obtained for the first coding scheme. For the chaotic data, it can be seen that as expected configuration prediction becomes better when the number of hidden neurons and maximum epochs increases: an improvement by a factor two is observed (from 36.10% for 10 neurons and 125 epochs to 70.97% for 25 neurons and 500 epochs). We also notice that the learning of outputs (2) and ( 3) is more difficult. Conversely, for the non-chaotic case the simplest training setup is enough to predict configurations. For all these feedforward network topologies and all outputs the obtained results for the nonchaotic case outperform the chaotic ones. Finally, the rates for the strategies show that the different feedforward networks are unable to learn them.

For the second coding scheme (i.e., with Gray Codes) Table 2 shows that any network learns about five times more non-chaotic configurations than chaotic ones. As in the previous scheme, the strategies cannot be predicted. Figures 3 and4 present the predictions given by two feedforward multilayer perceptrons that were respectively trained to learn chaotic and non-chaotic data, using the second coding scheme. Each figure shows for each sample of the test subset (577 samples, representing 25% of the 2304 samples) the configuration that should have been predicted and the one given by the multilayer perceptron. It can be seen that for the chaotic data the predictions are far away from the expected configurations. Obviously, the better predictions for the non-chaotic data reflect their regularity.

Let us now compare the two coding schemes. Firstly, the second scheme disturbs the learning process. In fact in this scheme the configuration is always expressed as a natural number, whereas in the first one the number of inputs follows the increase of the Boolean vectors coding configurations. In this latter case, the coding gives a finer information on configuration evolution.

Unfortunately, in practical applications the number of components is usually unknown. Hence, the first coding scheme cannot be used systematically. Therefore, we provide a refinement of the second scheme: each output is learned by a different ANN. Table 3 presents the results for this approach. In any case, whatever the considered feedforward network topologies, the maximum epoch number, and the kind of iterations, the configuration success rate is slightly improved. Moreover, the strategies predictions rates reach almost 12%, whereas in Table 2 they never exceed 1.5%. Despite of this improvement, a long term prediction of chaotic iterations still appear to be an open issue.

Conclusion

In this paper, we have established an equivalence between chaotic iterations, according to the Devaney's definition of chaos, and a class of multilayer perceptron neural networks. Firstly, we have described how to build a neural network that can be trained to learn a given chaotic map function. Secondly, we found a condition that allow to check whether the iterations induced by a function are chaotic or not, and thus if a chaotic map is obtained. Thanks to this condition our approach is not limited to a particular function. In the dual case, we show that checking if a neural network is chaotic consists in verifying a property on an associated graph, called the graph of iterations. These results are valid for recurrent neural networks with a particular architecture. However, we believe that a similar work can be done for other neural network architectures. Finally, we have discovered at least one family of problems with a reasonable size, such that artificial neural networks should not be applied in the presence of chaos, due to their inability to learn chaotic behaviors in this context. Such a consideration is not reduced to a theoretical detail: this family of discrete iterations is concretely implemented in a new steganographic method [?]. As steganographic detectors embed tools like neural networks to distinguish between original and stego contents, our studies tend to prove that such detectors might be unable to tackle with chaos-based information hiding schemes.

In future work we intend to enlarge the comparison between the learning of truly chaotic and non-chaotic behaviors. Other computational intelligence tools such as support vector machines will be investigated too, to discover which tools are the most relevant when facing a truly chaotic phenomenon. A comparison between learning rate success and prediction quality will be realized. Concrete consequences in biology, physics, and computer science security fields will then be stated.

Figure 1 :

 1 Figure 1: A perceptron equivalent to chaotic iterations

Figure 2 :

 2 Figure 2: Summary of addressed neural networks and chaos problems

Figure 3 :

 3 Figure 3: Second coding scheme -Predictions obtained for a chaotic test subset.

Figure 4 :

 4 Figure 4: Second coding scheme -Predictions obtained for a non-chaotic test subset.

  In the context of computer science different topic areas have an interest in chaos, as for steganographic techniques[?, ?]. Steganography consists in embedding a secret message within an ordinary one, while the secret extraction takes place once at destination. The reverse (i.e., automatically detecting the presence of hidden messages inside media) is called steganalysis. Among the deployed strategies inside detectors, there are support vectors machines [?], neural networks [?, ?], and Markov chains[?]. Most of these detectors give quite good results and are rather competitive when facing steganographic tools. However, to the best of our knowledge none of the considered information hiding schemes fulfills the Devaney definition of chaos[?]. Indeed, one can wonder whether detectors continue to give good results when facing truly chaotic schemes. More generally, there remains the open problem of deciding whether artificial intelligence is suitable for predicting topological chaotic behaviors.

