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Chaotic difference equations in two variables and their multidimensional perturbations
View the table of contents for this issue, or go to the journal homepage for more 2008 Nonlinearity 21 1019
(http://iopscience.iop.org/0951-7715/21/5/007)
Nonlinearity 21 (2008) 1019–1040 doi:10.1088/0951-7715/21/5/007
Chaotic difference equations in two variables and their
multidimensional perturbations
Jonq Juang1, Ming-Chia Li1and Mikhail Malkin2
1Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan 2Department of Mathematics, Nizhny Novgorod State University, Nizhny Novgorod, Russia
E-mail:[email protected],[email protected]@unn.ru Received 6 September 2007, in final form 10 March 2008 Published 8 April 2008
Online atstacks.iop.org/Non/21/1019 Recommended by L Bunimovich
Abstract
We consider difference equations λ(yn, yn+1, . . . , yn+m) = 0, n ∈ Z, of order m with parameter λ close to that exceptional value λ0 for which the function depends on two variables: λ0(x0, . . . , xm) = ξ(xN, xN+L)with
0 N, N + L m. It is also assumed that for the equation ξ(x, y) = 0, there is a branch y = ϕ(x) with positive topological entropy htop(ϕ). Under these assumptions we prove that in the set of bi-infinite solutions of the difference equation with λ in some neighbourhood of λ0, there is a closed (in the product topology) invariant set to which the restriction of the shift map has topological entropy arbitrarily close to htop(ϕ)/|L|, and moreover,
orbits of this invariant set depend continuously on λ not only in the product topology but also in the uniform topology. We then apply this result to establish chaotic behaviour for Arneodo–Coullet–Tresser maps near degenerate ones, for quadratic volume preserving automorphisms ofR3and for several lattice models including the generalized cellular neural networks (CNNs), the time discrete version of the CNNs and coupled Chua’s circuit.
Mathematics Subject Classification: 39A05, 37B45, 37B10, 54F15, 54H20
1. Introduction
In this paper, we continue the study from [13] on the chaotic behaviour of solutions for perturbed singular difference equations. Consider a family of difference equations of the form
λ(yn, yn+1, . . . , yn+m)= 0, n∈ Z, (1)
where λ is a parameter from some metric space. In [13], we assumed that at an exceptional value of the parameter, say λ0, the difference equation depends on only one variable, i.e.
λ0(x0, . . . , xm)= ϕ(xN),
where 0 N m, and we showed that among solutions for perturbed difference equation (1) with λ close to λ0, there are topological k-horseshoes (full Bernoulli shifts with k symbols), provided that ϕ has k 2 simple zeros; moreover, we proved that orbits in these horseshoes change continuously in the uniform topology as λ varies (see theorem11in the next section). In this paper, we consider similar problems in the situation when the difference equation at the exceptional value of the parameter depends on two variables, i.e.
λ0(x0, . . . , xm)= ξ(xN, xN+L),
where N and N + L are two distinct integers between 0 and m, and ξ(x, y) is a function such that for the equation ξ(x, y)= 0 there is a branch y = ϕ(x) with positive topological entropy, i.e. ξ(x, ϕ(x)) = 0 and htop(ϕ) > 0. Notice that in the case when L = 1, the solutions of difference equation (1) with λ= λ0contain orbits of the one-dimensional map x→ ϕ(x). On the other hand, if L > 1, the solutions of (1) with λ= λ0contain orbits of a generalized one-dimensional transformation which can be regarded as the ‘Lth root’ of ϕ (see subsections2.1 and2.4for details). For many cases, solutions of difference equations can be considered as orbits of a high dimensional map.
In view of more applications, we allow the functions λand the local map ϕ to be not
defined in some regions; more precisely, we suppose that λand ϕ are defined on domains Q
and Qm+1, respectively, where Q = [s1, s
2]\ V for some fixed real numbers s1, s2and open set V , the latter being the union of finitely many open intervals in [s1, s2]. Here s1and s2can be regarded as some fixed bounds (from below and from above, respectively) for coordinate projections of orbits we are interested in, while V stays for an escaping region which is never visited by those interesting orbits (for example, if ϕ(x) = ax(1 − x) with a > 4 on the interval [0; 1], then V could contain the escaping interval (1
2 − √ a2−4a 2a , 1 2 + √ a2−4a 2a )). Also, one may include in V those intervals where the functions under consideration are nonsmooth or discontinuous, whenever one is interested only in the orbits (solutions for (1)) which never visit V . In this situation, the topological entropy for (1) (as a quantity to estimate chaotic behaviour of solutions) is defined as htop(σ )for the shift map σ restricted to the set of bi-infinite solutions (xn)∞n=−∞ for (1) (with respect to the product topology) satisfying xn ∈ Q for all n∈ Z. Also, htop(ϕ)is meant as the topological entropy of ϕ restricted to∞n=0ϕ−n(Q)(see the next section for more precise definitions and comments). Our main result shows that if
htop(ϕ) > 0, then for λ sufficiently close to λ0, one can find a closed σ -invariant subset λ
of the set of solutions for (1) such that htop(σ|λ)is arbitrarily close to htop(ϕ)/|L|. Roughly
speaking, the perturbed multidimensional difference equations are chaotic provided that the one-dimensional map at the unperturbed value of the parameter has enough chaotic orbits which avoid prescribed regions. More precisely, we will prove the following.
Theorem 1. Consider a family of difference equations of the form
λ(yn, yn+1, . . . , yn+m)= 0, n∈ Z, (2)
with the function λ : Qm+1→ R which is C1for each λ and is continuous in λ along with
the partial derivatives ∂iλ, i= 1, . . . , m + 1, Q = [s1, s2]\ V for some (fixed) real numbers
s1< s2and V is the union of finitely many open intervals in [s1, s2], while parameter λ is from
some neighbourhood of the unperturbed value λ0in some metric space. Assume that at λ0,
the function depends on exactly two variables: λ0(x0, x1, . . . , xm)= ξ(xN, xN+L),
0 N, N + L m. Assume, in addition, that for the equation ξ(x, y) = 0 there is a branch
piecewise analytic function3. Then for any > 0 there exists δ > 0 such that for each λ in the δ-neighbourhood of λ0, there is a closed (in the product topology) σ -invariant subset λ
of the set of solutions for (1) with htop(σ|λ) > htop(ϕ)/|L| − . Moreover, solutions from λ
depend continuously on λ both in the product and in the uniform topologies.
In comparison with the mentioned result from [13], which can be regarded as a multidimensional perturbation of zero-dimensional systems, the presented result is in a sense a multidimensional perturbation (in the difference equations settings) of generalized one-dimensional maps (also see [18] for a topological approach to perturbations of one-one-dimensional maps to multidimensional ones). To establish the persistence of chaotic behaviour from low dimensional systems to perturbed high dimensional ones, we find a suitable hyperbolic repelling invariant set4carrying almost all topological entropy of the low dimensional system, and we show how to ‘continue’ these hyperbolic orbits to orbits for the perturbed systems.
So the first problem that appears here is in dimension one. It is well known that in contrast to higher dimensions, for smooth one-dimensional maps one has commonly axiom A and hyperbolicity. In [15], Ma˜n´e showed that for a C2interval map f whose periodic points are hyperbolic, any compact f -invariant set away from critical points is hyperbolic repelling (see proposition3in the next section for the case of piecewise C2 interval maps). Nevertheless, given a C2interval map, it is not easy to check the above assumption whether all the periodic orbits are hyperbolic. Also, given a compact invariant set, in order to ensure its hyperbolicity, one needs to check that this set is disjoint from some neighbourhood of the critical set. On the other hand, if the map f has positive topological entropy, it is reasonable to ask whether there is a compact f -invariant hyperbolic set whose topological entropy approximates htop(f )
with required accuracy (see similar problems in [26] for piecewise monotone piecewise C1 intervals maps without critical points and in [10] for C1+ε surface diffeomorphisms; let us mention in this connection that for merely continuous surface homeomorphisms such an approximation need not take place, because Rees in [23] has constructed a minimal positive entropy homeomorphism of the 2-torus). In this context, we prove that given a piecewise monotone piecewise continuous map f on the interval with htop(f ) >0, for any > 0, there exists a compact f -invariant set M = Maway from some neighbourhood of critical points
and discontinuity points, such that htop(f|M) > htop(f )− (see theorem4). Then, with the help of Ma˜n´e’s theorem, we can get rid of the redundant assumptions (both on the map and on the orbits of points from the invariant set) to ensure hyperbolicity for the restriction of a piecewise analytic map f to an appropriate set carrying entropy bigger than htop(f )− . Finally, by using our technique from [13], we are able to ‘continue’ this hyperbolic set for perturbed difference equations.
In many applications, solutions of the difference equations for nonexceptional values of λ are actually associated with orbits of well-defined maps on a high dimensional space; refer to definition 3.1 of [13] and the appendix. For instance, polynomial maps onRm, under generic
algebraic conditions, can be written as difference equations for xis inR. In [11], the description
of the elimination process for polynomial maps which leads to such difference equations is given, and the problems on the uniqueness of the obtained difference equations are discussed by using the algebraic hypersurface approach by Milnor [17]. In section3, we apply theorem1 to establish chaotic behaviour for Arneodo–Coullet–Tresser maps near degenerate ones and
3 The condition on piecewise analyticity of ϕ can be weakened to be piecewise C2with piecewise monotone derivative;
but, for such an extension, some cumbersome techniques from [12] are needed because there might be an interval consisting of nonhyperbolic periodic points for ϕ.
4 An invariant set J of an interval map f is said to be hyperbolic repelling if there exist constants C > 0 and µ > 1
for quadratic volume preserving automorphisms ofR3at anti-integrable limits5. We also apply
theorem1together with some results from [13] to study chaotic structures in stationary and travelling waves of several models including the generalized cellular neural network (CNN), the time discrete version of the CNNs, the coupled Chua’s circuit and lattice models of an evolution equation.
The paper is organized as follows. In section2, we prove our main result (theorem1), and along with the proof, in section2.1, we recall some techniques from [13]. In sections2.2 and2.3, we prove proposition 3and theorem4, the important one-dimensional ingredients for the main result. In section2.4, we give a comparison of the continuation schemes for perturbations of singular difference equations depending on one variable (as in [13]) with those of two variables (as in this paper). Section3collects several applications. In the appendix, we give some basic terminologies.
2. Perturbations of difference equations in one and two variables
2.1. Proof of theorem1
First we introduce the necessary notation. Let ∞denote the Banach space of bounded real bi-sequences endowed with the normy = supn∈Z|yn|, where y = (yn)∞n=−∞∈ ∞. In what
follows, we will consider both ∞and its subsets not only in the above (uniform) topology but also in the product topology onRZ, i.e. in the topology of pointwise convergence. In the latter case, to avoid misunderstanding, we will sometimes supply the notation of appropriate sets with the subscript prod, for example: ∞,prodand Bprod. Let σ denote the shift map on ∞, i.e. σ (y)= ywith yn = yn+1and n∈ Z, for any y ∈ ∞. We will denote by U (c, r) the open ball of radius r centred at c in an appropriate metric space.
For each λ from the parameter space, let Yλ be the set of solutions of the difference
equation (2), i.e. the set of bi-sequences y = (yn)∞n=−∞ such that for any n ∈ Z, one has
yn ∈ Q and, moreover, (m + 1) consecutive components yn, yn+1, . . . , yn+mof y satisfy (2). In [13], we have shown that Yλis a σ -invariant closed subset of the space ∞in the topology
of uniform convergence, Yλ,prod is compact as a closed subset of QZprod and the restriction
σ|Yλ,prodis a homeomorphism. Thus, one can define the topological entropy for solutions of the
difference equation (2) as htop(σ|Yλ,prod).
The following special version of the implicit function theorem is proved in [13]; it serves as the main theoretical background for our construction of symbolic continuation both in [13] and in this paper.
Theorem 2 ([13, theorems 2.1 and 2.5]). Let E be a metric space with metric ρ and let a
be a point in E. Let B be a σ -invariant compact subset of [s1, s2]Zprod for some real numbers
s1< s2. Denote V0= U(a, δ0) and W0 =
b∈BU (b, η0) (the latter balls being with respect
to the ∞metric) for some δ0 > 0 and η0 > 0 and assume that F : V0× W0 → ∞is a
function such that the following conditions hold: (i) F (a, b)= 0 for all b ∈ B;
(ii) F is continuous and, moreover, the family of functions F (·, y) with the domain V0and
parameter y∈ W0is equicontinuous, i.e. for any > 0 there is δ > 0 such that for any y∈ W0one hasF (x1, y)− F (x2, y) < whenever ρ(x1, x2) < δ;
5 The latter problem was motivated by a question posed to us in the referee report to our paper [13]. The referee asked
whether the results of [13] could apply for quadratic volume preserving automorphisms onR3in the generic form
at anti-integrable limits. Actually, the difference equations for such limits depend on two variables (see section3.2), and so, in order to establish chaotic behaviour in this situation, we have to use the presented new approach.
(iii) the partial derivative operator with respect to the second variable, D2F (x, y), at any
point (x, y) ∈ V0× W0 exists and is continuous at {a} × B uniformly in b ∈ B in
the following sense: for any > 0 there exist δ > 0 and η > 0 such that for any b∈ B, D2F (x, y)− D2F (a, b) < whenever x ∈ U(a, δ) and y ∈ U(b, η);
(iv) the operator D2F at any point (a, b) ∈ {a} × B is invertible, and the inverse,
(D2F )−1, is uniformly bounded, i.e. there is a constant M > 0 such that for any
b∈ B, (D2F (a, b))−1 M and
(v) for any x∈ U(a, δ0), the function F (x,·) commutes with σ and is continuous with respect
to the product topologies on the domain W0and codomain ∞.
Then there exist 0 < ˆδ < δ0 and 0 < ˆη < η0such that for any x ∈ U(a, ˆδ), there is a
map from B to U (b,ˆη), given by b → ¯ψx(b):= ψb(x), which conjugates σ|Bprodto σ|¯ψx(B)prod.
Moreover, the conjugacy map depends continuously in x not only in the product topology but also in the ∞topology; more precisely, the family of maps x → ψb(x) from U (a, ˆδ) to ∞
forms an equicontinuous family in b∈ B.
In order to apply theorem2, we consider E to be the parameter metric space and a to be the unperturbed parameter value λ0in theorem1. Define F : E× QZ→ ∞by
F (λ, y)= (λ(yn, yn+1, . . . , yn+m))∞n=−∞.
Then, Yλis precisely the zero-set of F (λ,·), i.e. Yλis the set of bi-sequences y∈ QZsatisfying
F (λ, y) = 0. Consider B to be a σ -invariant and compact subset of Yλ (in the product
topology), which will be specified later. Then assumption (i) of theorem2is satisfied. Since
E× Qm+1 is compact, both F (λ, y) and D2F (λ, y)are uniformly continuous on E × QZ. Therefore assumptions (ii), (iii) and (v) of theorem 2 are also satisfied. As for the most delicate assumption (iv), we need to specify a suitable subset B of Yλin order to guarantee
this assumption.