	6 Suitability of Feedforward Neural Networks
	for Predicting Chaotic and Non-chaotic Be-
	haviors

Table 1 :

 1 Prediction success rates for configurations expressed as boolean vectors.

	Networks topology: 6 inputs, 5 outputs, and one hidden layer
	Hidden neurons		10 neurons
		Epochs	125	250	500
		Output (1)	90.92% 91.75%	91.82%
	Chaotic	Output (2) Output (3) Output (4) Config.	69.32% 78.46% 68.47% 78.49% 91.53% 92.37% 36.10% 51.35%	82.15% 82.22% 93.4% 56.85%
		Strategy (5)	1.91%	3.38%	2.43%
	Non-chaotic	Output (1) Output (2) Output (3) Output (4) Config.	97.64% 98.10% 95.15% 95.39% 100% 100% 97.47% 97.90% 90.52% 91.59%	98.20% 95.46% 100% 97.99% 91.73%
		Strategy (5)	3.41%	3.40%	3.47%
	Hidden neurons		25 neurons
		Epochs	125	250	500
		Output (1)	91.65% 92.69%	93.93%
	Chaotic	Output (2) Output (3) Output (4) Config.	72.06% 88.46% 79.19% 89.83% 91.61% 92.34% 48.82% 67.80%	90.5% 91.59% 93.47% 70.97%
		Strategy (5)	2.62%	3.43%	3.78%
	Non-chaotic	Output (1) Output (2) Output (3) Output (4) Config.	97.87% 97.99% 95.46% 95.84% 100% 100% 97.77% 97.82% 91.36% 91.99%	98.03% 96.75% 100% 98.06% 93.03%
		Strategy (5)	3.37%	3.44%	3.29%

Table 2 :

 2 Prediction success rates for configurations expressed with Gray code Networks topology: 3 inputs, 2 outputs, and one hidden layer

		Hidden neurons		10 neurons
		Epochs	125	250	500
	Chaotic	Config. (1) Strategy (2)	13.29% 13.55% 13.08% 0.50% 0.52% 1.32%
	Non-Chaotic	Config. (1) Strategy (2)	77.12% 74.00% 72.60% 0.42% 0.80% 1.16%
		Hidden neurons		25 neurons
		Epochs	125	250	500
	Chaotic	Config. (1) Strategy (2)	12.27% 13.15% 13.05% 0.71% 0.66% 0.88%
	Non-Chaotic	Config. (1) Strategy (2)	73.60% 74.70% 75.89% 0.64% 0.97% 1.23%

Table 3 :

 3 Prediction success rates for split outputs. Networks topology: 3 inputs, 1 output, and one hidden layer

	Epochs	125	250	500
	Chaotic	Output = Configuration
	10 neurons	12.39% 14.06%	14.32%
	25 neurons	13.00% 14.28%	14.58%
	40 neurons	11.58% 13.47%	14.23%
	Non chaotic	Output = Configuration
	10 neurons	76.01% 74.04%	78.16%
	25 neurons	76.60% 72.13%	75.96%
	40 neurons	76.34% 75.63%	77.50%
	Chaotic/non chaotic		Output = Strategy
	10 neurons	0.76%	0.97%	1.21%
	25 neurons	1.09%	0.73%	1.79%
	40 neurons	0.90%	1.02%	2.15%
	Epochs	1000	2500	5000
	Chaotic	Output = Configuration
	10 neurons	14.51% 15.22%	15.22%
	25 neurons	16.95% 17.57%	18.46%
	40 neurons	17.73% 20.75%	22.62%
	Non chaotic	Output = Configuration
	10 neurons	78.98% 80.02%	79.97%
	25 neurons	79.19% 81.59%	81.53%
	40 neurons	79.64% 81.37%	81.37%
	Chaotic/non chaotic		Output = Strategy
	10 neurons	3.47%	9.98%	11.66%
	25 neurons	3.92%	8.63%	10.09%
	40 neurons	3.29%	7.19%	7.18%


Vibration of a Surface-Coated Elastic Block Subject to

Bending

and appropriate incremental boundary conditions are derived in a convenient form. This generalizes to the dynamic situation the work of Dryburgh and Ogden [START_REF] Dryburgh | Bifurcation of an elastic surface-coated incompressible isotropic elastic block subject to bending[END_REF] concerned with the (static) bifurcation of a rectangular block subject to bending. In section 6 time-harmonic vibrations are considered and appropriate non-dimensionalizations for the various parameters are established. The resulting equations are solved numerically for a representative range of parameter values and the results are discussed in section 6.1. The frequency is plotted against a measure of the underlying static deformation, namely the stretch on the long curved side of the block. The results illustrate the effect of changing the bending stiffness of the coating and the aspect ratio of the block, and a comparison is made with corresponding results for an uncoated block. The dependence of the frequency on the ratio of coating to bulk density and the relative importance of the coating rotatory inertia term are also discussed briefly.

'<1$0,&6 2) $ 685)$&(&2$7(' %2'<

We consider the plane motion of a body whose plane section occupies the region F 5 U 5 in the reference (natural) configuration. The current configuration, which changes with time U, is denoted by a F and the motion is described by the invertible and appropriately regular function

= F L $ a F +[ U, $ { @ +[ U, (1)
where [ and { are points of F and a F, respectively, and L is a time interval. The deformation gradient I+[ U, is given by I @ Judg, where Grad is the two-dimensional gradient operator on F.