Given y= (yn)∈ QZ, λ∈ E, and integers n ∈ Z, 1 i m + 1, we denote for brevity ∂iλ(˜yn) = ∂iλ(yn, yn+1, . . . , yn+m). Then by our assumptions on λ, we have that the
partial derivative operator D2F (λ, y)exists at any point (λ, y)∈ E × QZand is represented by the following bi-infinite band matrix.
D2F (λ, y) = · · · ... ... ... ... ... ... · · ·
· · · ∂1λ(˜yn) ∂2λ(˜yn) · · · · ∂m+1λ(˜yn) 0 · · ·
· · · 0 ∂1λ(˜yn+1) ∂2λ(˜yn+1) · · · · ∂m+1λ(˜yn+1) · · ·
· · · ... ... ... ... ... ... · · · the ← nth. row ↑ the nth column
Without loss of generality, we assume that L > 0, the case when L < 0 needs no additional treatment, because σ is a homeomorphism on the compact space Yλ,prod and
htop(σ|S) = htop(σ−1|S)for any compact invariant set S ⊂ Yλ,prod. Note that under our assumption on the branch y = ϕ(x) of the equation ξ(x, y) = 0, at the exceptional value of the parameter, the difference equation (2) reads yn+L = ϕ(yn), n∈ Z. Hence the difference
equation at λ= λ0corresponds to the map fλ0 : Q
L→ [s1, s2]Lof the form fλ0(x1, x2, . . . , xL−1, xL)= (x2, x3, . . . , xL, ϕ(x1));
see the appendix for what we mean by ‘a difference equation corresponds to a map’. So fλ0
can be regarded as the ‘Lth root’ (or ‘1/Lth iterate’) of the one-dimensional map ϕ. Also, it is easy to see that for each k∈ N, the iterate fkL
λ0 of fλ0is of the form
fλkL
0 (x1, x2, . . . , xL−1, xL)= (ϕ
k(x
1), ϕk(x2), . . . , ϕk(xL))
and corresponds to the difference equation
ykL+n− ϕk(yn)= 0, n∈ Z. (3)
To guarantee assumption (iv) of theorem2, we need a uniform estimate of the inverse of
D2F (λ0, y)on a suitable σ -invariant and compact subset B of Yλ.To this end, we shall find a compact ϕ-invariant hyperbolic set (carrying enough topological entropy) which should be ‘continued’ by orbits of difference equation (2) for λ close to λ0.
To estimate the topological entropy carried by such orbits, we will need some general results on the entropy of piecewise monotone (possibly discontinuous) maps. Let us give certain definitions and agreements about piecewise monotone maps on the interval. Without loss of generality, we put I = [0, 1] for the interval. Let f : I → I be a piecewise monotone piecewise continuous map and letZ be its partition, i.e. I =Z∈Z ¯Z and Z consists of finitely
many, say k, disjoint open intervals, denoted by (0, d1), (d1, d2), . . . , (dk−1,1), on each of which the restriction of f is monotone and continuous. It is also assumed that a piecewise monotone map can have only finitely many constancy intervals, i.e. maximal subintervals at which f takes a constant value. Let us remark that the map f need not be strictly monotone on each interval inZ and moreover dis need not be critical or discontinuity points. We say that a piecewise monotone map g : I → I is piecewise C2(respectively, piecewise analytic) if its partition can be chosen so that on each interval of the partition, g is C2(respectively, analytic). We will refer to the intervals of such a partition simply as monotonicity intervals. We will need the following lemma whose proof is given in section2.2.
Proposition 3. Let g : I → I be piecewise C2and let U be a neighbourhood of the set which consists of all critical points of g and all endpoints of monotonicity intervals. Then
1. all periodic orbits of g contained in I\U of sufficiently large periods are hyperbolic
repelling,
2. if M ⊂ I is a compact forward invariant set which contains neither attracting nor
nonhyperbolic periodic points of g and is disjoint from U , then M is a hyperbolic repelling set.
To define the topological entropy for piecewise continuous piecewise monotone maps, we use here the approach by doubling points construction, as in [19]; see section2.3for details. There are other definitions of topological entropy for these maps (via separated or spanned sets and also by counting the growth number for preturning points) and they are equivalent, as shown in [20].
The following result is an important ingredient for the proof of theorem1. In its statement, by strict f -invariance of a set M we mean the equality f (M)= M.
Theorem 4. Let f be a piecewise monotone piecewise continuous map on I with the partition
Z = {(0, d1), (d1, d2), . . . , (dk−1,1)}. If htop(f ) > 0 then for any > 0 there is a
compact strictly f -invariant set M ⊂ I and an open set J ⊃ {0, d1, . . . , dk−1,1} such that htop(f|M) > htop(f )− and M ∩ J = ∅.
The proof of theorem4is postponed to section2.3. We continue the proof of theorem1. Without loss of generality, we assume that the interval [s1, s2] in theorem1is I = [0, 1]. Since we allowed the local map ϕ : I\ V → I to be not defined on V , which consists of finitely
many open intervals, let us extend ϕ to become a self-map ϕ : I → I by taking at those ‘exceptional’ intervals the constant value equal to 1, the right endpoint of I (it is easily seen that this extension does not influence the value of the topological entropy of ϕ, see also [20]). We may apply theorem4to our piecewise analytic self-map ϕ : I → I because it is surely piecewise monotone and has only finitely many constancy intervals. Since ϕ has only finitely many critical points away from constancy intervals, we may include these points into the set
Dof endpoints of monotonicity and constancy intervals. Finally, let us agree that the (finitely many) values which ϕ takes at constancy intervals are also included in D. Then D induces a partition consisting of finitely many disjoint open intervals, on each of which the restriction of
ϕis monotone and continuous.
Given any small ε > 0, by theorem4, one gets a compact strictly ϕ-invariant set, which is denoted now by ¯M, and an open set J ⊃ D such that the ϕ-orbits of ¯Mare disjoint from J , and htop(ϕ|M¯) > htop(ϕ)− ε/2. Note that the ϕ-orbits of ¯Mare disjoint from each constancy
interval for ϕ (if it exists). For convenience, redenote the open set J by adding all constancy open intervals. Then htop(ϕ|I\J) htop(ϕ|M¯) > htop(ϕ)− ε/2. By applying proposition3
to ϕ with the neighbourhood J of the set of all critical points and endpoints of monotonicity intervals, one gets that all attracting and nonhyperbolic periodic points in I\ J have bounded periods, say k0. Thus, ϕ has finitely many attracting and nonhyperbolic periodic points in I\J . Redenote the partition D by adding those attracting and nonhyperbolic periodic points in I\J . Modify ϕ to get a new map ˜ϕ on I by taking the value 1 on the set J. Then ˜ϕ|I\J = ϕ|I\J and htop(˜ϕ) = htop(ϕ|I\J). By applying theorem4to ˜ϕ with the new partition D, we get a compact
˜ϕ-invariant set M and an open set ˜J ⊃ D such that the ˜ϕ-orbits of M are disjoint from ˜J, and
htop(˜ϕ|M) > htop(˜ϕ) − ε/2. Then M is ϕ-invariant, the ϕ-orbits of M are disjoint from ˜J and
htop(ϕ|M)= htop(˜ϕ|M) > htop(˜ϕ) − ε/2 = htop(ϕ|I\J)− ε/2 > htop(ϕ)− ε.
By the second item of proposition3, M is a hyperbolic repelling set. Moreover, it can be shown (see lemma 2.1 of chapter III in [16]) that there is an integer k1> k0and a real number
η >0 such thatDϕk1(x) > 1 + η for all x ∈ M.
Now that we have found the set M:= M, we are in a position to check assumption (iv)
of theorem2. Let B = lim
←−(M, ϕ) = {(xn)∞n=−∞ : xn+1 = ϕ(xn)and xn∈ M for all n ∈ Z}.
By replacing ϕ by ϕk1 and the difference equation (2) at λ0by (3) with ϕk1, we get that the
partial derivative operator D2F (λ0, y)for all y∈ B has the matrix of the form σk1L◦ (I + ), where σ is the matrix of the shift operator, I is the identity matrix and is a (shifted) one-diagonal matrix with entries bigger than 1 + η in absolute value. By using the following lemma on the norm of two-diagonal infinite matrices, assumption (iv) of theorem2will be satisfied.
Lemma 5. Let A : ∞→ ∞be a linear operator given by A= σk◦ (I + ), where k ∈ Z, σ is the shift operator and is associated with matrix of the form
ij =
qi, if j = i + L, 0, otherwise,
for some sequence (qi)∞i=−∞ satisfying q := infi∈Z|qi| > 1. Then A is invertible and
A−1 < 1/(q − 1).
Proof. Note that the operator is invertible, and its inverse is represented by the matrix of
the form
−1ij =
1/qi−1, if j = i − L,
Hence−1 q1 <1. Then A−1= (I + )−1◦ σ−k= −1◦ (I + −1)−1◦ σ−k= −1◦ ∞ i=0 (−1)i−i ◦ σ−k. Since the shift operator σ is invariant with respect to the ∞-norm, i.e.σ ◦T = T ◦σ−1 = T for any T , and since −1 < 1, it follows that the last series converges in the operator norm. So we have A−1 = −1◦ ∞ i=0 (−1)i−i ◦ σ−k −1 · ∞ i=0 −1i 1 q · 1 1− q−1 = 1 q− 1.
The proof of the lemma is completed.
Hence, theorem2implies that we may use the conjugacy ¯ψλ|Bto get a closed σ -invariant
subset λ := ¯ψλ(B)of Yλ,prodsuch that σ|B is topologically conjugate to σL|λ (both in the
product topology). For details about the conjugacy map ¯ψλand its properties see section2.4, especially diagram (15). Therefore, htop(σ|λ) = htop(σ|B)/|L| = htop(ϕ|M)/|L|, which is
arbitrarily close to htop(ϕ)/|L|. This completes the proof of theorem1.
2.2. Proof of proposition3
Proposition3itself and its proof are adapted from the following results by Ma˜n´e.
Theorem 6 ([15], see also [16]). Let I be a compact interval in R and g : I → I be a C2
map. Let U be a neighbourhood of the set of critical points of g. Then
1. all periodic orbits of g contained in I\U of sufficiently large periods are hyperbolic
repelling,
2. if all the periodic orbits of g contained in I\U are hyperbolic repelling, then there exist C >0 and µ > 1 such thatDgn(x) Cµn, whenever gi(x) ∈ I\(U ∪ B0) for all
0 i n − 1, where B0is the union of the immediate basins of the attracting periodic orbits of g contained in I\U.
The above theorem implies the following important corollary.
Corollary 7 ([16, corollary III.5.1]). Let I be a compact interval inR, g : I → I be a C2
map and M ⊂ I be a compact forward invariant set. If M does not contain critical points,
attracting periodic points and nonhyperbolic periodic points of g, then it is a hyperbolic repelling set.
Let us prove proposition3by applying theorem6and corollary7as follows. In order to obtain a C2 map, we can modify the map g inside small neighbourhoods of endpoints of monotonicity intervals so that such neighbourhoods are contained in U . Denote the obtained map by G. By applying theorem6to G, we get that any periodic orbit of G contained in I\ U of sufficiently large period is hyperbolic repelling. On the other hand, by the construction of
G, any g-periodic orbit contained in I\ U coincides with the G-periodic orbit and is away from U . This proves the first item. The second item follows from corollary7being applied for G, because here we use again that G and g coincide on I\U.
2.3. Proof of theorem4
The proof of theorem4contains several lemmas. We will need the so-called doubling points construction (see [19] for instance). Let I = [0, 1] and D = {d1, d2, . . . , dk−1}. Let us emphasize that we do not care about values of f at the points from D because only one-sided limits of f at these points are of use (see also [20], where the authors proved that the values of the map at the endpoints of intervals of continuity are irrelevant for calculation of topological entropy). Define the set
W:= ∞ i=0 f−i(D) \ {0, 1}
(which could be thought of as ‘the set of preturning points’). Now consider the following set ˆI which contains ‘doubling preturning points’ rather than single ones:
ˆI := (I \ W) ∪ {w−, w+: w∈ W}.
This means that we have doubled (i.e. separated by moving apart) all points of D along with all their inverse images; the order and the topology in this new set are as follows. The set ˆI = ˆI(f ) is endowed with the natural (full) order so that if y < w < z in I and w ∈ W , then
y < w−< w+< z. Then it is supposed that ˆIis endowed with the order topology (note that ˆI
is a totally disconnected space provided f has no homtervals). It is also convenient sometimes to include the points{0, 1} in D, in which case the (‘half-open’) intervals [0, 0+)and (1−,1] are included in ˆI. We will call ˆI the doubling construction space for f .
Let π : ˆI → I denote the map by
π(y)= w for y ∈ {w−, w+} with w ∈ W and π(y) = y for y ∈ I \ W. (4) For a subset A ⊂ I, let closˆIA denote the closure of π−1(A \ W) in ˆI. Let
ˆ
Z = {closˆIA: A∈ Z}. The restriction f |I\W can be uniquely extended to a continuous
piecewise monotone map ˆf : ˆI → ˆI. We will call ˆf the doubling extension of f .
For x, y∈ ˆI, let (x, y) be the minimal nonnegative integer such that ˆf (x)and ˆf (y)
belong to different elements of ˆZ, and set (x, y)= +∞ if for any n, ˆfn(x)and ˆfn(y)belong
to the same element (depending on n) of ˆZ. Then the order topology on ˆI is induced by the metric ˆρ on ˆI defined by the formula
ˆρ(x, y) := 1
(x, y)+ 1 +|π(x) − π(y)|. (5)
We remark that ˆρ(d−, d+)= 1 for any d ∈ D (even in the case when f is continuous at d). The map π is continuous on ˆI and it is a semiconjugacy from ˆf to f in the following sense:
f◦π(x) = π ◦ ˆf (x)for all x∈ ˆI\{d−, d+: d∈ D}, and if x = d+(respectively, x= d−) for some d∈ D, then π ◦ ˆf (x)= limydf ◦ π(y) (respectively, π ◦ ˆf (x)= limydf ◦ π(y)).
So, if P is an ˆf-invariant set disjoint from{w−, w+: w∈ W} then π conjugates ˆf|
Pto f|π(P ).
According to the full order in ˆI, we can consider intervals in ˆI of the form (a, b), (a, b], [a, b), or [a, b] (the latter possibly with a = b), a, b ∈ ˆI. Let c ∈ ˆI, > 0 and let U(c)
denote the open ball of radius centred at c : U(c)= {x ∈ ˆI : ˆρ(x, c) < }. Note that U(c)
is an interval in ˆI which might have any of the four above forms, and π(U(c))is an interval in I which contains π(c), but π(c) need not be the middle point of this interval. Nevertheless, it is easily seen that π(U(c))tends to{c} as → 0.