Let the boundary !F be parametrized by arclength 4, taken in the counterclockwise sense such that the outward unit normal Q is rightward following the orientation of the curve. Similarly, points of ! a F are identified by the arclength T and the outward unit normal on ! a F is denoted by q, which is related to Q by the classical Nansonòs formula, which, for the present plane-strain specialization, has the form qgT @ I 5 Qg4

(2)

Let part of the boundary 1 !F be coated with a thin elastic film that deforms as a material curve to become a 1 ! a F. The motion of each point [+4, 5 1 is described by the position vector

u+4 U, @ +[+4, U, (3)
In terms of the notation defined by + , 3 @ !+ ,!4, the unit tangent W @ [ 3 +4, to 1 maps to

u 3 +4 U, @ I+[+4, U,W+4, @ u +4 U, +4 U, (4)
where u +4 U, @ mI+[+4, U,W+4,m @ T 3 +4 U, is the stretch of 1 induced by the motion and +4 U, is the unit tangent to a 1 at the point corresponding to arclength 4 on 1. The unit tangent can be decomposed into the directions singled out by a Cartesian basis ih 4 h 5 j through the counterclockwise angle +4 U, between its direction and that of h 4 . Thus +4 U, @ frv +4 U,h 4 . vlq +4 U,h 5 [START_REF] Dryburgh | Bifurcation of an elastic surface-coated incompressible isotropic elastic block subject to bending[END_REF] We denote by D+4 U, @ n +4 U, the leftward unit normal to a 1, where n @ h 4 h 5 . Differentiation of Equation ( 5) with respect to 4 yields

3 +4 U, @ +4 U,D+4 U, +4 U, @ 3 +4 U, (6)
and we note that the curvature of a 1 is u 4 . The material in F is assumed to be hyperelastic, with strain-energy function 8+I,. It is also assumed to be incompressible, so that the constraint ghw I @ 4 is satisfied. The nominal stress tensor V is then given by

V @ !8 !I Q I 4 ( 7
)
where Q is the usual Lagrange multiplier.

The surface coating is taken to be characterized by the strain energy 6+u , per unit length [START_REF] Steigmann | Plane deformations of elastic solids with intrinsic boundary elasticity. 3URF 5 6RF /RQG[END_REF] which accounts for elastic resistance to both extension and flexure of 1. The tangential force ' and moment . on a 1 are given respectively by

' @ 6 u . @ 6 (8)
where a comma indicates a partial derivative.

In general, the force, denoted I, at a point on a 1 has tangential component equal to ' and also a normal component, ( say, so that we can write

i @ ' . (D (9)
The quantity ( may be interpreted as the transverse shear force on the film. It is a Lagrange multiplier associated with the constraint that no transverse shear strain is admitted because of the assumed geometrical nature of the coating. It is not determined by a constitutive equation and is therefore an unknown of the problem requiring a specific equation for its determination.

In the absence of body forces the equations of motion in F take the form

Gly V @ zG 5 GU 5 ( 10
)
where z is the mass density of the bulk material per unit area of F, Gly is the divergence operator in F and GGU denotes the material time derivative. If we denote by !F Y and !F G parts of !F without coating on which, respectively, displacements and tractions are prescribed, the appropriate boundary conditions are

{ @ 1+U, rq !F Y (11) V 5 Q @ w+U, rq !F G (12)
Further boundary conditions are provided by derivation of the equations governing the motion of the surface coating. Linear momentum and moment-of-momentum balances for the film give the following local equations [START_REF] Ogden | Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation. -0[END_REF]

i 3 @ V 5 Q . z 3 G 5 uGU 5 rq 1 (13) . 3 . u ( @ * G 5 GU 5 rq 1 ( 14
)
where z 3 and *+4,, respectively, are the mass density and the mass moment of inertia per unit reference length of the film, the latter taken to be independent of the motion. Equation ( 13) couples the response of the bulk material with that of the film, while (14) determines the Lagrange multiplier (+4 U,, so that the force i+4 U, at each point of the coating is fully determined. In deriving (14) the rotatory inertia term has been taken into account. The importance of this term will be emphasized in the application considered in sections 5 and 6, and situations where its neglect can be justified will be identified. We refer to [START_REF] Ogden | Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation. -0[END_REF] for a detailed discussion of this point.

,1&5(0(17$/ 027,216

We now consider an incremental motion superimposed on the dynamic deformation described in section 2. Incremental displacements are given by x+{ U, @ b {+[ U,, where b + , denotes a small increment in the quantity concerned. The incremental versions of the equations of motion referred to the reference and current configurations, respectively, are

Gly b

V @ zG 5 b {GU 5 lq F (15) and gly @ zG 5 xGU 5 lq a

F ( 16
)
where div is the divergence operator in a F, @ I b V. We note that because of the incompressibility constraint the density z of the bulk material is unaffected by the change of reference configuration.

The definition of the tensor originates from the equality b V 5 Q = u 5 q, which expresses the incremental traction vector in terms of the two incremental stress tensors, and the two-dimensional specialization u q @ I 5 Q of Nansonòs formula, obtained from (2) with u @ gTg4. The linearized constitutive equation for the bulk material is

@ F 3 b QL ( 17
)
where @ judg x (grad being the gradient operator in a F), L is the two-dimensional identity tensor and F 3 is the fourth-order tensor of instantaneous elastic moduli (possessing the major symmetry $ 3JKLM @ $ 3LMJK , J K L M 5 i4 5j). Incompressibility requires that wu @ 3.

Let j denote the Cauchy stress tensor and { 4 and { 5 its principal components. Since @ b j j, the balance of rotational momentum yields 45 54 @ { 4 54 { 5 45 , and a comparison of this with (17) shows that $ 3JKJK @ $ 3KJJK . { J J K 5 i4 5j J 9 @ K (18) Specific expressions for $ 3JKLM are given in section 5, where isotropic materials will be examined.

The incremental boundary conditions for the field equation ( 16) can be derived from (11)ï(14). In particular, for points of ! a F where there is no coating, we have

x @ b 1+U, rq ! a F Y (19) 5 q @ b w+U, rq ! a F G (20)
On a 1, by taking the increments of ( 13) and ( 14), updating from the variable 4 to T, and noting that u @ gTg4, we obtain

b i 3 +T U, @ 5 q . u 4 z 3 G 5 z+T U,GU 5 rq a 1 (21) b . 3 +T U, . u 4 b u ( . b ( @ u 4 * G 5 b +T U,GU 5 rq a 1 (22)
where + , 3 now indicates differentiation with respect to T and z+{ U, @ x+{ U, for { 5 a 1. As pointed out in [START_REF] Ogden | Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation. -0[END_REF], if the underlying deformation is purely static then the form of the inertia term in Equation ( 22) is correct whether or not * is strain dependent because the equations are linearized with respect to a fixed configuration of the film. Note also that u 4 z 3 and u 4 *, respectively, are the mass density and mass moment of inertia of the film per unit current length. Just as ( 14) determines (, Equation ( 22) essentially provides an expression for b (.

Since the incremental form of (9) may be written b

i @ b ' . b (D . ' b . ( b D, use of the expressions b @ b D and b D @ b yields b i @ + b ' ( b , . +' b . b (,D (23)
The term b ( is given by ( 22), while ( can be obtained from the updated counterpart of (14), namely

. 3 +T U, . (+T U, @ u 4 * G 5 +T U,GU 5 (24)
Furthermore, use of 3 @ 3 D @ u 4 D, D 3 @ 3 @ u 4 and b 3 @ u 4 b enables us to obtain an expression for b i 3 , which is needed in (21). Thus

b i 3 @ 5 . /D rq a 1 (25)
where

5 @ b ' 3 . u 4 b . 3 . . 33 b u 4 +' b . . 3 u 4 b u b . 3 . u 4 * G 5 b GU 5 , . u 4 * G 5 GU 5 +u 4 u 3 b . u 5 b u u 4 b , u 4 * b G 5 3 GU 5 / @ u 4 b ' . ' 3 b . u 4 . 3 + b u 3 u 4 u 3 b u , . u 4 b u . 33 b . 33 (26) . u 4 + b ' . . 3 b , . u 4 *+G 5 b 3 GU 5 u 4 u 3 G 5 b GU 5 , . u 5 * G 5 GU 5 +5u 4 u 3 b u b u 3 b , u 5 b u * G 5 3 GU 5
For future reference we record the incremental kinematical expressions

b u @ u z 3 b @ u D z 33 z 3 b @ D z 3 (27)
which follow directly from ( 4) and [START_REF] Ogden | Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation. -0[END_REF].

%(1',1* 2) $ 685)$&(&2$7(' (/$67,& %/2&.

In the application considered in sections 5 and 6 the effect of an incremental motion superimposed on the quasi-static finite deformation of a coated rectangular isotropic elastic block under flexure will be examined, so that time dependence in the equations that govern the finite motion is omitted. Here, we summarize the equations describing this type of inhomogeneous deformation are summarized. In a Cartesian coordinate system +9 4 9 5 , the undeformed block F occupies the region

" 9 4 " ) 9 5 ) (28)
The bending deformation is described, in terms of polar coordinates +S r,, by

S @ G +9 4 , r @ H+9 5 , (29)
so that the deformed block a F is defined by

G + ", S G +", H+ ) , r H+) , (30)
Imposition of the incompressibility constraint reduces the deformation to

S @ 59 4 k . l 45 r @ k9 5 (31)
where k and l are constants which are fixed by the boundary conditions (see, for example, Ogden [START_REF] Ogden | 1RQOLQHDU[END_REF]).

With respect to a reference rectangular Cartesian basis ih 4 h 5 j and to a current polar basis ih S h r j, the (in-plane) deformation gradient takes the form

I @ u 4 h S h 4 . u h r h 5 (32)
where u S @ u 4 and u r @ u @ kS are the principal stretches associated with the Eulerian principal directions h S and h r , respectively.

For an isotropic elastic material the strain-energy function 8+I, can be written, in the present plane context, as a function of the principal stretches, namely 8+u S u r ,. Hence, through [START_REF] Ogden | 1RQOLQHDU[END_REF], the nominal stress tensor may be written

V @ U 4 h 4 h S . U 5 h 5 h r (33)
where

U 4 @ !8 !u S Qu 4 S U 5 @ !8 !u r Qu 4 r ( 34
)
The equilibrium equations obtained from (10) reduce to the following single ordinary differential equation in 9 4 (see [START_REF] Ogden | 1RQOLQHDU[END_REF] for details)

gU 4 g9 4 @ kU 5 ( 35
)
the integration of which, with the help of (34), yields

U 4 @ + 8 . q,u (36)
In (36) the integration constant q and the notation 8+u , @ 8+u 4 u , have been introduced. In terms of 8, the difference between the Cauchy principal stresses { 5 @ { r @ u U 5 and