Following [19], we define the topological entropy, htop(f ), of the initial map f to be
an > 0 and a∈ ˆI, let M,adenote the set M,a:= ˆI \ ∞ n=0 ˆ f−n(U(a)). (6)
So M,ais a compact ˆf-invariant subset of ˆI (in the order topology), and ˆf-orbits of points
from M,anever visit U(a). Let us fix (for a while) an with
0 < < min{1, min{|d − d|/2 : d, d∈ D, d = d}} (7) and take a point a∈ {d−, d+} for some d ∈ D. We now consider, without loss of generality, the
case when a= d−. Then, because of (7), the subinterval U(a)of ˆIis of the form either (y, d−] with y ∈ I \ W or [y, d−] with y = w+for some w∈ W. We denote such a left endpoint y by d−. Note that (7) also implies that these 2(m− 1) subintervals {U(d−), U(d+): d∈ D}
are disjoint (because the distance ˆρ between points on ˆI is bigger than or equal to the distance on I between the π -image of these points).
Define the following map ˜f,d−on ˆI by
˜ f,d−(x)= ˆ f (x), if x /∈ U(d−), ˆ f (d−), otherwise, (8)
and call it the left -truncation of ˆf at d.
Lemma 8. The map ˜f,d− : ˆI → ˆI is continuous and htop( ˜f,d−) = htop( ˜f,d−|M,d−) = htop( ˆf|M,d−).
Proof. The map ˜f,d−differs from ˆf only in the interval U(d−), which equals either (d−, d−]
or [d−, d−]. Since ˆf is continuous, it follows that in both cases ˜f,d− is continuous at d−
(because of the definition by (8)). Moreover, ˜f,d−is continuous at d−because d−is isolated
in ˆI from the right.
Let ( ˜f,d−)be the nonwandering set of ˜f,d−. If ˆfn(d−
)∩ U(d−) = ∅ for every n 1, then ( ˜f,d−)⊂ M,d−because U(d−)consists of a wandering point for ˜f,d−, and
if we suppose, in contrast, that there is a point in ( ˜f,d−)\ M,d−, then we would have a
contradiction to the fact that the nonwandering set is invariant. Therefore,
htop( ˜f,d−)= htop( ˜f,d−|( ˜f,d−))= htop( ˜f,d−|M,d−)= htop( ˆf|M,d−), (9)
where the last equality holds because the restriction of ˆfto M,d− coincides with ˜f,d−.
In the case when ˆfn0(d−
) ∈ U(d−) for some n0 1, it is easily seen that the set
( ˜f,d−)\ M,d− consists precisely of one periodic orbit of period n0. Thus we have as
before, htop( ˜f,d−|( ˜f
,d−))= htop( ˜f,d−|M,d−)because the topological entropy on a finite set is
zero. So in this case the equalities in (9) are true.
Similar to the left -truncation, we can consider the right -truncation, ˜f,d+, and get
htop( ˆf|M,d+) = htop( ˜f,d+|M,d+). Furthermore, we consider the -truncation for all d ∈ D
simultaneously. To do this, we define the map ˜fon ˆIby
˜ f(x)= ˆ f (x), if x /∈ U(d−)∪ U(d+)for all d ∈ D, ˆ f (d−), if x∈ U(d−)with d∈ D, ˆ f (d+), if x∈ U(d+)with d∈ D. (10)
Also let ˜M:=
d∈D(M,d−∩ M,d+). Then ˜Mis a compact ˆf-invariant subset of ˆI whose
ˆ
f-orbits never visit the -neighbourhood of the setd∈D{d−, d+} . Since D is a finite set, a
result similar to lemma8readily follows.
Lemma 9. The map ˜f: ˆI → ˆI is continuous and htop( ˜f)= htop( ˜f|M˜)= htop( ˆf|M˜).
In order to relate the above properties of continuous maps on ˆIto the properties of piecewise continuous maps on I (in other words, ‘to project’ the constructed truncations to maps on I ), we need to introduce an intermediate space. To do this, we identify those pairs of points {w−, w+}, w ∈ W, which under some iterate of ˆfbelong to the same -neighbourhood of either
d−or d+for some d∈ D. More precisely, consider the following equivalence relation ∼ on ˆI:
x∼ y ⇐⇒ x= y or {x, y} = {w
−, w+}, w ∈ W, and there exist
n 0 and ˜d ∈d∈D{d−, d+} such that { ˆ˙ fn(x), ˆfn(y)} ⊂ U( ˜d).
Note that by the above definition, the relation ∼ is preserved by ˜f, i.e. if x ∼ y then
˜
f(x)∼ ˜f(y). Let ˆIbe the quotient space with respect to this relation∼ and let ˆπ: ˆI → ˆI
be the corresponding quotient map. Clearly, ˆπis at most two-to-one, order preserving, and is
continuous with respect to the order topologies on ˆIand ˆI. If two points w−, w+are collapsed
by ˆπ(i.e. w− ∼ w+), we will denote their common image simply by w. Let Wdenote the
subset of W which consists of ‘noncollapsed’ points by ˆπ, i.e. W= {w ∈ W : { ˆπ−1(w)} = 1}.
Then ˆIcan be represented as ˆI = (I \ W)∪ {w−, w+: w∈ W
}. By the definition of ˆπ,
one easily gets π◦ ˆπ= π, where π: ˆI→ I is defined just as π by (4) (for πwe use the
subscript in order to mention that it acts on the space different from ˆI).
Let g : ˆI → ˆI be a continuous map which preserves the relation ∼, i.e. g satisfies the
assumption that g(w−)= g(w+)for every w ∈ W with ˆπ(w−)= ˆπ(w+). Then g projects to a continuous map on ˆI. Indeed, consider the map g†: ˆI→ ˆIdefined by
g†(x)= ˆπ◦ g( ˆπ−1(x)),
where by ˆπ−1(x)we mean the full preimage of x; note that although ˆπ−1(x)may consist of two points, g(ˆπ−1(x))is a single point because of the above assumption on g. By its definition,
g†is continuous and satisfies g†◦ ˆπ
= ˆπ◦ g. We may apply the above construction to the
map g = ˜fbecause it is continuous on ˆI and preserves the relation∼; hence we have the
continuous map ˜f†
on ˆI.
Now we return to the ‘initial phase space’ I . Let M := π( ˜M).
Lemma 10. The following statements hold:
1. the set M is a compact f -invariant subset of I ;
2. there is a neighbourhood of D such that for any point x0∈ M
, its f -orbit is disjoint from this neighbourhood.
Proof. Let us recall that π : ˆI → I is continuous and the restriction of π to any subset of ˆI
disjoint from{w−, w+ : w∈ W} is a one-to-one map which satisfies f ◦ π(x) = π ◦ ˆf (x). This implies the first statement of the lemma. Let
δ0= min
d∈Dmin{d − π(d
−
), π(d+)− d}.
0
Χ
π(dε+)
π(dε-) d d' d" Figure 1. The graph of the -truncation map f.
So, in particular, M is disjoint from W and therefore, by notation in (4), M is the same as ˜M(they are homeomorphic metric spaces with respect to the usual length on Mand the
metric ˆρ on ˜M). Next, we define the -truncation map f: I→ I by
f(x)= f (x), if x /∈ (π(d−), π(d+))for all d∈ D, f (π(d−)), if x∈ (π(d−), d)with d∈ D, f (π(d+)), if x∈ (d, π(d+))dwith D (11) (see figure 1).
It is easily seen that fis continuous at the points π(d−), π(d+)for any d ∈ D. Hence, f is a piecewise continuous and piecewise monotone map with the partitionZ. Then it is easily checked that the doubling construction space for fis precisely ˆI(i.e. if we consider fas the initial map then ˆIf, its doubling construction space, coincides with ˆI), while the doubling extension of fis precisely ˜f†, i.e. ˆf = ˜f†. So we have the following commutative
diagrams: ˆI ˆπ −−−−→ ˆI π −−−−→ I ˜ f f˜†fˆ f. ˆI −−−−→ ˆπ ˆI −−−−→ π I (12)
Now we are in position to prove theorem4.
Proof. Since ˜f is semiconjugate to ˆf by the map ˆπ, which is at most two-to-one, we
have htop( ˜f)= htop( ˆf). Note that if we consider the restrictions ˜f|M, ˆf|ˆπ(M)and f|M
in diagram (12), then the semiconjugacies ˆπ and π become in fact conjugacies. Hence htop( ˜f|M)= htop(f|M). So by using lemma9, we have
Since ˆf is the doubling extension of f, it follows from the definition of topological
entropy for piecewise monotone maps that htop(f)= htop( ˆf). Now, using the fact that the
restrictions to Mof the maps f and fcoincide, we have
htop(f)= htop(f|M)= htop(f|M). (13)
Since each interval (π(d−), π(d+))tends to{d} as → 0, we have that the Hausdorff distance
between graphs of fand f tends to zero. Thus by the lower semi-continuity property of the
topological entropy function on the set of piecewise monotone piecewise continuous maps with the given number of monotone intervals (see [20]), we have lim inf→0htop(f) htop(f ). On the other hand, it is easily seen that for any > 0, htop(f) htop(f ). So we get that lim→0htop(f|M) = htop(f ) and thus, by (13), lim→0htop(f|M) = htop(f ). So,
given a δ > 0 we can find 0 > 0 such that for 0 < < 0, htop(f|M
) > htop(f )− δ. Finally, we let M := ∞n=0f n(M )and J := d∈D(π(d−), π(d+))according to notations
in the statement of theorem4. Then the strict f -invariance of M follows from compactness of M and continuity of the restriction f|M. Thus, by corollary 8.6.1 of [25], we have htop(f|M)= htop(f|
∞
n=0f n(M
))= htop(f|M).
2.4. Properties of the conjugacy ¯ψλand comparison with the result in [13]
In [13], we considered the case when the unperturbed difference equation involved only one variable: λ0(x0, x1, . . . , xm)= ϕ(xN), and the local map ϕ had multiple simple zeros, say
{d1, . . . , dk} ∈ int Q. In this case, we let B = {d1, . . . , dk}Z, then the infinite matrix of
D2F (λ, y)for λ = λ0 and y ∈ B contains only one nonzero diagonal with finitely many
values of the form ϕ(di). This implies that assumption (iv) of theorem2is satisfied. As a
result, we got the following (some terminologies used in the following statement are given in the appendix).
Theorem 11 ([13]). Under the above assumptions, there exists ¯δ > 0 such that for any
λ ∈ U(λ0, ¯δ) there is a closed (in the product topology) σ -invariant subset λ ⊂ Yλand the following holds.
(i) σ|λis topologically conjugate to σ|k by the conjugacy map ¯ψλ: k→ λ, where ¯ψλ
is from theorem2and k= {1, 2, . . . , k}Z; moreover, one has the commutative relations in the first three columns in diagram (14).
(ii) The conjugacy map ¯ψλis the identity map for λ= λ0and is continuous in λ; moreover, the map λ→ ¯ψλ(x) from U (λ0, ¯δ) to ∞(in the uniform topology) forms an equicontinuous
family in x∈ k.
(iii) If, in addition, the difference equation (2) for given λ corresponds to a map fλ: Pλ→ Rm, then one has the following commutative diagram:
k −−−−→ ¯ψλ λ i → Yλ Tλ −−−−→ ˜Pλ −−−−→ Kπ0 λ σ σ σ σm fλ, k −−−−→ ¯ψλ λ→ i Yλ −−−−→Tλ ˜ Pλ −−−−→ π0 Kλ (14)
where all the maps involved are continuous (in the product topology on the symbolic spaces), ¯ψλis injective, Tλis bijective, the notation→ denotes the embedding and π0i is a (surjective) projection which is entropy preserving and is bijective when restricted to the set of periodic points; here ˜Pλ = {p = (pn)∞n=−∞ ∈ PλZ : pn+1 = fλ(pn)} and
Kλ= ∞ i=0f i λ( ∞
n=0fλ−n(Pλ)). In particular, the map θλ:= π0◦Tλ◦ ¯ψλsemiconjugates σ|k with the restriction of fλto the closed fλ-invariant set λ:= θλ(k). Moreover, the
inverse limit of fλ|λis conjugate to σ|k. If fλ : Pλ→ R
mis injective, then the above semiconjugacy θλ|k is in fact a conjugacy.
In the settings of this paper, we show similar results using the same approach by theorem2. The only difference is that instead of the full horseshoe k, we continue with an appropriate
hyperbolic, strictly ϕ-invariant set M(i.e. ϕ(M)= M) with enough entropy and consider the
set B in theorem2to be lim
←−(M, ϕ), the inverse limit of ϕ|M (rather than Mitself); because,
by the definition of Yλ, it consists of bi-infinite sequences, i.e. we need to recover all preimages
inside M. Moreover, the set M is chosen so that it is away from some neighbourhood of
the intervals of V and endpoints of Q. It follows that, in terms of notations in theorem2, the function F is well defined in some uniform neighbourhood of λ0× B, i.e. on the domain of the formb∈BU (λ0, δ0)× U(b, η0)for some fixed positive radii δ0and η0. As a result we will have, instead of (14), the following commutative diagram:
lim ←−(M, ϕ) ¯ψλ −−−−→ λ i → Yλ Tλ −−−−→ ˜Pλ −−−−→ Kπ0 λ σ σ σ σm fλ, lim ←−(M, ϕ) −−−−→¯ψλ λ→ i Yλ −−−−→Tλ ˜ Pλ −−−−→ π0 Kλ (15)
where λ ⊂ Yλis a closed (in the product topology) σ -invariant set, which is a continuation
of B= lim
←−(M, ϕ)for perturbations.
3. Applications
In this section, we apply our perturbation results to several families of maps. We also obtain chaotic structures in stationary and travelling waves in lattice models and consider spatially homogeneous solutions of these systems by checking their local maps6.
3.1. The H´enon map
First let us discuss a simple (and well known) example of the standard H´enon map H (x, y)=
(y, ay(1− y) − bx); it has a corresponding difference equation of the form
xn+2− axn+1(1− xn+1)+ bxn= 0.
In order to apply theorems1and11, one can fix s1 0, s2 1 (and if the inequalities are strict then the ‘exceptional’ set V must contain the intervals [s1,0) and (1, s2]). Here, we give two choices for the exceptional value of λ0 depending on how the values of a and b play the role of a parameter.