{ 4 @ { S @ u 4 U 4 is { r { S @ u ! 8 !u (37)
On the curved boundaries of the deformed body the traction vector is given by

V 5 Q @ U 4 h S rq 9 4 @ " (38)
while on the straight sides r @ k) the resultant (in modulus) of the normal stress U 5 , denoted O, and its moment calculated about the origin, denoted P, are

O @ " ] " U 5 g9 4 @ 4 k ^U4 `" " P @ " ] " SU 5 g9 4 (39)
The boundaries 9 4 @ ", denoted 1 and 1 . (with 1 @ 1 ^1 . ), are coated with an elastic thin film with strain energy 6 per unit length. Parametrization of 1 is set so that 4 @ ) 9 5 on 1 and 4 @ ) . 9 5 on 1 . . Therefore, since @ r . 6y5 on 1 and @ r . y5 on 1 . , the kinematical quantities introduced previously on the two coated sides of the block are u @ u + ", @ kS @ r 3 +9 5 , @ k D @ h S @ q @ h r

< A A @ A A > rq 1 u . @ u +", @ kS . . @ r 3 +9 5 , @ k D @ h S @ q @ h r < A A @ A A > rq 1 . (40)
where subscripts and . refer to 1 and 1 . respectively. Along each coated boundary, u and are constant and, since the coating material is homogeneous, it follows that ' 3 @ . 3 @ 3 and, from (14) specialized to the static case, that ( @ 3. Consequently, i+4, @ 6 u and i 3 +4, @ 6 u D on 1 and 1 . . From (13), ( 36) and (38) it then follows that the boundary conditions that couple the response of the block to that of the elastic coating are

U 4 @ + 8 . q,u @ 6 u rq 1 dqg 1 . (41)
These conditions, together with (39), provide sufficient information to determine k l q O and P. For an uncoated block, whatever the strain-energy function 8+u ,, the condition u u . @ 4 holds, leading to

l @ s 4 . 7" 5 k 5 k O @ 3 (42)
Steigmann and Ogden [START_REF] Steigmann | A necessary condition for energy-minimizing plane deformations of elastic solids with intrinsic boundary elasticity. 0DWK 0HFK 6[END_REF] have shown that a necessary condition for a static deformed configuration +u , to be an energy minimizer is that the Hessian matrix

6 u u 6 u 6 u 6 is positive semi-definite. ( 43
)
For a membrane film, for which bending stiffness is disregarded, a necessary condition is that ' is non-compressive, and hence 6 u 3 (44)

0DWHULDO PRGHOV

We assume that the bulk material is characterized by a two-dimensional form of the general class of strain-energy functions introduced in [START_REF] Ogden | Large deformation isotropic elasticity ï on the correlation of theory and experiment for incompressible rubberlike solids. 3URF 5 6RF /RQG[END_REF]. Thus, we may write 8+u , @

/ [ J@4 v J n J +u nl . u nl 5, (45)
where v J and n J are constant parameters satisfying v K n K 3 (no summation over index K) and the ground-state shear modulus is v @ S / J@4 v J n J 5. In particular, a three-term form of (45) is used in the results shown in section 6.1, with values of the parameters given by n 4 @ 46 n 5 @ 83 n 6 @ 53

v 4 @ 47<4 v v 5 @ 3336 v v 6 @ 3356: v (46)
For the coating material we take 6 to have the simple form

6+u , @ 4 5 N+u 4, 5 . 4 5 O 5 (47)
where the material constants N and O, for consistency with the requirement discussed in (43), are non-negative. The current axial force and couple in the coating (cf ( 8)) are given by

' @ N+u 4, . @ O (48)
As discussed in [START_REF] Steigmann | Plane deformations of elastic solids with intrinsic boundary elasticity. 3URF 5 6RF /RQG[END_REF], from a comparison with engineering plate theory it can be deduced that the quantity s ON, which has dimensions of length, may be interpreted as being of the order of magnitude of the coating thickness. Therefore, since the theory of surface coating applies to problems where the coating thickness is very small compared with any other length scale we shall restrict our attention to values of N and O such that ON " 5 .

Having defined the strain-energy functions for the block and the coating, it is now possible to find the parameters that characterize the finite deformation of the coated block. In [START_REF] Dryburgh | Bifurcation of an elastic surface-coated incompressible isotropic elastic block subject to bending[END_REF] explicit results for a specific simple version of (45) are given for the same boundary-value problem.

,1&5(0(17$/ (48$7,216 )25 7+( &2$7(' %/2&.