First, in the case when b plays the role of the parameter λ with λ0 = 0, it is well known that for small b the standard H´enon map behaves like the one-dimensional logistic map ϕ(x)= ax(1 − x), which is precisely the local map in our notation. In particular, the H´enon map is chaotic for a close to 4, and if a > 4, its nonwandering set is the Smale horseshoe. This agrees with our results. Indeed, for a > 3.569 . . ., we may use theorem1 to assure the chaotic behaviour of orbits for the H´enon map (with b small enough), because
htop(ϕ) > 0. Also, the statement of theorem1on the continuous dependence of perturbed
solutions in the uniform topology assures that the orbits starting in [0, 1]2under H = H
a,bwith
3.569 . . . < a < 4 and small b will stay infinite times in a narrow strip around the parabola
y= ax(1 − x), x ∈ [0, 1]. On the other hand, for a > 4 sufficiently big, in order to establish
a full 2-horseshoe structure we may use theorem11also (see [8] and [13] for more details on generalized H´enon maps).
Next, consider other regions of parameters for the H´enon map. Let a and b be sufficiently big. More precisely, let a and b tend to∞ in such a way that a/b → λ0for some constant
λ0 >0. Then after dividing the difference equation by b, we will have the limit equation of the form
xn− λ0xn+1(1− xn+1)= 0.
Note that this equation corresponds to the value of delay L= −1 in our notation, and it is easy to see that by reversing the time, we have again the same situation, with the roles of x and y interchanged. So in the regions of parameters with a and b sufficiently big, we also have a chaotic structure of orbits when a/b is bigger than 3.569 . . ., and for a/b > 4 we have again the full 2-horseshoe.
3.2. The Arneodo–Coullet–Tresser maps
Consider the family of the so-called ACT maps f :R3 → R3 (due to Arneodo, Coullet and Tresser; refer to [6]); they are of the form
f (x, y, z)= (ax − b(y − z), bx + a(y − z), cx − dxk+ ez),
where a, b, c, d, e∈ R are parameters and k 2. If (a2+b2)e= 0, then f is a diffeomorphism with the inverse
f−1(x, y, z)= ˆx,−bx + ay a2+ b2 +ˆz, ˆz , whereˆx = ax+ by a2+ b2 andˆz = z− c ˆx + d ˆxk e .
If bd = 0, then there are interesting dynamical properties and bifurcations in several regions of the parameter space; see [6]. For an initial point p= (x0, y0, z0), denote the nth iteration of p under f by (xn, yn, zn). Then we have the following difference equation which in fact is an equivalent form for defining the ACT map (see [7,13]):
dxnk+1−a 2e+ b2e b xn+ a2+ b2− bc + 2ae b xn+1− 2a + e b xn+2+ 1 bxn+3= 0.
Now we consider the pair of coefficients (a, e) as the parameter λ, and for the singular value λ0= (0, 0) we have the difference equation in two variables with L = 2:
dxkn+1+ (b− c)xn+1+ 1
bxn+3= 0.
At λ = λ0 = (0, 0) the ACT map fλis not invertible and its Jacobian becomes zero,
while for small nonzero a, e, fλis a diffeomorphism. So we are able to apply theorem1in
order to show chaotic behaviour of fλfor a, e small; namely, it is sufficient to check that the
one-dimensional map
ϕ(x)= (bc − b2)x− bdxk (16)
has positive topological entropy. In [6] and [7], sufficient conditions for the existence of full horseshoes (Bernoulli shifts on two or three symbols depending on the evenness of k) for ACT maps in the above situation, i.e. near degenerate ACT maps with a = e = 0, were obtained. Now our result provides other regions of parameters in which one has a chaotic structure with positive entropy, which need not be full horseshoe. For instance, it applies for small b whenever
Figure 2. The graph of map (16) with k = 4, b = 2, d = 1 and c = 1.5b and the topological entropy as a function of µ, when c= (1 + µ)b varies.
Figure 3. The graph of map (16) with k= 3, b = 2, d = 1 and c = 1.7b and its topological entropy as a function of µ, when c= (1 + µ)b varies.
as a continuation of ‘finer horseshoes’ (actually, they are horseshoes under some iterates of the map, and it is well known that such horseshoes are contained necessarily in the nonwandering set of one-dimensional maps with positive topological entropy). See figures2 and3which show some regions of parameters with positive htop(ϕ); note that since L= 2, the topological entropy of the perturbed ACT maps is bounded below approximately by htop(ϕ)/2.
3.3. Quadratic volume preserving maps
In this subsection, we consider the family of maps f :R3→ R3defined by
f (x, y, z)= (η + αx + βy + z + Q(x, y), x, y), (17)
where η, α, β, γ are real parameters and Q(x, y)= ax2+ bxy + cy2is a quadratic form. As shown by Lomeli and Meiss in [14], generically every quadratic automorphism, i.e. volume preserving diffeomorphism ofR3which has a quadratic inverse, is topologically conjugate to a map (17) with a + b + c= 1 (note that in [14], it is also shown that the parameter β can be
eliminated by an appropriate change of coordinates; however, we do not assume β = 0 because this parameter will be of use for our considerations). It is easily seen that the corresponding difference equation for (17) is of the form
η+ αxn+ βxn−1+ xn−2+ axn2+ bxnxn−1+ cx
2
n−1− xn+1= 0.
By using theorem11, we have proved in [13] that if the parameters η, β, b, c in (17) are fixed while α→ ∞ and a → ∞ in such a way that a/α = constant = 0, then for |a| sufficiently big, the map f = fahas a closed invariant set asuch that fa|a is conjugate to the full shift
on two symbols. (In fact, to have the same conclusion it is sufficient that all the parameters
η, β, b, cin (17) are o(α) while α→ ∞ and a → ∞ in such a way that a/α → constant = 0).
Note that theorem11does not apply to quadratic automorphisms in the generic form with
a+ b + c= 1 because the latter equality implies that both coordinates xnand xn−1should be
involved in the limit difference equation. But now by using theorem1, we are able to get a chaotic structure of quadratic automorphisms in the generic form. Indeed, for the case when
β = 0, let the parameter β tend to ∞, b tend to 0 and a + c tend to 1 with a > 0, c > 0
and a + b + c = 1, while other parameters remain constant; or more generally, let us have (as β → ∞) the following: b = o(1), α = o(β), η = o(β2)and a + c converges to 1 with
a > 0, c > 0 and a + b + c = 1. Then after scaling by β we get for the new coordinates
¯xn= −xn/β, the following difference equation at the limit as β→ ∞ and b → 0:
(1− c) ¯xn2+ c¯xn2−1− ¯xn−1= 0, (18)
whose upper branch is the map
ϕ(x)= 1 4(1− c)c − c 1− c x− 1 2c 2 . (19)
It is a unimodal map (with an upper semi-ellipse for the graph) and, thus, one has a positive topological entropy whenever c is close to 0.8 (and also for c > 0.8, in which case we have the full 2-horseshoe). In figure4, it is shown that the topological entropy of the local map ϕ becomes positive beginning with c≈ 0.791. Therefore, by using theorem1with L= 1, the system in (17) has chaotic dynamics for sufficiently large β and sufficiently small b.
Furthermore, we can consider the case when β = 0 in the generic form of quadratic volume preserving automorphisms. Indeed, there is chaotic dynamics for sufficiently large α and sufficiently small b by using theorem1 with L = −1, because in the new coordinates ¯xn= −xn/α, we have at the limit as α → ∞ and b → 0, the following difference equation
(1− a) ¯xn2−1+ a¯x2n− ¯xn= 0,
which is the same as (18) with only the roles of a and c and also the roles of ¯xn−1and ¯xn
interchanged. So the chaotic dynamics here takes place for a near 0.8.