,QFUHPHQWDO HTXDWLRQV RI PRWLRQ

The incremental equations of motion ( 16) are coupled with the constitutive law (17) to obtain a description in terms of displacement x+{ U,, which, in the polar coordinate system introduced in a F, takes the form x @ V S h S . V r h r . With respect to the same basis the tensor is

@ V SS h S h S . V Sr V r S h S h r . V rS h r h S . V S . V rr S h r h r (49)
The incompressibility condition (tr @ 3) requires that

SV SS . V S . V rr @ 3 (50)
For an incompressible isotropic elastic material the components of the constitutive fourthorder tensor F 3 can be written in terms of two incremental moduli, denoted m 4 and m 5 , that depend on the deformation. The non-vanishing components of F 3 in (17) may be expressed in the form

$ 3SSSS @ $ 3rrrr @ 5m 4 . Q $ 3SrrS @ m 5 . Q $ 3SrSr @ m 5 { r { S 5 $ 3rSrS @ m 5 . { r { S 5 (51)
where Q @ +{ S . { r ,5 and b Q in ( 17) is adjusted for consistency. For hyperelastic materials, m 4 and m 5 are given in terms of the strain-energy function 8+u , by

m 4 @ u 7 + 8 u . u 8 u u , m 5 @ u 5 u 7 . 4 u 7 4 8 u (52)
The incremental constitutive equations ( 17) can now be written as

SS @ +5m 4 . Q, V SS b Q rr @ +5m 4 . Q, V S . V rr S b Q Sr @ +m 5 . Q, V Sr V r S . m 5 { r { S 5 V rS rS @ m 5 . { r { S 5 V Sr V r S . +m 5 . Q,V rS (53)
Since the underlying deformation is static, the incremental equations of motion, in polar coordinates, can be written as

where here and henceforth U denotes the material time derivative. By substituting Equations ( 53) into (54) we obtain

b Q S @ +5m 4 . Q, S . 5+5m 4 . Q, S V SS . +5m 4 . Q,V SSS . +m 5 . Q, V rSr S . +m 5 . , V Srr V rr S 5 zV SUU b Q r @ ^S+m 5 , S . m 5 `V rS . V Sr V r S . +m 5 ,SV rSS Let z 3 @ z D 7
, where z D is the mass density of the film per unit reference area and 7 is the film thickness in the reference configuration. Then

* @ z D W 5 7 45 (70)
For simplicity, we assume that the coating material is incompressible. This leads to the equality W @ 7u and hence

* @ z D 7 6 45u 5 (71)
'<1$0,& %(+$9,285

Separable solutions of (57) of time-harmonic form are now examined. We represent the function +S r U, as

+S r U, @ ~+S, frv Rr h l U (72)
where ~+S, is a real function, R is a real number to be determined and is the frequency of vibration. Substitution of (72) into (73) yields a fourth-order linear ordinary differential equation for ~+S,, not given explicitly here, which is solved numerically. To this end, it is transformed into a system of four first-order ordinary differential equations in the form

[ 3 J @ p J +S [ 4 [ 5 [ 6 [ 7 , J 5 i4 5 6 7j (73)
where [ 4 +S, @ ~+S, [ 5 +S, @ ~3+S, [ 6 +S, @ ~33 +S, [ 7 +S, @ ~333 +S,. A general solution of (73) is given in terms of four independent solutions obtained by solving, using a standard RungeïKutta method, four initial-value problems with initial conditions [ J +S , @ n JK J K 5 i4 5 6 7j, where K is associated with the relevant initial-value problem and S is the value of S on the boundary 1 . When the general solution is introduced into the boundary conditions (66) and (67) a homogeneous linear system of fourth order is obtained. This admits nontrivial solutions if the determinant of the associated matrix of coefficients vanishes, and this yields the ñfrequencyò equation.

As in [START_REF] Dryburgh | Bifurcation of an elastic surface-coated incompressible isotropic elastic block subject to bending[END_REF], we impose the incremental conditions

V S @ 3 rr @ 3 ( 74
)
on r @ k). The first condition requires that r @ 3 on r @ k) and is satisfied provided R is given by

R @ Ly k) ( 75
)
where L is an integer (taken to be positive without loss of generality). With the choice (72) for +S r U,, ) and L appear in Equation ( 57) and the boundary conditions ( 66) and (67) only through R. It therefore suffices to set L @ 4 in (75). Results for L 4 can be obtained from those for L @ 4 by the scaling )L of ).

As already pointed out in section 4.1, the bulk solid is modelled by a three-term strain energy of the form (45) and the coating behaviour is described by the strain energy (47). All parameters appearing in the formulation are non-dimensionalized using the shear modulus v and the length scale ", and an overbar signifies a dimensionless quantity. The dimensionless parameters for the coating material are

N @ N v" O @ O v" 6 (76)
The dimensionless form of the squared frequency is defined as @ z 5 " 5 v (77)

and the dimensionless form of the moment of inertia *, given by (71), is * @ z D 7 6 45 zu 5 (78) where 7 @ 7". As previously discussed, 7 is proportional to s ON. In particular, it can be shown that, in dimensionless form, 7 @ 5

u 6 O N (79)
In dimensionless form, the requirement ON " 5 becomes O N 4 (80)

1XPHULFDO UHVXOWV

The dynamic behaviour of a coated block subject to bending is now illustrated with some numerical examples. For this purpose we concentrate on results where the thin film occupies only one boundary, 1 or 1 . . The response of the structure depends on the following parameters:

the initial dimensions of the block through the ratio )"; the extent of the bending deformation given by k; the material properties of the block fixed by the constitutive assumptions described in section 4.1; the density of the coating relative to that of the bulk material; N and the ratio O N.

When the values of these parameters are given it remains to determine the dimensionless frequency from the frequency equation. Numerical results for a coating on 1 . are reported in Figures 1ï3. In Figures 1 and2, for )" @ 4 5 respectively, the parameter is plotted against the stretch u . on the longer boundary. Illustrative plots are reported for N @ 4 with O N @ 3334 334 335. Results for lower and higher values of O show little qualitative difference from those considered here. Variations in N have been explored but no significant differences have been found for N @ 34 334. For convenience " is set equal to 1 throughout. The influence of the relative densities of the bulk and film materials is investigated by considering density ratios z D z @ 34 4 43. In general, continuous curves pertain to the coated block, while dashed curves describe the behaviour of the uncoated block. Several solution branches are shown. Their cut-off point on the right of the figures corresponds to the value of u . for which the plane section of the block is deformed into a circular annulus, at which point k) @ y. In particular, with N @ 4, the limiting values are u . 688 586 for )" @ 4 5, respectively.

The first branch in each plot intersects the axis @ 3 at a point coinciding with the critical value of u . corresponding to bifurcation. Dryburgh and Ogden [START_REF] Dryburgh | Bifurcation of an elastic surface-coated incompressible isotropic elastic block subject to bending[END_REF] have found that a coating on 1 . promotes bifurcation and that the value of the critical stretch at bifurcation, denoted u eli . , does not depend significantly on the flexural stiffness. For )" @ 4 5, u eli . 499 4;7, respectively, which can be compared to the corresponding values u eli . 4;; 549 for an uncoated block. Solution branches beyond a critical value of u . cannot be reached in a stable quasi-static deformation starting from the undeformed rectangular configuration.

Figure 1 is for )" @ 4. The first row of plots shows that, when the density of the bulk material dominates that of the coating, an increase of bending stiffness has no effect on the frequency of vibration. In the vicinity of the undeformed configuration (u . @ 4) the frequency for the coated block is close to that in the uncoated case, especially for the lower solution branches. In general, the coating has the effect of lowering the frequencies of the first branch and, as indicated above, bifurcation occurs earlier than for the uncoated block. When the densities of the substrate and the film are equal ( z D z @ 4), an increase in the bending stiffness influences primarily the upper branches, whereas the lower branch frequencies are only slightly reduced. When the film density is greater the values of for the first three branches decrease significantly as O N increases while the other solution branches are virtually unaffected. Figure 2 shows the results for )" @ 5 for z D z @ 4 43. Apart from a necessary re-scaling of the abscissa axis, the plots are quite similar to those of Figure 1.

The effect of varying the mode parameter R @ Ly+k) , on at u . @ 46 is shown in Figure 3. By means of R different modes may be selected through the ratio )L. Moreover, it allows for varying the ñfanò described by the flexed block through k) (cf Equation (31)). Since the block deforms into an annulus when k) @ y, the R-axis starts from 1. Instead of , the variable plotted on the vertical axis is R 5 , which is a dimensionless measure of the square of the speed of propagation of a wave along a curved boundary. Since the wavelength is @ 5)L @ 5y+kR, the frequency , which, by definition, is @ 5yD , can be written as @ kRD, where D is the wave speed. It turns out that The first branch in each plot is affected by the pathological situation that @ 3 when R @ 4, corresponding to an apparent bifurcation. A straightforward calculation shows that, for L @ 4 and u . @ 46, R @ 4 corresponds to )" @ 4477 43;5 for the uncoated and coated blocks, respectively. But from the results given in [START_REF] Dryburgh | Bifurcation of an elastic surface-coated incompressible isotropic elastic block subject to bending[END_REF] it is evident that no bifurcation occurs for specimens of these dimensions. In fact, for R @ 4 the equations can be solved exactly and the resulting solution corresponds to a rigid rotation, which does not qualify as an acceptable bifurcation solution. The point + R 5 R, @ +3 4, is therefore associated with the trivial solution. Of course, in the limit R @ 4 the boundary conditions (74) are not applicable. The general trend noticed in Figures 1 and2 is also apparent in Figure 3. In particular, the behaviour of the block is sensitive to changes in bending stiffness only if the coating density is greater than that of the block material.

R 5 @ zk 5 " 5 v D 5 (81)
Figures 4 and5 are for a coating on 1 . Figure 4 results for a block with )" @ 4, and density ratios z D z @ 4 43 are given. In contrast to the case when the coating is on 1 . , bifurcation is advanced or delayed depending primarily on the ratio O N. In this case the . lies in the range ^48< 54:`for both values of the density ratio. Also, when the coating is on 1 changes in N do not alter significantly the general behaviour, as found for a coating on 1 . . Figure 5 is the counterpart of Figure 3 for u . @ 46. Compared to the latter, the most apparent difference is that, as the mode wavelength decreases, a coating on the compressed side appreciably lowers the frequencies of vibration when the bending stiffness is small ( O N @ 3334) and for all density ratios. This can be interpreted as an indication of the tendency of the compressed film to be unstable when its behaviour is close to that of a membrane.

In Figure 6, the influence of the coating rotatory inertia term on the frequencies of vibration of an undeformed block is considered. To this end, the boundary conditions (21) and ( 22) are specialized for the case in which the block remains rectangular with u @ 4. On 1 . we then have @ h 5 D @ h 4 @ 3 . 3 @ 3 ( @ 3 (82) As expected, the differences are more pronounced when the wavenumber (inversely proportional to )L) is relatively high, but are essentially independent of the bending stiffness of the coating. In each plot, the first three solution branches are shown. For )L @ 38 and for the first branch, the discrepancy between results for * @ 3 and * 9 @ 3 increases up to 100% when z D z @ 43. For )L 4 the results are indifferent to inclusion of the coating inertia.

$FNQRZOHGJPHQWV 7KH DXWKRUV DFNQRZOHGJH WKH 3K' 7KHVLV RI 'U *LOOLDQ 'U\EXUJK ZKLFK LV UHODWHG WR WKH FXUUHQW ZRUN 0 * LV JUDWHIXO IRU VXSSRUW IURP WKH 8QLYHUVLW\ RI 7UHQWR ,WDO\

7 l 6 ' 7 l 6 '

 7676 DJDLQVW bn IRU DQ 2JGHQ PDWHULDO EORFN FRDWHG RQ n VROLG FXUYHV ZLWK M* ' 7 GDVKHG FXUYHV DUH IRU DQ XQFRDWHG EORFN DJDLQVW b n IRU DQ 2JGHQ PDWHULDO EORFN FRDWHG RQ n VROLG FXUYHV ZLWK M* ' 2 7 GDVKHG FXUYHV DUH IRU DQ XQFRDWHG EORFN

6 ' 7 l 6 '

 676 DJDLQVW ^IRU DQ 2JGHQ PDWHULDO EORFN FRDWHG RQ n VROLG FXUYHV DW bn ' 7 GDVKHG FXUYHV DUH IRU DQ XQFRDWHG EORFN DJDLQVW b n IRU DQ 2JGHQ PDWHULDO EORFN FRDWHG RQ 3 VROLG FXUYHV ZLWK M* ' 7 GDVKHG FXUYHV DUH IRU DQ XQFRDWHG EORFN critical stretch u eli

2 ) 6 'DJDLQVW 4 S 6 '

 2646 LJXUH 3ORWV RI 7 l*^2 DJDLQVW ^IRU DQ 2JGHQ PDWHULDO EORFN FRDWHG RQ 3 VROLG FXUYHV DW b n ' 7 GDVKHG FXUYHV DUH IRU DQ XQFRDWHG EORFN ρ *4 IRU DQ HODVWLF XQGHIRUPHG EORFN b n ' b 3 ' VROLG FXUYHV 7 GDVKGRWWHG FXUYHV DUH IRU 7 U ' f

. +m 5 5m 4 ,V SSr zSV rUU (55)

where @ +{ r { S ,5. In (55) [START_REF] Dryburgh | Bifurcation of an elastic surface-coated incompressible isotropic elastic block subject to bending[END_REF] the connection (18) has been employed together with the equilibrium equation in terms of the principal Cauchy stresses, which takes the form { SS @ +{ r { S ,S.

The incompressibility condition (50) admits the introduction of a displacement function @ +S r U, such that

Substitution of (56) into (55) followed by elimination of b Q leads to a single incremental equation of motion in terms of , namely $S 7 SSSS . 5%S 5 SSrr . & rrrr . 5+S$ S . $ ,S 6 SSS . 5+S% S %,S Srr .+S 5 $ SS . S$ S $,+S 5 SS S S , +5S% S 5% . S 5 $ SS . S$ S $ &, rr zS 5 +S 5 SSUU . S SUU . rrUU , = 3 (57)

In terms of the incremental moduli m 4 and m 5 and strain energy 8+u ,, the parameters $, % and & introduced in (57) are given by

Equation ( 57) is strongly elliptic if

%RXQGDU\ FRQGLWLRQV

Through (25), the incremental boundary conditions (21) can be written in component form as

The general expressions for 5 and /, given in Equation ( 26), are now specialized for the problem under consideration. Since the underlying deformation is static, we have G 5 GU 5 @ 3. Moreover, u and are constant on a 1 . and a 1 , so that u 3 @ 3 @ 3 and hence, from Equation (48), ' 3 @ . 3 @ 3.

Finally, from Equation ( 8 Equations (60) 4 and (61) 4 involve SS , which includes the unknown b Q. This can be eliminated by differentiating the two conditions with respect to r and substituting (55) [START_REF] Dryburgh | Bifurcation of an elastic surface-coated incompressible isotropic elastic block subject to bending[END_REF]

The final form of the boundary conditions on a 1 for the incremental equation of motion (57) can be obtained through the specialization of ( 27), which yields

and noting that, from Equation (41), { S @ u 4 6 u . Substitution in the expressions (62) for 5 and / and the use of + , 3 @ u 4 + , r then yield 6 u u u 5 + rr S Srr , u 6 u +S Srr . 5 rrrr S Srrrr , . 5

For an uncoated block, boundary conditions on the two curved sides correspond to imposition of traction-free surface conditions, namely $S 6 SSS +5% . $,+ rr S Srr , S 6 z SUU @ 3 rq a 1 (68)

rr . S S S 5 SS @ 3 rq a 1 (69)

The moment of inertia * (per unit length in 1) can be calculated using the analogy with plate theory by considering the coating to be a rod of length W (the deformed coating thickness). The moment of inertia about the axis n through its centre is then * @ z 3 W 5 45.