3.4. The generalized CNN models
We consider the one-dimensional generalized CNNs of the following form, which was introduced by Itoh, Juli´an and Chua [9],
dx
dt = −G(x) + AF(x) + Bu + z, (20)
where
x= (xi)i∈Z, G(x)= (g(xi))i∈Z, A= (a−m, . . . , a0, . . . , am),
F(x)= ((f (xi−m), . . . , f (xi), . . . , f (xi+m))T)i∈Z, B= (b−m, . . . , b0, . . . , bm),
Figure 4. The graph of map (19) with c= 0.795 and its topological entropy as a function of c (which is presumed to be continuous at finer scales of c).
Here, x denotes the state, G(x) denotes the v–i characteristic, F(x) denotes the output, A, B and z denote the feedback, control and threshold template parameters, respectively, and m is the neighbourhood radius of each cell. As usual, the output function F(x) is related through the piecewise-linear saturation function
f (x)=12(|x + 1| − |x − 1|),
and the v–i characteristic function G(x) is related to the piecewise-linear function
g(x)= αx + γ (|x − Vp| − |x − Vv|) − γ (|x + Vp| − |x + Vv|),
where α and γ are constants and Vpand Vvare the peak and valley voltages.
Here, we will allow the output and v–i characteristic functions to have more deviations. One can rewrite the generalized CNN model (20) as a system of state equations:
dxi dt = −g(xi)+ m j=−m ajf (xi+j)+ m −m bjuj + zi.
Let h(x)= g(x) − a0f (x). Then we arrive at the form dxi dt = −h(xi)+ m j=−m,j=0 ajf (xi+j)+ m j=−m bjuj + zi. (21)
Standard CNNs and resonant tunnelling diode (RTD) based CNNs are special cases of the generalized CNN model (21). For example, model (21) with h(x) = x − a0f (x) is the original standard CNN model, with h(x) = x − a0f (x)˜ the modified standard CNN model, with h(x)= g(x) − a0xand f (x)= x the original RTD-based CNN model and with
h(x)= g(x) − ax and f (x) = g(x) − αx − β the modified RTD-based CNN model. Circuit
implementations of these CNNs have been explicitly illustrated in [9].
Here, we assume that the template parameters have a common parameter λ; more precisely, the generalized CNN model (21) becomes of the form
dxi dt = −h(xi)+ λ m j=−m,j=0 ajf (xi+j)+ m j=−m bjuj + zi . (22)
In order to demonstrate the spatial chaos phenomenon, we consider the set of stationary solutions, i.e. solutions which do not depend on ‘time’: xi(t )= xi. Thus letting the right-hand
side of (22) be zero (i.e. the exceptional value λ0= 0), we have the stationary solutions of (22) given by h(xi)− λ m j=−m,j=0 ajf (xi+j)+ m j=−m bjuj + zi = 0.
Thus, by using theorem 11, we have the following result on the chaotic structure of stationary solutions.
Corollary 12. If h has k simple zeros and both f and h are C1 in a neighbourhood U of
these zeros, then for any sufficiently small λ, there exists a closed (in the product topology) σ -invariant sunset λof the set of stationary solutions for (22) such that σ|λis topologically
conjugate to σ|k, the full shift on k symbols and the conjugacy map can be chosen to be
continuous in λ in the uniform topology. 3.5. Time discrete version of the CNN system
One can also consider the time discrete version of the CNN system introduced by Sbitnev and Chua in [24] as follows xin+1= xin− δh(xin)+ δλ m j=−m,j=0 ajf (xin+j)+ m j=−m bjunj+ z n i , (23)
where δ and λ are nonzero constants. Stationary solutions do not depend on the ‘time’ coordinate n, i.e. xn
i = xi for these solutions. In this case, equation (23) is reduced to the
equalities h(xi)+ λ m j=−m,j=0 ajf (xi+j)+ m j=−m bjuj + zi = 0 .
Theorem11implies exactly the same result on the chaotic structure of stationary solutions for (23) as corollary12.
Let p, q ∈ Z with (p, q) = 1. A travelling wave solution with velocity q/p of equation (23) is a solution of the form
xin= xpi+qn, where y :Z → Rkand write y = y( ).
Let r= pi + qn + q be the travelling wave coordinate, i.e. set xin+1= yr= ypi+qn+q. Then the
travelling wave solutions are given by
yr = yr−q − δh(yr−q)+ δλ m j=−m,j=0 ajf (ynr−q−pj)+ m j=−m bjur−q−pi+pj+ zr−q . Spatially homogeneous solutions do not depend on the ‘space’ coordinates i, i.e. xn
i = ynfor
these solutions. In this case, equation (23) is reduced to the equalities:
yn+1= yn− δh(yn)+ δλ m j=−m,j=0 ajf (yn)+ m j=−m bjun+ zn . Thus, by theorem1, we have the following result.
Corollary 13. Suppose that the function x → x − δh(x) from Q = [s1, s2]\V to [s1, s2],
for some s1 < s2 and V ⊂ [s1, s2] open, is piecewise analytic and has positive topological
entropy. If f is C1on Q, then for any sufficiently small λ, there exists a subset λof travelling
wave solutions (or spatially homogeneous solutions) for (23) such that λis invariant under the spatial translation σ and htop(σ|λ) >0.
3.6. Steady state of Chua’s circuit
The equations for the coupled Chua’s circuit in [21] are ˙ xi = αyi− α(xi+ g(xi))+ d(xi+1− 2xi+ xi−1), ˙ yi = xi− yi+ zi, ˙zi = −βyi− γ zi, (24)
where α, β, γ , d are positive parameters and g(x)= m1x+m0−m1
2 (|x+1|−|x−1|). Considering the stationary solutions, equation (24) yields
− αβ
γ+ βxi− αg(xi)+ d(xi+1− 2xi+ xi−1)= 0. From theorem11, we have the following corollary.
Corollary 14. If the function x → γβ+βx + g(x) has k simple zeros and g is C1 in a neighbourhood U of these zeros, then for any sufficiently small d, there exists a closed (in the product topology) σ -invariant sunset d of the set of stationary solutions for (24) such that σ|dis topologically conjugate to σ|kand the conjugacy map is continuous in d in the uniform
topology.
3.7. Lattice models of an evolution equation
Finally, we consider solutions in lattice models of an evolution equation of which some motions can be described by discrete versions of reaction–diffusion equations (for lattice models and their chaotic and stability properties see [1,2,3,22]). Consider the lattice models of an evolution equation of the form
utn+1= f (utn)+ g(utn−s, utn−s+1, . . . , utn+s), (25) where t ∈ Z is the time variable, n ∈ Z is the space one and 0 usually stands for the diffusion coefficient. The function f is called the local map and g is called the interaction of finite size s.
If we look for the steady state (or stationary) solutions ut
n of (25), then utn must be
independent of the time coordinate t, i.e. ut
n:= xnfor all t ∈ Z. In this case, equation (25)
can be reduced to the difference equation
xn= f (xn)+ g(xn−s, xn−s+1, . . . , xn+s), n∈ Z.
Thus, by using theorem11, we have the following result on the chaotic structure of steady state solutions.
Corollary 15. Let the function x → x − f (x) have k simple zeros and be of class C1 in a neighbourhood U of these zeros and let g be of class C1in U2s+1. Then for sufficiently small , there exists a closed (in the product topology) σ -invariant subset of the set of steady state solutions for (25) such that σ| is topologically conjugate to σ|k and the conjugacy map is
continuous in in the uniform topology.
Let p, q ∈ Z with (p, q) = 1. A travelling wave solution with velocity q/p of equation (25) is a solution of the form
unm= xpm+qn, where x :Z → Rkand write x = x( ).
Let i= pm + qn + q be the travelling wave coordinate, i.e. set un+1
m = xi = xpm+qn+q. Then
we obtain the following equation for xi:
