Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations

We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDSs with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive.


Introduction
We demonstrate that if an evolutionary system of partial differential equations (PDE) in unbounded domain has a solution localized in space and chaotic in time, then one should expect both temporal and spatial chaotic behavior in the system. Namely, one may observe a formation of non-trivial spatial patterns that evolve in an irregular fashion with time, and the corresponding space-time entropy [CoEc99,Zel07] is strictly positive. In other words, the number of solutions which are essentially different from each other on a finite space-time window grows exponentially with the window volume.
As a tool for finding the spatially-localized, temporally-chaotic solutions one may try, as we do it here, to look for special types of both spatially and temporally localized solutions. Thus, like Shilnikov homoclinic loop and Lorenz butterfly serve as a criterion for chaos formation in systems of ODE's [Shi65,Shi70,ShST,TuSh], the existence of the Shilnikov homoclinic loop in the dynamical system generated by the PDE on the space of spatially localized solutions implies the space-time chaos in the extended system that corresponds to uniformly bounded solutions of the same PDE.
We do not prove this principle in full generality here. Instead, we decided to show how it works for a class of Ginzburg-Landau equations with a broken phase symmetry. The main motivation for such approach is that despite a huge amount of numerical and experimental data on different types of space-time irregular behavior in various systems, there are very few rigorous mathematical results on this topic and mathematically relevant models describing these phenomena. Therefore, we made an effort of providing a free from numerics, completely analytic proof of the existence of space-time chaos in an important equation of mathematical physics. The basic mathematical model for the space-time chaotic behavior is the so-called Sinai-Bunimovich chaos in discrete lattice dynamics, see [AfrFe00,BuSi88,PeSi88,PeSi91]. This model consists of a Z n -grid of discrete-time chaotic oscillators coupled by a weak interaction. The single chaotic oscillator of this grid is described, say, by the Bernoulli scheme M 1 := {0, 1} Z , so the uncoupled system naturally has an infinite-dimensional hyperbolic set homeomorphic to the multi-dimensional Bernoulli scheme M n+1 := {0, 1} Z n+1 = (M 1 ) Z n . The temporal evolution operator is then conjugate to the shift in M n+1 along the first coordinate and the other n coordinate shifts are associated with the spatial translations on the grid. Due to the structural stability of hyperbolic sets, the above structure survives under a sufficiently weak coupling. Thus, in this model, the space-time chaos is described by the multi-dimensional Bernoulli scheme M n+1 . Importantly, the space-time entropy in the Sinai-Bunimovich model is strictly positive. We know from the general theory of dissipative systems in unbounded domains (see e.g. [CoEc99,MirZe08,Zel03,Zel04,Zel07]) that under some reasonable dissipativity assumptions this entropy is finite for systems of evolutionary PDE's, therefore the Sinai-Bunimovich model carries "enough complexity" to be able to capture certain basic features of spatio-temporal chaos in systems of various nature. In particular, it is well established by now (see e.g. [Pom86,Man09,BaT05]) that the transition from regular to chaotic space-time behavior often happens via the emergence of well spatially separated and long living "turbulent spots". As the interaction between such spots seems to be weak, the Sinai-Bunimovich chaos paradigm can be relevant for the analysis of these near-threshold phenomena.
Yet, a direct application of the Sinai-Bunimovich construction to systems with continuous time and space is not possible, in general. Even the existence of one PDE which possesses an infinite-dimensional Bernoulli scheme was a long-standing open problem.
The first examples of such PDEs (in the class of reaction-diffusion systems), have been recently constructed in [MieZe07]. The method used in that paper is based on a strong and explicit spatio-temporal modulation of the equation right-hand sides, which effectively transforms the systems into a discrete-time lattice dynamical system. The disadvantage is that very special (and artificial) nonlinear interaction functions emerge in the result, which are far from the usual nonlinearities arising in physics models.
A different approach to the problem is suggested in [MieZe09], where a theory of weak interaction of dissipative solitons was developed and, as an application, a space-time chaotic pattern has been constructed for the perturbed 1D Swift-Hohenberg equation (1.1) ∂ t u + (∂ 2 x + 1) 2 u + β 2 u + f (u) = µh(t, x, µ), f (u) = u 3 + κu 2 , µ ≪ 1. Here µh(t, x, µ) is a space-time periodic forcing. Its exact form is quite non-trivial, however the amplitude µ can be taken arbitrarily small. The idea is to create a spatially localized spot of chaotic temporal behavior, and to build then a grid of such spots, well separated in space. The spots are pinned down to the prescribed locations at the grid points by spatial oscillations in the forcing µh. If the spots stay sufficiently far apart, their interaction is small, so a small amplitude forcing occurs to be sufficient to sustain the grid for all times (the wave length of the forcing has, however, to grow as the amplitude decreases).
Equation (1.1) at µ = 0, like many other important equations, does have a spatially localized solution, a soliton, u = U(x) with exponentially decaying tails. One may therefore look, at all small µ, for multi-soliton solutions in the form u(t, x) = j U(x − ξ j (t)) + "small corrections", where ξ j (t) is the position of the j-th soliton; the well-separation condition reads as L := inf j =k ξ j −ξ k ≫ 1. Due to the "tail" interaction and the small forcing, the solitons' positions ξ j (t) may move slowly, and this motion is described by a lattice dynamical system (LDS), see [MieZe09] for details. The obtained LDS is not in the form one needs for establishing the Sinai-Bunimovich chaos (a grid of chaotic maps with weak coupling), since the individual solitons u = U(x) are equilibria at µ = 0 and do not have their own (chaotic) dynamics. However, as it is shown in [MieZe09], a pair of weakly interacting solitons in the 1D Swift-Hohenberg equation can be forced to oscillate chaotically in time by an appropriate choice of the time-periodic perturbation µh(t, x, µ). For a wellseparated grid of such soliton pairs, one obtains a time-periodic LDS, and the period map for this system is the sought discrete lattice of weakly coupled chaotic maps, i.e the space-time chaos is established.
The scope of [MieZe09] is much more general than the Swift-Hohenberg equation: by developing the center manifold approach proposed in [San02], the paper derives the LDS that governs the evolution of weakly coupled multi-soliton configurations for a large class of systems of evolutionary PDE's. It also proposes a method for constructing spatially localized and temporally chaotic solutions which are obtained as a system of finitely many weakly coupled stationary solitons. Note that, although spatially localized solutions with non-trivial temporal dynamics have been observed numerically and experimentally in various physical systems (see e.g. [AkSCT01,BlWe02,TuVZ07] and references therein), the direct analytic detection and study of such solutions is obviously a very difficult task. However, when a finite system of well-separated solitons is considered, the descriprion provided by [MieZe09] for the evolution of such object is often a low-dimensional system of ODE's which can exhibit a chaotic dynamics [TuVZ07] and can be studied analytically, so the chaotic temporal behavior of such localized patterns can be rigorously proven.
In the present paper we show how a space-time chaotic lattice can be built out of these chaotic multi-soliton systems in the case where no spatial nor temporal modulation is imposed. Two problems immediately appear in this setting: 1. with no external forcing, the LDS which describes the multi-soliton dynamics is an autonomous system with continuous time, so the Sinai-Bunimovich chaos construction (for which the discreteness of time is very essential) is not applicable; 2. with no spatial modulation, there is no pinning mechanism which would keep the solitons eternally close to any given spatial grid, therefore the infinite-time validity of the LDS description is no longer guaranteed.
We resolve here both the issues. As an application, we consider the 1D quintic complex Ginzburg-Landau equation with slightly broken phase symmetry: where β, δ, ρ, ε 1,2 , µ are some real parameters, and µ ≪ 1. We mention that, in contrast to the previous on the Swift-Hohenberg equation, we do not have here any artificial functions, and the only freedom we have is the choice of the numeric parameters. Note also that the Ginzburg-Landau equation serves as a normal form near an onset of instability, i.e. it very often appears in applications as a modulation equation for various more complicated problems. The phase symmetry in the modulation equation appears as an artefact of closeness to the instability threshold, so if there is no such symmetry in the original problem, then the effects of small symmetry breaking also need to be considered, see [Mie02] and references therein. While we introduce only the simplest symmetry breaking term ("+µ") in (1.2), the general case is also covered by the theory (see Section 2).
The main result of the paper is the following theorem (Section 3).
Equation (1.2) at µ = 0 has the additional phase symmetry u → e iφ u. Therefore, for each stationary soliton u = V (x) of this equation, u = e iφ V (x) is also a stationary soliton. Therefore, the multi-soliton configurations are given by where ξ j and φ j are the coordinate and phase of the j-th soliton. For a soliton pair with the states (ξ 1 , φ 1 ) and (ξ 2 , φ 2 ), the evolution is governed, to the leading order with respect to the distance |ξ 2 − ξ 1 |, by the following system of ODE's: TuVZ07,MieZe09]. Here τ is a scaled slow time, R := (ξ 2 − ξ 1 )/2, Φ := φ 1 − φ 2 , Ψ := (φ 1 + φ 2 )/2 and a, b, ω, θ 1,2 , c, ν and Ω are parameters whose exact values depend on the values of the original parameters of (1.2) (see the corresponding expressions, as well as asymptotic expansions near the exactly solvable nonlinear Schrödinger equation, in Sections 2,3). While the variables R, Φ and Ψ can be treated as the "internal variables" of the two-soliton pattern, the variable p := (ξ 1 + ξ 2 )/2 marks the spatial position of the soliton pair. To the leading order, it is governed by the equation A numerical study of system (1.3) undertaken in [TuVZ07] revealed various chaotic regimes for different parameter values. In order to provide an analytic proof (see Lemma 2.3) of the chaotic behavior (i.e. the existence of a nontrivial hyperbolic invariant set) in this system for an open set of parameter values, we have found a degenerate equilibrium of that system with 3 zero eigenvalues. The normal form equation for this bifurcation (see [ArCST85]) leads to the following 3rd order equation: where E (a certain combination of the parameters of the original system) can take any real values. In [IbRo05], the existence of a Shilnikov homoclinic loop for this system was proven at certain E values, which implies [Shi65,Shi70] chaos for some interval of the E values and, hence, for an open set of parameter values for system (1.3). Chaotic solutions of system (1.3) correspond to a chaotically oscillating soliton pair, which is a temporally chaotic and spatially localised solution, by construction. After that, according to the program described above, we build a well spatially separated lattice of such time-chaotic solitons. The center manifold reduction theorem proved in [MieZe09] ensures that the evolution of this lattice is governed by a system of infinitely many weakly coupled copies of the ODE's (1.3),(1.4).
Even when every individual ODE-subsystem in the continuous time LDS is hyperbolic, the LDS itself is not hyperbolic (this is a principal difference with the Sinai-Bunimovich chaos in the discrete-time LDS's where the countable product of hyperbolic sets for the individual maps is hyperbolic again). Each constituent ODE contributes a neutral direction corresponding to the time shift, so for the linearized flow of the continuous time LDS we have infinitely many neutral directions. Therefore, after a weak coupling is switched on, the dynamics is not preserved (the LDS can hardly be topologically conjugate to the uncoupled one). Still, the invariant manifold theorem of Section 4 shows that if, given any orbit of the uncoupled LDS, we consider the family of all orbits obtained by all possible time-reparametrizations in each of the constituent ODE's, then this family continues in a unique way as an invariant manifold of the weakly coupled LDS. This fact allows to show the strict positivity of space-time topological entropy for the countable systems of weakly coupled chaotic oscillators with continuous time.
In fact, results of Section 4 cover LDS's of a more general type. The problem we have to deal with is that, although system (1.3) for the internal variables (R, Φ, Ψ) of the chaotic soliton does have a uniformly hyperbolic set, the full system describing the motion of the chaotic soliton includes equation (1.4) for the soliton position p, and is clearly non-hyperbolic (so we have to consider the LDS's built of partially-hyperbolic individual ODE's). The neutral directions appear because the right-hand sides of (1.3),(1.4) are p-independent, which is a mere consequence of the translational symmetry of the PDE under consideration, i.e. their presence is an inherent property of the soliton-interaction equations in systems without a spatial modulation.
Since the internal variables of the soliton change chaotically with time, the soliton position p(t) performs, essentially, an unbounded random walk (as an integral of a chaotic input, see (1.4)). When the chaotic solitons are well spatially separated, the contribution of the neighboring solitons to the p-equation is small, so we have essentially independent random walks for each of the chaotic solitons in the lattice. This makes it impossible for us to ensure that the well-separation condition is fulfilled for all times and all initial multi-soliton configurations. We, in fact, believe that the majority of these configurations do break up this condition in a finite time, so the corresponding solutions cannot be completely described by the weak soliton interaction paradigm.
However, the weak soliton interaction theory of [MieZe09] is the only tool we have here for the analysis of dynamics of the multi-soliton patterns. As we are unable to control the soliton's random walk, we devise a method of keeping track of those configurations for which the well-separation condition holds eternally (i.e. the LDS description is applicable). This method allows us to verify (Section 6) that the number of such solutions is large enough to ensure the positivity of the space-time entropy. It is worth to emphasize that, instead of fighting with the random walks, our method exploits them in a crucial way.
Roughly speaking, assume that the hyperbolic set for (1.3) contains two periodic orbits Γ 1 and Γ 2 and a number of heteroclinics which connect them. Assume that, according to equation (1.4), the soliton pair moves to the left if (R, Φ, Ψ) belongs to Γ 1 and to the right when it belong to Γ 2 . The direction of this motion is determined by the sign of is enough, as we show). Then, our orbit selection method works as follows: assume that initially the j-th soliton is in the interval [L − j , L + j ] with L + j − L − j large enough; then until it remains in that interval, we allow the internal state (R, Φ, Ψ) of the soliton to jump randomly between Γ 1 and Γ 2 along the heteroclinic orbits (thus we gain the complexity which is enough to have the positive entropy); however, when the soliton reaches the bound (say, L + j ), we stop allowing jumps and consider only orbits that stay near Γ 1 until the soliton position p j (t) arrives close to (L + i + L − i )/2 (when the bound L − j is achieved, the orbit must stay near Γ 2 ); after p j (t) is driven to the middle of the interval, the random motion is allowed again, and so on.
We proved in Section 6 that the above described procedure can be implemented simultaneously for all chaotic solitons on the grid, and it allows indeed for a selection of a set of spatially non-walking solitons with positive space-time entropy. In order to do this, we need a further development of the theorem on normally-hyperbolic manifolds in the countable product of partially hyperbolic sets which is proved in Section 4; namely we prove certain "asymptotic phase" results in Section 5.
As the above discussion shows, the theory we build is readily applicable to any dissipative PDE for which the weak soliton interaction system for some finite multi-soliton configuration exhibits a chaotic dynamics. In analogy to the finite-dimensional case, we are now able to analyze localized structures and effectively use them for the understanding of space-time dynamics generated by PDEs.
We are grateful to A.Mielke and A.Vladimirov for useful discussions, and to WIAS (Berlin) and BenGurion University for the hospitality.

Space-time chaos in complex Ginzburg-Landau equation with broken phase symmetry
Consider the one-dimensional complex Ginzburg-Landau equation where u = u 1 + iu 2 is an unknown function of x ∈ R and t ∈ R, the function H : R → C is smooth, H(0) = 0, and parameters β, δ are real; the symmetry-breaking parameter µ is assumed to be small, and the function G is smooth.
Let K be a set of solutions of (2.1) which are defined and uniformly bounded for all (t, x) ∈ R 2 (under certain standard dissipativity assumptions, equation (2.1) will have global attractor; in this case one can choose as the set K the set of all solutions that lie in the attractor, see more after Theorem 2.1). The complexity of spatio-temporal behavior of the solutions can be characterized by the space-time topological entropy defined as where K |t|≤T,|x|≤R stands for the set of functions from K restricted on the space-time window {|t| ≤ T, |x| ≤ R}, and h ε denotes the Kolmogorov ε-entropy of this set, i.e. the logarithm of a minimal number of ε-balls in the space which are necessary to cover 1 the set; see [KoTi61]. It is well-known (see, e.g., [CoEc99,MirZe08,Zel04,Zel07]) that the space-time topological entropy h s−t (K) is well-defined and finite in our case. Thus, if h s−t (K) is strictly positive for some set K, then the number of various spatio-temporal patterns that are supported by the equation grows exponentially with the volume of the space-time window. In our construction of spatio-temporal chaos we assume that the nonlinearity H is such that for some β = β 0 and δ = δ 0 the Ginzburg-Landau equation (2.1) possesses at µ = 0 a stationary, spatially localized solution u = U(x): (1 + iβ 0 )∂ 2 x U − (1 + iδ 0 )U − UH(|U| 2 ) = 0; for the existence results see [AfMi99,AfMi01] and references therein, and Theorem 3.1.
Equation (2.1) is invariant with respect to spatial translations x → x − ξ and, at µ = 0, with respect to phase shifts u → e iφ u. So, along with the given soliton U(x), equation (2.1) possesses at µ = 0 a family of stationary solitons: Because of the symmetry with respect to x → −x, along with the soliton u = U(x), equation (2.3) also has a localized solution u = U(−x). Equation (2.3) is an ODE with 4-dimensional phase space. A localized solution corresponds to a homoclinic intersection of the stable and unstable manifolds of the zero equilibrium of this system. Since these manifolds are 2-dimensional and family (2.4) is 2-parametric, all the localized solutions of (2.3) are contained in family (2.4). Thus, U(−x) ≡ e iφ 0 U(x − ξ 0 ) for some φ 0 , ξ 0 , which immediately implies that U ξ/2,0 (−x) = ±U ξ/2,0 (x). In other words, we may from the very beginning assume that our soliton is chosen such that it is either symmetric: or antisymmetric (U(−x) ≡ −U(x)). In this paper we consider the symmetric case, i.e. we assume that (2.5) holds (in the antisymmetric case the soliton interaction equations are different; however one can show that a small perturbation of an equation with antisymmetric soliton creates symmetric solitons -cf. [AfMi99,AfMi01], so the results of our paper can be applied in this way). Since every function in (2.4) is a stationary solution of (2.3) at µ = 0, it follows that the functions ϕ 1 := −∂ ξ U ξ,0 ξ=0 = ∂ x U and ϕ 2 := ∂ φ U 0,φ φ=0 = iU belong to the kernel of the linearization L U of (2.3) at U: L U ϕ 1,2 = 0, where We assume that the soliton U is non-degenerate in the sense that the rest of the spectrum of L U is bounded away from the imaginary axis; e.g. the algebraic multiplicity of the zero eigenvalue is two (note that since U(x) → 0 as x → ±∞, the operator L U is a compact perturbation of the operator ϕ → (1 + iβ 0 )∂ 2 x ϕ − (1 + iδ 0 )ϕ, so the essential spectrum is bounded away from the imaginary axis; however, one should check that the eigenvalues stay away from the imaginary axis as well).
Theorem 2.1. Let, along with (2.14),(2.15), the following conditions be satisfied for a non-degenerate, symmetric stationary soliton U(x): Then, arbitrarily close to µ = 0 and δ = δ 0 there exist an interval of values of µ and an interval of values of δ such that the corresponding equation (2.1) has a uniformly bounded set of globally defined solutions with strictly positive space-time entropy.
Proof. Each of the solutions of equation (2.1) that belong to the large (of positive entropy) set we are going to construct can be viewed as a slowly evolving multi-soliton configuration. Namely, we choose a sufficiently large L and consider solutions u(x, t) which for every t ∈ R stay close, in the space C b (R) of bounded continuous functions of x, to the multi-soliton manifold M L defined as the set of all functions u(x) of the form For sufficiently large L, the multi-soliton manifold is indeed an infinite-dimensional submanifold of C b (R) which is parameterized by the sequences m := {ξ j , φ j } of the soliton positions and phases (see [MieZe09]). The boundary ∂M L is given by inf j∈Z (ξ j+1 − ξ j ) = 2L. We will seek for solutions of equation (2.1) in the form u(t) := u m(t) + w(t) where m(t) is a slow trajectory in M L and w(t) is a small corrector. Recall a result from [MieZe09].
Theorem 2.2. For all L large enough there exists a C k -map S : where α > 0 is defined by (2.9)) and that the manifold S := {u = u m + S(u m ), m ∈ M L } is invariant with respect to equation (2.1). Namely, there exists a C k -vector field F on M L such that given any solution of solves equation (2.1).
Moreover, system (2.22) has the following form: where α and ω are the same as in (2.9), the constants r, s, q are defined by (2.10), the function F is defined by (2.12), and the dots stand for terms which are The complete proof of formulas (2.24) occupies a substantial part of [MieZe09]. For reader convenience, we provide a brief heuristic derivation of the equations in Appendix.
According to Theorem 2.2, the evolution of well-separated multi-soliton configurations in the driven Ginzburg-Landau equation is governed by system (2.24). Therefore, in order to prove the positivity of space-time entropy in equation (2.1), it is enough to find a large set of solutions of system (2.24) which satisfy the separation condition (2.20). 2 The corresponding theory for a class of lattice dynamical systems which includes system (2.24) is built in Sections 4-6. In particular, Theorem 6.1 gives a general result on the existence of a setK of non-walking trajectories of a lattice dynamical system such that h s−t (K) > 0. In what follows we will show that a certain subsystem of (2.24) indeed satisfies conditions of Theorem 6.1.
For any v and any sufficiently large L we may define a sequence L n , n ∈ Z, as follows: We will look for pulse configurations which satisfy ξ where the constant C is independent of L, j and t. In other words, we have a grid of weakly interacting pulse pairs with the distance between the pulses in the pair of order 2L and the distance between pairs of order 4L. Assumption (2.26) then means that we should ensure that this structure is preserved for all t although a uniform spatial drift of the whole grid is allowed (ve −αL is the velocity of the drift).
Further, we introduce the scaling τ := te −2αL , Ω := (δ − δ 0 )e 2αL , ν := µe 2αL . We will consider a region of bounded Ω and ν, which corresponds to µ and δ − δ 0 of order O(e −αL ). We also assume L = πn ω , n ∈ N. Equations (2.24) recast as follows (see (2.11)): As we see, only interaction inside the soliton pairs gives a contribution into the leading terms of equations (2.27): since the distance between pairs is, in our configuration, of order 4L, the leading term for the interaction between solitons from different pairs will be of order O(e −4αL ) in the non-rescaled time t, so after the time rescaling it is of order O(e −2αL ), i.e. it is absorbed in the O(e −αL )-terms in (2.27). Let us rewrite the system in the coordinates R j , Φ j , Ψ j := (φ 2j+1 + φ 2j+2 )/2, and p j := (η 2j+1 + η 2j+2 )/2 (i.e. p j is the center of the soliton pair, R j is the distance between the solitons in the pair, Φ j and Ψ j describe the soliton phases). We obtain At large L this system is a lattice dynamical system of form (4.9): at L = +∞ the subsystems that correspond to different j are independent and identical, and the equations for variables y j := (R j , Φ j , Ψ j ) (equations (2.29)) are independent of the p-equation (2.28). Therefore, in order to prove Theorem 2.1, it is enough to check that system (2.28),(2.29) satisfies conditions of Theorem 6.1 at some v. According to that theorem, we will then obtain, for all sufficiently large L, the existence of a setK of solutions of system (2.28),(2.29) which has a positive space-time entropy and is uniformly bounded by a constant independent of L (i.e. condition (2.26) is fulfilled -this, in turn, ensures that the separation condition (2.20) holds, with somewhat smaller L, for all the solutions from K). Now, lifting the setK by formula (2.23), we obtain a uniformly bounded set K of globally defined solutions of the perturbed Ginzburg-Landau equation, and the positivity of the space-time entropy of the set K follows from the smallness of S and the positivity of the space-time entropy ofK. Thus, to finish the proof we need the following behaves chaotically, i.e. it has a basic (=non-trivial, uniformly-hyperbolic, compact, locally-maximal, transitive, invariant) set Λ. Moreover, in Λ one can find two periodic orbits, y = y − (τ ) and y = y + (τ ), of periods T − and T + , respectively, such that where g(y) := v + a 2 e −αR cos(ωR + θ 1 ) sin(Φ). One may check that condition (2.31) implies that (2.32) for an appropriately chosen v. Hence, the lemma indeed establishes the required fulfilment of conditions of Theorem 6.1: chaotic system (2.30) coincides with the y-subsystem (2.29) at L = +∞ (for every j), and condition (2.32) coincides with condition (6.22) (the function g is the right-hand side of the p-equation (2.28)). So, it remains to prove the lemma.
We note that numerically the existence of chaos in system (2.30) with F = c cos φ as well as different scenarios of its emergence for various parameter values were established in [TuVZ07]. In our analytic proof of chaotic behavior we use one of the scenarios mentioned in [TuVZ07]. Namely, we find an equilibrium of system (2.30) with 3 zero characteristic eigenvalues. It is known [ArCST85,IbRo05] that bifurcations of such equilibrium lead to a Shilnikov saddle-focus homoclinic loop, hence to chaos.
Proof of Lemma 2.3.
Scale the parameters as follows: (2.39) for a sufficiently small s, and for some bounded E. By scaling the time and the variables: we bring system (2.38) to the form (where ′ denotes the differentiation with respect to the new, slow time σ).
The limit equation has two hyperbolic equilibria: O + : Y = 1, with a one-dimensional stable manifold W s + and a two-dimensional unstable manifold W u + , and O − : Y = −1, with a twodimensional stable manifold W s − and a one-dimensional unstable manifold W u − . At E < 3 these equilibria are saddle-foci, i.e. each of them has a pair of complex characteristic exponents. By [KuTs76], equation (2.41) has, at which connects the saddle-focus O − with O + . This solution corresponds to a curve Γ −+ along which the one-dimensional manifolds W u − and W s + coincide. By [IbRo05], at the same E there exists another heteroclinic curve, Γ +− , which corresponds to a transverse intersection of the two-dimensional manifolds W u + and W s − . By the transversality, the heteroclinicnic orbit Γ +− persists for all E close to E * . The other heteroclinic orbit, Γ −+ , splits as E varies, and this results [Byk99,Byk00] in the sequence of values E k → E * which correspond to the existence of homoclinic loops to the saddle-foci O + and O − (equation (2.41) is time-reversible, so homoclinic loops to the both saddle-foci appear simultaneously). One can view the one-parameter family (2.41) as a smooth curve in the space of smooth flows in R 3 ; then the parameter values E k correspond to the intersections of this curve with smooth codimension-one surfaces filled by systems with a homoclinic loop to, say, the saddle-focus O − . Importantly, these intersections are transverse. Therefore, fixing any arbitrarily large k, we will have at some E close to E k a homoclinic loop to a saddle-focus close to O − , for any one-parameter family which is sufficiently close to (2.41).
To finish the proof we need to show that the set Λ can be chosen in such a way that it will contain a pair of periodic orbits for which (2.31) is satisfied. According to Remark 6.6, it is enough to check that the integral of the function g − g| O − along the homoclinic loop to the saddle-focus O − is non-zero. In order to verify this condition, let us rewrite the function g in the new variables (Y, Y ′ , Y ′′ ): . Let Y = Y (σ; k, s) be the solution of (2.40) that corresponds to the homoclinic loop Γ k s ; note that As k → +∞, the homoclinic loops of equation spends at large k a large time in a neighborhood of the other equilibrium, O + : {Y = +1}. Therefore, the integral of (Y (σ; k, 0) + 1) tends to +∞ as k → +∞. Thus, for sufficiently large k and sufficiently small s, By Remark 6.6, this proves the lemma, which finishes the proof of the theorem as well.
The proof of the following proposition is standard, cf. [Zel03].
Proposition 2.4. Let the non-linearity H satisfy for some constant C independent of z. Then for all sufficiently small µ equation (2.1) is well-posed in the space C b (R) of uniformly bounded continuous functions and generates a dissipative semigroup S(t) t≥0 in C b (R), and this semigroup possesses a global attractor A.
The attractor is defined here as follows. Let S(t), t ≥ 0, be a semigroup acting on the 3) as t → ∞, the set A attracts, in the topology of C loc (R), the images of all bounded subsets B ⊂ C b (R), i.e. for every neighborhood O of A in the local topology and for every bounded Remark 2.5. It is well-known (see e.g. [MirZe08,Zel04]) that, in contrast to the case of bounded domains, the global attractor is usually not compact in C b (R) if the underlying domain is unbounded. However, attractor's restrictions to every bounded subdomain remain compact. The attraction property itself holds, too, in this local topology only.
A characteristic property of the global attractor is that it consists of all initial conditions which give rise to globally defined solutions. Namely, a function u 0 (x) ∈ C b (R) belongs to the attractor if and only if there exists a function u(t, x) ∈ K such that u 0 (x) ≡ u(0, x). Note that due to the invariance of the equation with respect to temporal and spatial translations, the boundedness and local compactness of the attractor mean also that the set K of the solutions which are defined and bounded for all (t, x) ∈ R 2 is bounded in C b (R 2 ) and compact in C loc (R 2 ).
Thus, we may define the space-time entropy of the attractor as the space-time entropy of the set K: h s−t (A) := h s−t (K) (see (2.2); more discussion and a comparison with other definitions can be found e.g. in [MirZe08,Zel04]). As we mentioned (see [CoEc99,Zel07]), the space-time entropy of the attractor of the Ginzburg-Landau equation is finite: The next Section gives an explicit example of a scientifically relevant equation with 3. Attractor of positive space-time entropy in a perturbed nonlinear Shrödinger equation Here we prove the following Theorem 3.1. Given any sufficiently large β, there exist (continuously depending on β) intervals of values of δ, ρ, ε 1 > 0, ε 2 and µ such that the attractor of the equation Proof. The global attractor of equation (3.1) exists at ε 1 > 0 according to Proposition 2.4. By theorem 2.1, in order to prove (2.45) it is enough to show that the equation has a non-degenerate symmetric localized solution at some δ that depends on the other parameters β, ρ 1,2 , ε 1,2 , and that conditions (2.14),(2.17),(2.18) are satisfied at γ = 0. The localized solution of the ODE (3.2) corresponds to an intersection of the two-dimensional stable and unstable manifolds of the hyperbolic equilibrium at U = 0. Because of the phase-shift symmetry, when these manifolds intersect they coincide. The soliton nondegeneracy conditions imply (among other things) that as δ changes the manifolds split with a non-zero velocity. It follows that a non-degenerate soliton will persist at small perturbation of the nonlinearity, provided a small adjustment to the value of δ is made (see more in [AfMi99,AfMi01]). Thus, it is enough to consider the cubic equation (1 + iβ)∂ 2 x U − (1 + iδ)U + (i + ρ)|U| 2 U = 0; once the existence of a non-degenerate soliton is established for this equation, it can be carried on to the equation (3.2) for all sufficiently small ε 1,2 , and since conditions (2.14),(2.17),(2.18) are open, they will persist as well.
, we obtain the following equation: The linearization operator L is given by (the odd and, respectively, the even one) belong to the kernel of L.
We introduce a scalar product as (ϕ, ψ) = Re +∞ −∞ ϕ(x)ψ(x)dx, so the conjugate to (3.7) operator is As L has two zero modes, one even and one odd, the same holds true for the conjugate operator L † . At w = B = 0 the equation for zero eigenfunctions of L † reads as It is easy to see that the odd and even localized solutions of (3.10) are We will look for asymptotic expansions of these solutions at small w and B. By (3.9), the localized zero modes of L † satisfy (3.14) (we take into account that U * depends on w as well: by (3.6),(3.11)Ū 2 * = Γ 2 (1−2iw ln Γ+ O(w 2 )), while |U * | 2 = Γ 2 ). By (3.14), we have By (3.15),(3.16),(3.12) the two sought localized solutions of (3.14) are given by where S and Q are real, decaying to zero, as x → ±∞, functions which satisfy To find S(x), we multiply (3.18) to Γ(x). The equation will take the form (see (3.12)): By integrating this equation with respect to x, we find (there is no integration constant in the right-hand side, since both S and Γ tend to zero as x → ±∞). By solving the first-order equation, we finally obtain Similarly, by multiplying (3.19) to Γ ′ (x) and integrating the obtained equation, we find, with the use of (3.12), that It is immediately seen that functions S and Q given by (3.20),(3.21) are localized indeed. Moreover, S is odd and Q is even, so by plugging (3.20) and (3.21) in (3.17), we obtain the odd (ψ 1 ) and even (ψ 2 ) zero eigenfunctions of L † . One can also compute (see (3.8)) that As we see, these inner products are non-zero for the values of B and w that we consider here (small B, w such that B > 0 and B > 2w 1−w 2 ). This shows that there are no adjoint functions to the eigenfunctions (3.8). The absence (at small B, w) of eigenvalues on the imaginary axis follows from [KaSa98]. Thus, the pulse U = U * (x) is non-degenerate.

Normally-hyperbolic manifolds for lattice dynamical systems
In this and the the next Sections we study a class of lattice dynamical systems which includes systems describing weak interaction of solitons localized in space and chaotic in time, e.g. system (2.28),(2.29). We start with a skew-product system of ODE's where f, g are C r , r ≥ 1. We assume that y ∈ R n , p ∈ R m ; for more clarity we will denote the space of y variables as Y and the space of p variables as P . We will further assume that the y-part of our system: possesses a bounded, uniformly-hyperbolic invariant set Λ.
Recall that the hyperbolicity means that for every point of Λ there are two subspaces, N s (y) and N u (y), such that the following holds: 1) N s (y) and N u (y) depend continuously on y ∈ Λ, 2) the direct sum of N s (y), N u (y) and N c (y) := Span(ẏ) := {λf (y)|λ ∈ R} constitutes the whole of R n , 3) given any orbit y(t) from Λ, each of the families of subspaces N s (y(t)) and N u (y(t)) is invariant with respect to the flow of system (4.2) linearized about the orbit y(t), 4) the linearized flow is exponentially contracting in restriction onto N s (y(t)) as t → +∞ and in restriction onto N u (y(t)) as t → −∞ (the flow, then, is expanding on N s (y(t)) as t → −∞ and on N u (y(t)) as t → +∞).
The linearized system is Since v(t) =ẏ(t) = f (y(t)) is a uniformly bounded solution of it, there exists a uniformly bounded non-zero solution y * (t) for the conjugate system As y * (t) solves (4.4), it follows that d dt y * (t) · v(t) = 0 for every solution v(t) of (4.3), i.e. y * (t) · v(t) stays constant. Thus, since the solutions of (4.3) which lie in N s (y(t)) ⊕ N u (y(t)) tend to zero either as t → +∞ or as t → −∞ and y * (t) is bounded, we find that this constant is zero for every v ∈ N s (y(t)) ⊕ N u (y(t)), i.e. the vector y * (t) is orthogonal to N s (y(t)) ⊕ N u (y(t)) for all t. This condition defines y * up to a scalar factor; we fix it by normalizing y * in such a way that (4.5) y * (t) ·ẏ(t) ≡ 1.
The exponential dichotomy for system (4.3) restricted to v(t) ∈ N s (y(t)) ⊕ N u (y(t)) implies that the equation has a unique uniformly bounded solution v(t) ∈ N s (y(t)) ⊕ N u (y(t)) for any uniformly bounded function h(t) ∈ N s (y(t)) ⊕ N u (y(t)). It is more convenient for us to express this property in the following equivalent way: the equation has a unique uniformly bounded solution v(t) given any uniformly bounded function h(t).
More precisely, equation (4.6) defines a linear operator L y : h → v such that The assumed uniform hyperbolicity of the set Λ means that the constant C Λ in (4.7) can be taken the same for all orbits y ∈ Λ.
Take a countable set of equations of type (4.1). This produces an uncoupled LDS (lattice dynamical system): (4.8) y ′ k (t) = f k (y k ), p ′ k (t) = g k (y k ), k ∈ Z We assume that the derivatives of f k and g k up to the order r are uniformly continuous and bounded for all k, and that for each k the k-th individual ODE's in the LDS has a hyperbolic set Λ k , all these sets are uniformly bounded and uniformly hyperbolic for all k (the uniform hyperbolicity means in our approach that the constant C Λ in (4.7) can be taken the same for all k). In the example considered in Section 2, the individual ODE's are identical to each other, so the uniformity with respect to k holds trivially.
By introducing Banach spaces Y := l ∞ (Y ), y Y := sup k∈Z y k Y , y := {y k } k∈Z , P := l ∞ (P ), p P := sup k∈Z p k P , p := {p k } k∈Z , we may write the LDS as The subject of our study will be a coupled LDS, obtained by a small smooth perturbation of this system. Namely, we consider (4.9) y ′ (t) = f(y) + εF ε (y, p), p ′ (t) = g(y) + εG ε (y, p), where ε is a small parameter, and F ε and G ε are C r -functions Y×P → Y and, respectively, Y × P → P; by "C r " we mean, here and below, that all the derivatives up to the order r exist, are uniformly continuous and uniformly bounded. We also assume continuity (in C r ) with respect to ε, so (4.10) Let y 0 (t) := {y 0 k (t)} k∈Z be a sequence of arbitrary orbits y 0 k (t) ∈ Λ k ; we will say that y 0 (t) is an orbit from Λ ∞ . Each orbit y 0 k (t) defines a curve in the Y -space. The direct product of these curves, times the space P, is a C r -submanifold of Y × P, we will denote is as W 0 y 0 . Given an orbit y 0 , the corresponding manifold W 0 y 0 is given by the equation (4.11) y k = y 0 k (φ k ), k ∈ Z, where the "phases" φ k run all real values, independently for different k. If we introduce a Banach space Ψ of the bounded sequences Φ := {φ k } k∈Z with the uniform norm Φ := sup k∈Z |φ k |, then W 0 y 0 is a C r -embedding of Ψ × P into Y × P. Obviously, W 0 y 0 is invariant with respect to the non-coupled LDS (4.8). Moreover, this manifold is normally-hyperbolic (as each of the orbits y 0 k is uniformly-hyperbolic). It is a well-known general principle that normally-hyperbolic invariant manifolds persist at small smooth perturbations (see [Fen71,HiPS77]). The next theorem shows that this principle holds true in our setting.
Theorem 4.1. For all sufficiently small ε, given any orbit y 0 ∈ Λ ∞ there exists a uniquely defined C r -manifold W y 0 ,ε ⊂ Y × P, which is invariant with respect to system (4.9), depends continuously on ε (in C r , uniformly with respect to y 0 ), and coincides with W 0 y 0 at ε = 0. Namely, W y 0 ,ε is given by uniformly for all k ∈ Z and all y 0 ∈ Λ ∞ .
Proof. We start with some preliminary constructions. Define the exponential α-norm h α := sup t∈R e −α|t| h(t) on the space of continuous, uniformly bounded functions h; e.g. · 0 is just the C 0 -norm.  |φ ′ (t) − 1| ≤ ν for all t ∈ R, and for any A(t) and b(t) sufficiently close (in C 0 ) to f ′ (y(t)) and, respectively, to y * (t), the equation is uniquely solvable for any uniformly bounded function h(t), and the corresponding linear operator L φ : h → v satisfies Moreover, the operator L φ is Lipshitz with respect to φ: if v 1 (t) and v 2 (t) are the solutions of equation (4.15) which correspond to two different functions φ 1 (t) and φ 2 (t) (and to the same right-hand side h), then for some constant K, proportional to the C 1 -norms of A, b and f .

Proof.
A uniformly small continuous perturbation of the time-dependent coefficients in the left-hand side of (4.6) does not destroy its unique solvability property. Hence, equation has a unique uniformly bounded solution v(t) given any uniformly bounded function h(t); moreover, for the corresponding operator L : h → v estimate (4.7) holds (we assume that the constant C Λ in (4.7) was taken with a margin of safety, so all our small perturbations of the equation do not change C Λ ). Note also, that given any function φ(t) that satisfies (4.14), if we introduce a new time τ = φ(t) in the equation (4.15) and a new function v new by the rule v new (φ(t)) ≡ v(t), then the left-hand side of equation will be O(ν)-close to the left-hand side of (4.18). For sufficiently small ν this gives us the unique solvability of (4.15) and estimate (4.16) with α = 0. Next, we note that a multiplication of the functions v and h in (4.15) to any smooth function of t with uniformly small derivative just results in a uniformly small correction to A(φ(t)). This immediately shows the unique solvability of equation (4.15) in any weighted space with a sufficiently slowly growing weight; e.g. we obtain (4.16) for all small α.
In order to show the Lipshitz property of L φ with respect to φ, we note that Now, since A(φ), b(φ), f (y(φ)) are smooth -hence, Lipshitz -with respect to φ, and since v 1 (t) is uniformly bounded by (4.7), we immediately get (4.17) from (4.16).
Further we will use where µ is a small constant and ξ ≥ 0 is such that +∞ −∞ ξ(s)ds = 1. At µ = 0 we have b ≡ y * and A ≡ f ′ (y); at small µ the functions A(t) and b(t) are close, respectively, to f ′ (y(t)) in C r−1 and to y * (t) in C r (we have y * (t) ∈ C r as it satisfies equation (4.4)). Thus, uniformly for all t, we have (see (4.5),(4.4)). By taking ξ ∈ C ∞ and such that +∞ −∞ |ξ ′ (s)|ds < ∞, we will make A(t) and b(t) at µ = 0 more smooth then f ′ (y(t)) and, respectively, y * (t), namely we will use A which is at least C r and b which is at least C r+1 ; the price is that the last derivatives do not stay bounded as µ → 0, however we have en estimate (4.21) The next proposition describes the way we coordinatize a small neighborhood of the curve w 0 k : y = y 0 k (t) in the Y -space (y 0 k (t) is an orbit from the hyperbolic set Λ k of the k-th subsystem of the uncoupled LDS (4.8)). Let w k : y = y k (t) be a curve, γ-close to w 0 k on some, finite or infinite, interval I of t, i.e. there exists a smooth time-reparametrization ψ(t) such that y 0 k (ψ(t)) − y k (t) Y < γ at t ∈ I. Lemma 4.3. There existsγ > 0 (independent of the choice of the orbit y 0 k ∈ Λ k and independent of k) such that if γ <γ, then there exists a uniquely defined on I function given by (4.19) at some small µ.
Proof. The derivative of the left-hand side of (4.22) with respect to φ at constant y . By (4.20), it is bounded away from zero, provided y k − y 0 k (φ) is sufficiently small. Thus, by the implicit function theorem, for any point y k from the (sufficiently small) γ-neighborhood of the point y 0 k (ϕ), we have a uniquely defined φ(y k ) which satisfies (4.22) and condition φ(y 0 k (ψ)) = ψ. Moreover, φ depends smoothly on y and the derivatives are uniformly bounded. So, as y 0 , as required (we denote φ(t) := φ(y k (t))).
Let us now proceed to the proof of the theorem. Let y 0 (t) be an orbit from Λ ∞ . The sought invariant manifold W y 0 ,ε consists of all solutions of the LDS (4.9) which stay for all times in a small neighborhood of the manifold W 0 y 0 . This means that, for every k ∈ Z, the k-th component of y(t) stays uniformly close to the corresponding curve w 0 k : y = y 0 k (φ) in the Y -space. In other words every trajectory (y(t), p(t)) ∈ W y 0 ,ε satisfies (4.23) y k (t) = y 0 k (φ k (t)) + v k (t), k ∈ Z, where the functions v k (t) are uniformly small. By Lemma 4.3, we may always assume that the parametrization φ k (t) is chosen so that (4.22) is fulfilled. By differentiating (4.22) with respect to t we get . Now, plugging (4.23) into the first equation of (4.9) gives , where we denote z k (t) := y 0 k (φ k (t)). By multiplying both sides of this equation to b(φ k (t)), and taking (4.22),(4.24) and (4.20) into account, we obtain the following equation for the evolution of φ k :

Equation for
the v-components can now be obtained by plugging (4.26) into (4.25): Equation for the evolution of p(t) is given by the second equation of (4.9): . We remark that if we choose µ = 0 in (4.19), equations (4.27) and (4.29) are simplified and reduce to (see (4.20)). However, the functions q k and Q k will be only C r−1 with respect to Φ, therefore we use small non-zero µ -in order not to lose the last derivative (and to be able to treat the case r = 1). By multiplying both sides of (4.28) to b k (φ k (t)) and using (4.26),(4.20) we find that This equation has only one bounded solution: b k (φ k (t)) · v k (t) ≡ 0; therefore, since b is uniformly bounded, we find that every uniformly bounded solution v(t) of the system (4.26),(4.28),(4.30) (with k running all integer values) satisfies (4.22). Hence, it satisfies (4.25). Thus, the solutions of system (4.26),(4.28),(4.30) whose v(t)-component is uniformly small give us all the solutions of system (4.9) which stay uniformly close to the manifold W 0 y 0 (i.e. all the solutions which comprise the sought invariant manifold W y 0 ,ε ). We show below that for all small δ > 0 the solution of (4.26),(4.28),(4.30) for which exists and is defined uniquely for any given initial condition Φ(0) and p(0). In order to prove the existence and uniqueness of the (small v) solution, we will show that it can be obtained as a fixed point of a contracting operator on an appropriate space. Namely, we consider the set X δ,ν of all functions (v(t), Φ(t), p(t)) belonging to space C loc (R, Y × Ψ × P) such that (4.32) and (4.14) hold for all k and t for which the following norm is finite: where α > 0, and κ > 0 is assumed to be sufficiently small. Obviously, the set X δ,ν is a complete metric space with respect to that norm. Note that in the limit lim ε→0,v→0 the functions q k , Q k given by (4.27),(4.29) tend uniformly to zero for all k ∈ Z, and in the limit lim µ→+0 lim ε→0,v→0 their first derivatives with respect to v, Φ and p tend uniformly to zero too (see (4.10),(4.20),(4.21); the order of the limits is important: ε and v first, then µ). The first derivative of the right-hand side of (4.30) with respect to p is also uniformly small. Thus, if we rewrite system (4.26),(4.28),(4.30) as (4.34) where (Φ 0 , p 0 ) ∈ P × Ψ is arbitrary and the operator L is defined by equation (4.15), then it is easy to check that the right-hand side of (4.34) (for every fixed Φ 0 and p 0 ) defines a contracting operator T : X δ,ν → X δ,ν for every exponential norm with a sufficiently small weight α 0 . Namely, we first fix small ν and α 0 such that the operators L φ k will all be defined and Lipshitz with respect to φ (see (4.17), actually, we may fix ν of order ε); the operator of integration t 0 (·)dt is also Lipshitz in the α 0 -norm, with the Lipshitz constant 1 α 0 ; then we choose a sufficiently small µ for which the Lipshitz constants of q k and Q k can become small enough as ε and v tend to zero; then we see that one may choose κ sufficiently small such that for all sufficiently small ε and δ the Lipshitz constant of the right-hand side of (4.34) on the space X δ,ν is less than 1, which means the operator T is contracting indeed (we need to introduce the small factor κ in the definition of norm on X δ,ν because the derivative of g k with respect to v k and φ k , though bounded, is not necessarily small). As at ε = 0 and v = 0 the v-component of the image by T vanishes, the contractivity of T implies that given any small δ the condition (4.32) is invariant with respect to T for all sufficiently small ε; i.e., the T X δ,ν ⊂ X δ,ν . By the Banach principle, the iterations by T of any initial element from X δ,ν converge to a uniquely defined limit in X δ,ν , the fixed point of T . Thus, we have shown that every solution which stays sufficiently close to the manifold W 0 y 0 for all times can be found as the uniquely defined solution of (4.34). Therefore, the union of all such solutions comprises the sought invariant manifold W y 0 ,ε given by (4.12) where the function V k is defined by the map (Φ(0), p(0)) → v k (0). Note that this map (hence the manifold W y 0 ,ε ) is Lipshitz continuous, since the contracting operator T is Lipshitz continuous with respect to (Φ 0 , p 0 ). We omit the proof of the smoothness of this map, as it is completely standard (yet laborious): one may show that the operator T is smooth on a scale of Banach spaces corresponding to different weighted α-norms on X δ,ν (cf. [GiVa87,ShSTC01]), or alternatively check, by fiber-contraction arguments, that the iterations of by T of an initial element from X δ,ν converge to the fixed point of T uniformly along with the derivatives with respect to (Φ 0 , p 0 ) (cf. [Mie86,Mie88]).
In order to finish the proof of the theorem, it remains to show estimate (4.13). The C r -part is obvious, as v = 0 solves (4.34) at ε = 0, and the fixed point of a contracting operator which depends on a parameter continuously must depend on the same parameter continuously. To show the C r−1 -estimate, we note that when the right-hand side of (4.34) depends smoothly on some parameter, the solution must also be smooth with respect to the same parameter. In particular, if we rewrite system (4.9) as y ′ (t) = f(y) + σF ε (y, p), p ′ (t) = g(y) + σG ε (y, p), then v(t) will depend C r -smoothly on σ as well, which immediately gives (4.13) if we note that v(t) = 0 at σ = 0 and plug σ = ε back.
Remark 4.4. The theorem also remains true if the number of systems coupled in the LDS is finite, i.e. if the index k runs a finite set instead of Z. Then the range of ε values for which the corresponding invariant manifolds exist will be independent on the number N of systems in the LDS -provided the constant C in the bound (4.10) on the norm of the coupling terms is independent of N. Note that condition (4.10) does not requires that the coupling is local, it just means that the "total coupling strength" for each subsystem in the LDS is bounded independently of the total number N of subsystems involved.
Remark 4.5. Given any symmetry in system (4.9), if the set Λ ∞ obeys the same symmetry, then system of invariant manifolds W y 0 ,ε inherits the symmetry for all small ε -by uniqueness. A basic example of such symmetry is invariance with respect to spatial translation k → k + 1 (in this case, the coupling terms F, G are shift-invariant, the individual ODE's (4.8) are the same for all k, and the sets Λ k should be chosen the same).
Theorem 4.1 allows us to construct a huge number of special solutions of the weakly coupled LDS (4.9). Indeed, for every y 0 ∈ Λ ∞ , one can construct the associated manifold W y 0 ,ε and then, for every (Φ 0 , p 0 ) ∈ Ψ × P, there exists a solution (y(t), p(t)) in the form (4.35) where the functions (Φ(t), p(t)) solve the reduced problem on the center manifold (see (4.26),(4.30)): (4.36) It is interesting to have an expansion in powers of ε for the system on the invariant manifold. In order to do this we need a sufficient smoothness of the right-hand sides: for instance, to find the first order in ε approximation to (4.36) we assume the original system to be at least C 2 with respect to all variables and ε. In this case we may take µ = 0 in formulas (4.19), so the function q will be given by (4.31). As V = O(ε) by (4.13), we immediately obtain the first-order in ε approximation to the φ-equation: Formula (4.37) describes the evolution of phases on the invariant manifolds W and can be useful in the study of phase synchronization in coupled chaotic systems (see e.g. [RoPK96]).
To obtain the approximate p-equation, we need the first-order approximation to v. By expanding the first equation in (4.34) in ε, we find that where the function u k (t) is given by ) . Since, by (4.36), φ k (t) is for all k uniformly O(ε)-close to φ k (0) + t in the exponential α-norm with α > 0, and p k (t) is for all k uniformly O(ε)-close to p k (0) + t 0 g k (y 0 k (s + φ k (0)))ds, also in the exponential α-norm, it follows from the Lipshitz property of the operator L φ (see (4.17),(4.16)) that where, given any constant Φ and p, we denote as w k (Φ, p) the value at t = 0 of the uniquely defined bounded solution w(t) of the equation , where F k := F k,0 (y 0 (t + Φ), p + t 0 g(y 0 (s + Φ))ds). By plugging the above formula for v k into the p-equation of (4.36) and dropping all o(ε)-terms, we find that the first-order approximation to the p-equation is p)) + εG k,0 (y 0 (Φ), p). Remark 4.6. Let the uniformly-hyperbolic sets Λ k be compact and locally-maximal, i.e. there exists γ 0 > 0 (independent of k) such that for each k every orbit of (4.8), which stays in the γ 0 -neighborhood of Λ k for all t, belongs to Λ k itself. Then, for ε sufficiently small, every solution of the coupled LDS (4.9) whose y-component stays for all times in a small neighborhood of Λ ∞ belongs to one of the manifolds W y 0 ,ε . Indeed, given any k the k-th component y k (t) of such solution must be close, after some reparametrization of time, to a γ-orbitỹ(ϕ(t)), which is a countable union of consecutive piecesỹ(ϕ)| φ∈[φ j ,φ j+1 ) of orbits from the set Λ k such that ỹ(ϕ j ) −ỹ(ϕ j − 0) ≤ γ, for some small γ. It is known that when Λ k is locally-maximal, any γ-orbit is shadowed by a true orbit, i.e. there exists an orbit in Λ k which is O(γ)-close toỹ(ϕ) (after a reparametrization of time). Thus, our solution y(t) of the coupled LDS stays for all times close to a (time-reparametrized) orbit y 0 ∈ Λ ∞ , i.e. we can write it in the form (4.23), and we showed in Theorem 4.1 that every such solution belongs to the invariant manifold W y 0 ,ε .

A theorem on asymptotic phase
In this Section we compare the behavior of orbits of the LDS (4.9) which belong to different invariant manifolds W y 0 ,ε . We start with the analysis of the dependence of the invariant manifold W y 0 ,ε on the choice of the trajectory y 0 ∈ Λ ∞ . Clearly, W y 0 ,ε depends on y 0 continuously: namely, the function V y 0 ,ε (Φ, p) (hence -the function U y 0 ,ε (Φ, p)) in (4.12) is found via an application of the contraction mapping principle, and the corresponding contracting operator (the operator T defined by the right-hand side of (4.34)) depends continuously on y 0 in some exponential weighted norm, so on any bounded set of values of Φ and p the functions U y 0,1 ,ε and U y 0,2 ,ε will be uniformly close provided the trajectories y 0,1 and y 0,2 are sufficiently close in the weighted norm. We need, however, a different statement about the closeness of U y 0,1 ,ε and U y 0,2 ,ε . Note that though the manifold W y 0 ,ε is defined uniquely (as the set of all solutions that for all t stay uniformly close to the manifold W 0 y 0 defined by (4.11)), the function U y 0 ,ε in (4.12) is defined up to an (arbitrary) reparametrization of the space Ψ of phases φ k . Therefore, when comparing functions U corresponding to two different trajectories y 0 (as we do it below), we should describe how the corresponding parametrization choices agree with each other.
In order to do so we recall the construction used in the proof of Theorem 4.1. Take any orbit {y = y 0 (t)} ∈ Λ ∞ ; its k-th component y 0 k (t) defines a smooth curve w 0 k in the space Y . Take any other curve w : {y = y k (t)} in Y . By Lemma 4.3, there existsγ > 0 (independent of the choice of the curves) such that if w stays in theγ-neighbourhood of w 0 k for a certain interval of time, then, for every t from this interval, condition (4.22) defines the projection y 0 k (φ k (t)) of the point y k (t) ∈ w onto the curve w 0 k uniquely. We will call φ k (t) the phase relative to y 0 . If we have two orbits, y 0,1 and y 0,2 , from Λ ∞ , and these orbits are γ-close (γ <γ) on some time interval, then for every curve y = y k (t) which stays at the distance less than γ from both w 0,1 k and w 0,2 k on this time interval we have two phases, φ 1 k (t) and φ 2 k (t), relative to y 0,1 and y 0,2 respectively. By Lemma 4.3 (with ψ standing for φ 2 k and φ for φ 1 k ), these two phases are related by a close to identity diffeomorphism: φ 2 . For a solution in the invariant manifold W y 0 ,ε the canonical phases φ k (which we used before) are the phases relative to y 0 . However, if two orbits y 0,1 and y 0,2 are γ-close (γ <γ) on some time interval, then for solutions in, say, W y 0,j ,ε the phase ϕ k relative to y 0,1 is also defined on this interval, along with the canonical phase φ k relative to y 0,2 .
Since both x 1 (t) and x 2 (t) satisfy system (4.26),(4.28),(4.30) at t ∈ [t 1 + S, t 2 − S], the function x 3 (t) satisfies the same system with a uniformly bounded correction to the right-hand sides which is localized at |t| ∈ [S − 1, S] (and which is denoted below as ρ). Since the initial conditions in Φ and p coincide for x 1 and x 2 by construction (recall that we choose y 1 (t) and y 2 (t) such that ϕ k (0) = φ k (0)), we find that x 3 (t) t∈[−∞,∞] satisfy the following equation (a perturbation of (4.34)) (5.8) where z 3 k := y 0,1 k (φ 3 k ) and the perturbations ρ k (t) satisfy ρ α 0 = O(e −α 0 S ). Recall that x 2 k (t) = (u k (t), ϕ k (t), p 2 k (t)) satisfies system (4.26),(4.28),(4.30) at |t| ≤ S, and u(t) is uniformly small on this interval (provided γ and ε are small enough). The smallness of u and ε implies the smallness of the functions q k in the right-hand side of the equation (4.26) for the phases ϕ k , therefore sup |t|≤S |ϕ ′ k (t) − 1| is uniformly small for all k. By (5.7), we find then that sup t∈R |φ 3′ k (t) − 1| is also uniformly small. This guarantees that the operator L φ 3 k is defined and Lipshitz in the α 0 -norm (see comments after (4.34) in the proof of Theorem 4.1). Since operator L φ is Lipshitz in the exponential α 0 -norm, and so is the operator of integration t 0 (·)dt, we may rewrite (5.8) as where T is the operator defined by the right-hand side of (4.34), i.e. x 1 = T x 1 , and we immediately get that x 3 − x 1 α 0 = O(e −α 0 S ), since the operator T is contracting (in the norm given by (4.33); note that, as we have shown in the proof of Theorem (4.1), in order to have contraction, both v 1 and v 3 must be uniformly small, i.e. must satisfy (4.32) with a sufficiently small δ, and this property indeed holds true when ε and γ are sufficiently small).
Remark 5.2. By a shift of time, we obtain that if for all p 0 ∈ P and Φ 0 such that for all k the components φ k of Φ 0 satisfy (5.11) We may now prove the following theorem, crucial for the next Section.
Remark 5.4. Absolutely analogously, for every two trajectories y 0,1 and y 0,2 from Λ ∞ satisfying the condition (5.14) sup given any solution (y 1 (t), p 1 (t)) from the manifold W y 0,1 ,ε , there exists a unique solution (y 2 (t), p 2 (t)) from W y 0,2 ,ε such that Proof. As we explained in Lemma 5.1, condition (5.12) (which is an analogue of condition (5.1) for the case of infinite time interval) implies that for all sufficiently small ε, for any two solutions (y 1 (t), p 1 (t)) and (y 2 (t), p 2 (t)) from the invariant manifolds, respectively, W y 0,1 ,ε and W y 0,2 ,ε , for all t ≥ t 0 and every k ∈ Z we have well-defined projections of the points y 1 k (t) and y 2 k (t) onto the curve y = y 0,1 k (ϕ k ) in the space Y . The position of the projection point is defined by its phase ϕ k , so we have two phases (relative to the same orbit y 0,1 ) defined for all t ≥ t 0 : ϕ 1 k (t) for the point y 1 k (t) and ϕ 2 k (t) for y 2 k (t). Thus, (5.16) here b k is given by (4.19) with y * = y * ,1 k . As the solution (y 1 (t), p 1 (t)) belongs to the invariant manifold W y 0,1 ,ε associated with the orbit y 0,1 relative to which the phase is defined, the phases ϕ 1 k are just the canonical phases φ 1 k . For the solution (y 2 (t), p 2 (t)), as we explained in the introduction to Lemma 5.1, the phases ϕ 2 k are related to the canonical phases φ 2 k by a close to identity diffeomorphism η k : is uniformly small for all t ≥ t 0 . Formula (5.16) is identical to (4.23),(4.22), hence (see the proof of Theorem 4.1) the functions (v i (t), ϕ i (t), p i (t)), both for i = 1 and i = 2, solve the same system (4.26),(4.28),(4.30) (where one should replace φ with ϕ and z k with y 0,1 k (ϕ k )), for all t for which v i (t) remains small. As the solution (y 1 (t), p 1 (t)) belongs to the invariant manifold W y 0,1 ,ε , we have that y 1 (t) stays close to y 1,0 (Φ 1 (t)) for all times, which guarantees the smallness of v 1 . The smallness of v 2 (t) at all t ≥ t 0 follows from the fact that (y 2 (t), p 2 (t)) belongs to the invariant manifold W y 0,2 ,ε , hence y 2 k (t) stays close to y 2,0 k (φ 2 k (t)) for all times, and because of the uniform closeness of φ 2 k (t) to ϕ 2 k (t) and y 2,0 k to y 1,0 k at t ≥ t 0 we obtain the uniform closeness of y 2 k (t) to y 1,0 k (ϕ 2 k (t)) at t ≥ t 0 . Thus, we have where q k is given by (4.27) (the only important thing for us is that q k is uniformly small along with its first derivatives), and . By Theorem (4.1), since the solutions (y i (t), p i (t)) belong to the respective invariant manifolds W y 0,i ,ε , we may put in equations (5.17), whereṼ i are certain functions of (ϕ, p) with the Lipshitz constant uniformly small. Namely, the functionṼ 1 is just the function V 1 that defines the manifold W y 0,1 ,ε by (4.12), while the functionṼ 2 is given byṼ(ϕ, p) = V 2 (η(ϕ), p) + y 0,2 (η(ϕ)) − y 0,1 (ϕ), where η is the diffeomorphism which sends the phase relative to y 0,1 to the phase relative to y 0,2 ; the required Lipshitz property ofṼ 2 follows from the Lipshitz property of V and η. Note that by (4.12) (5.20)Ṽ 1 (ϕ, p) = U 1 (ϕ, p) − y 0,1 (ϕ),Ṽ 2 (ϕ, p) = U 2 (η(ϕ), p) − y 0,1 (ϕ), hence, by Remark 5.2, when ϕ k → +∞ uniformly for all k, we have for some α ′ > 0. It follows from (5.21),(5.19),(5.16) that we will have the required exponential decay of y 1 (t) − y 2 (t) if the difference between the corresponding solutions (ϕ 2 (t), p 2 (t)) and (ϕ 1 (t), p 1 (t)) of (5.17),(5.19) tends exponentially to zero as t → +∞. Given (ϕ 1 (t), p 1 (t)), the sought, tending to it solution (ϕ 2 (t), p 2 (t)) corresponds to the fixed point of the operator (φ(t), p(t)) t≥0 → (φ(t),p(t)) t≥0 defined by the following equation: where we denote φ(t) := ϕ 2 (t)−ϕ 1 (t), p(t) := p 2 (t)−p 1 (t). Thus, it remains to prove the existence and uniqueness of the fixed point of this operator in the space of exponentially decreasing functions, and also to show that this fixed point tends to zero as γ → 0.
In order to do this, we first note that because ϕ 1 (t) and ϕ 2 (t) grow within linear bounds with time, estimate (5.21) along with the boundedness of the Lipshitz constants of the functions q and h implies that for some α > 0 the operator (5.22) takes exponentially decreasing functions (φ(t), p(t)) = O(e −αt ) into functions (φ(t),p(t)) which are exponentially decreasing as well, with the same exponent α.
Recall also that the Lipshitz constant of q is uniformly small (and tends to zero as ε → 0 and γ → 0). The Lipshitz constant ofṼ with respect to ϕ is uniformly bounded and the Lipshitz constant with respect to p is uniformly small (and tends to zero as ε → 0; see (5.20),(4.12),(4.13)). The Lipshitz constant of h with respect to p is of order ε and the Lipshitz constants with respect to ϕ and v are bounded (see(5.18),(4.10)). Thus, for the functions under the integrals in (5.22), the Lipshitz constants with respect to both ϕ and p in the first equation of (5.22) and with respect to p in the second equation are uniformly small, while the Lipshitz constant with respect to ϕ in the second equation is uniformly bounded. This immediately implies that operator (5.22) is contracting on the space of exponentially decreasing functions (φ(t), p(t)) t≥0 endowed with the norm φ, p = sup t≥0 e αt ( ϕ(t) Ψ + κ p(t) P ), for all sufficiently small κ, ε and γ. This gives us the required existence and uniqueness of the fixed point (φ(t), p(t)) t≥0 . Being the fixed point of a contracting operator, it depends continuously on every parameter on which the operator depends continuously, so it depends continuously on the functionṼ 2 . Note thatṼ 2 →Ṽ 1 as γ → 0 (by (5.20), this just means that the manifold W y 0 ,ε depends on y 0 continuously). Hence, in the same limit we have (φ(t), p(t)) → 0 (which is the trivial fixed point of (5.22) whenṼ 2 ≡Ṽ 1 ).

Spatially non-walking solutions and their entropy
In our application to Ginzburg-Landau equation, the p-component of the LDS (4.9) describes the temporal evolution of the centers of soliton pairs, namely the deviations of the pair centers from the points of a given spatial lattice. This description is valid only if the distances between the soliton pairs are large enough, i.e. the deviations of the soliton pairs from the lattice points stay uniformly bounded for arbitrarily large lattice sizes, see Section 2. Thus, it is crucial to be able to control the norm of p(t) = {p k (t)} +∞ k=−∞ , i.e. to keep all p k bounded.
On the other hand, according to (4.38),(4.37), in the zero order approximation with respect to ε we have where y 0 k is a trajectory from the given hyperbolic set Λ. Thus, an independent diffusivelike behavior of the coordinates p k (t) should be expected [TuVZ07] in the case Λ is nontrivial (chaotic), i.e. the quantities p k (t) are out of control in this case.
The main aim of the Section is to show, however, that under some natural assumptions on the set Λ there exists a set of solutions for which at all t ∈ R (6.2) p(t) P ≤ R 0 for some constant R 0 >> 1. Moreover, this set is large enough, so that it has positive space-time entropy. In what follows, in order to simplify notations, we assume that all individual ODE's in the uncoupled LDS (4.8) are identical, i.e. f k ≡ f , g k ≡ g for all k.
Theorem 6.1. Let the assumptions of Theorem 4.1 hold and let N := dim P . Let us also assume that the hyperbolic set Λ of system (4.2) is transitive and locally-maximal and contains N + 1 periodic orbits Z 1 : y = z 1 (t), Z 2 : y = z 2 (t),. . . , Z N +1 : y = z N +1 (t) with periods T 1 ,. . . , T N +1 respectively. Define the vectors b i ∈ P , i = 1, . . . , N + 1, as follows: and require the following properties to be satisfied: 1. linear combinations of vectors b i generate the whole space P : (6.4) P = span{ b 1 , . . . , b N +1 }; 2. there exist strictly positive numbers A i such that (6.5) Then, for all sufficiently small ε > 0, there exists a uniformly bounded set K of solutions of system (4.9) which has strictly positive space-time entropy: Proof. We start with the following observation.
Indeed, suppose there exists p such that (6.7) is wrong, i.e. p· b i ≥ 0 for all i = 1, . . . , N +1. Multiplying then equality (6.5) by this p and using that A i > 0, we conclude that p · b i = 0 for all i. By (6.4), this contradicts the assumption p = 0. Thus, (6.7) is verified and (6.8) follows immediately from (6.7) by compactness arguments. The idea of the proof of the theorem is as follows. As the set Λ is transitive and locallymaximal, for every two of the periodic orbits Z i and Z j we may choose two different heteroclinic orbits Z ijm : y = z ijm (t), m = 1, 2, that connect them, i.e.
for some constant θ ± ijm . The orbits Z ijm also belong to Λ; in fact, the number of different heteroclinics is infinite for each pair of periodic orbits in Λ, but we need only two of them for each i and j. The existence of the heteroclinics mean that we may build orbits in Λ which stay for some time near the orbit Z i , then "jump" along any two of the heteroclinics Z ij1,2 into a neighborhood of Z j , stay there, then jump again into a neighborhood of another periodic orbit, etc.. We will see that for sufficiently small ε one can build orbits y = {y k (t)} k=+∞ k=−∞ of system (4.9) with a similar behavior for every component y k (t): the component stays close to z i (t) for some time then jumps to z j (t), etc., moreover the choice of the sequence of the periodic orbits the component shadows can be made independently for different k. When the component y k is close to z i (t) for sufficiently long time, the p k -component of the associated solution will move in the direction close to b i as time grows (see (6.1),(6.3)). By (6.8), if the norm of p k becomes large enough we can always find a vector b j such that moving in its direction will lead to a decrease in the norm of p k . Thus, by jumping each time to a properly chosen periodic orbit Z j we may keep the norm of all p k bounded. As each jump can be made by at least two different ways (along the first or the second heteroclinic) the set of different solutions of system (4.9) we obtain in this way will have positive entropy.
As the first step in implementing this construction we recall the following standard result on the "shadowing" in hyperbolic sets. Lemma 6.3. There existγ > 0 and α > 0 such that for any two orbits y − (t) and y + (t) from the hyperbolic set Λ which satisfy whereγ > γ > 0, there exists an orbit y(t) ∈ Λ and a phase shift θ such that (6.10) where C γ > 0 depends only on γ and tends to zero as γ → 0.
A proof can be found e.g. in [KaHa95]. The orbit y(t) corresponds simply to the intersection of the local unstable manifold of y − (t) with the local stable manifold of y + (t); this intersection belongs to Λ because this set is locally-maximal.
Combining Lemma 6.3 with Theorem 5.3, we obtain an analogous result for the lattice dynamical system (4.9).
Proof. Indeed, in order to find the required solution (y(t), p(t)), we first construct a trajectory y 0 ∈ Λ ∞ , each component y k (t) of which is defined by y 0 k− (t) and y 0 k+ (t) by virtue of Lemma 6.3 such that (6.12) is satisfied (since the unperturbed system (4.8) is a Cartesian product of systems (4.2), we only need to apply Lemma 6.3 to every component in this product). Applying after that Remark 5.4, we find (in a unique way) the solution (y(t), p(t)) of the perturbed system (4.9), satisfying (6.13) for t ≤ t 0 .
We are now ready to complete the proof of the theorem. We will choose sufficiently large constants T and R and sufficiently small constants ν and µ and construct a sequence of sets K l of solutions of (4.9) and a sequence of sets K 0 l of orbits from Λ ∞ such that: 1) for each of the solutions from K l there exists an otbit y 0 ∈ K 0 l such that the solution belongs to the invariant manifold W y 0 ,ε ; 2) for every trajectory y 0 = {y k (t)} +∞ k=−∞ ∈ K 0 l , for every k ∈ Z there are periodic orbits Z i k+ : y = z i k+ (t) and Z i k− : y = z i k− (t) (from the set of periodic orbits Z 1 , . . . , Z N +1 under consideration) such that (6.14) y 0 k (lT ) − z i k+ (τ kl ) < ν, y 0 k (−lT ) − z i k− (τ k,−l ) < ν for some (irrelevant) constants τ ∈ [0, T ], where T = max i=1,...,N +1 T i (the periods of Z i ); 3) at t 0 = ±lT (6.15) p(t 0 ) P ≤ R; 4) for every solution (ỹ(t),p(t)) ∈ K l+1 there exists a solution (y(t), p(t)) ∈ K l such that (6.16) ỹ(t) − y(t) Y + p(t) − p(t) P ≤ µe −α(lT −|t|) for all |t| ≤ lT (the constant α > 0 depends on the set Λ only). By condition 4, the sequence of the sets K l converges, as l → +∞, to a certain set K of solutions of the LDS (4.9) (convergence is uniform on any bounded time interval). Moreover, conditions 2, 3 and 4 imply that solutions in the set K are uniformly bounded, in particular p(t) is uniformly bounded for all of the solutions. Thus, to prove the theorem, we need to actually construct the sequence K l and to do it in such a way that the sets K l would contain "sufficiently many" solutions -this would warrant the positivity provided T is taken sufficiently large with respect to µ and R is sufficiently large with respect to T (note that b j is bounded away from zero by virtue of (6.4), (6.5)). Analogously, one checks thatp 2 k (−(l + 1)T ) < R 2 .
Thus, we have shown that given any solution from the set K l 0 and any pair of sequences m k± (these sequences define which of the two heteroclinic connections is used to jump from the periodic orbit Z i k± to Z j k± ) we obtain a solution which satisfies above conditions 1-3 with l = l 0 + 1, i.e. the newly built solution can be included into the set K l 0 +1 ; we have also checked condition 4 that ensures the convergence of the sequence of sets K l as l → +∞. As we may choose the sequences m k± in an arbitrary way at each step of the procedure, the number of solutions in the set K l which stay at a bounded away from zero distance from each other at |t| ≤ lT and |k| ≤ n equals to 4 l(2n+1) . This immediately shows that the space-time entropy of the limit set K is strictly positive.
Note that the assumption that the set Λ is locally-maximal and transitive can be formulated in a more constructive way. Indeed, assume that we have a set of hyperbolic periodic orbits Z 1 , . . . , Z N +1 , which satisfy conditions 1 and 2 of the theorem. Build an oriented graph with N + 1 vertexes: the edge connects the vertex i with vertex j if we know there exists a heteroclinic orbit Z ij which corresponds to a transverse intersection of the unstable manifold of Z i with the stable manifold of Z j . If this graph is transitive, then the set Λ of all orbits which stay for all times in a sufficiently small neighborhood of the union of the hyperbolic periodic orbits Z i and the transverse heteroclinic orbits Z ij is uniformly-hyperbolic, transitive and locally-maximal [AfrS74], so Theorem 6.1 holds.
Note also that assumption (6.5) is really important for the proof of the theorem. Indeed, consider the case dim P = 1 for example. Here the integrals b 1 and b 2 are real numbers, and if condition (6.5) is violated, they both have the same sign, positive, say. In this case, when the component y k stays close to either of the periodic orbits Z 1,2 , the component p k will increase with time, so we cannot keep p k (t) bounded by mere switching between Z 1 and Z 2 . However, assumption (6.5) can be relaxed if we allow for a uniform drift, common for all p k (t). Namely, the following statement holds true.
Corollary 6.5. Let all of the assumptions of Theorem 6.1 be fulfilled except of (6.5). Assume the convex hull I b of vectors b 1 , · · · b N +1 have a non-empty interior: (6.20) Let p ⊂ I b . Then, for every sufficiently small ε, there exists a set K p of solutions (y(t), p(t)) of system (4.9) such that K p has positive space-time entropy and each solution from K p satisfies (6.21) where the constant R 0 depends on p, but is independent of t and the choice of the solution.
Indeed, for every p ∈ I b conditions (6.4),(6.5) hold for the vectors b 1 − p, b 2 − p, . . . , b N +1 − p. Then, applying Theorem 6.1 to the system obtained from (4.9) by substracting p from the function g, we immediately obtain the corollary.
Remark 6.6. In the one-dimensional case (dim P = 1), we only need two hyperbolic periodic orbits, Z + and Z − , connected by transverse heteroclinics. Conditions (6.4), (6.5) read now (6.22) T − 0 g(z − (t)) dt · T + 0 g(z + (t)) dt < 0; conditions (6.4), (6.20) read as (6.23) 1 T − In order to establish the existence of the heteroclinic cycle with two hyperbolic periodic orbits one may use Shilnikov criterion. Namely, it is enough to show the existence of a saddle-focus equilibrium state y = z 0 with a homoclinic loop y = z h (t), z h (t) → z 0 as t → ±∞, and to check that the so-called Shilnikov conditions of chaos are satisfied (we will not discuss a higher-dimensional case as in the application we consider in this paper we have y ∈ R 3 ; in the three-dimensional case the Shilnikov condition is that the nearest to the imaginary axis characteristic exponent is not real; the equilibrium state must be hyperbolic, i.e. it has characteristic exponents on both sides of the imaginary axis and no characteristic exponents on the axis). Then there exists a sequence Z n of hyperbolic periodic orbits which converge to the homoclinic loop as n → +∞, any two of them are connected by transverse heteroclinics [Shi65,Shi70]. The periods T m of Z m tend to infinity. One can always choose time parametrization such that sup t∈[− Tm 2 , Tm 2 ] |z m (t)−z h (t)−z 0 | → 0 as m → +∞. It follows that one can always choose among the orbits Z m a pair satisfying conition (6.23), provided (6.24) +∞ −∞ (g(z h (t)) − g(z 0 )) dt = 0 (the integral converges since z h (t) tends to z 0 exponentially -because of the hyperbolicity of z 0 ). Note that the homoclinic loop to a saddle-focus may split as we perturb the system, however the two hyperbolic periodic orbits that we find near the loop do not disappear, nor the transverse heteroclinics that connect them do, so by checking condition (6.24) for one parameter value we establish the existence of spatio-temporal chaos for an open set of parameter values; see Lemma 2.3 in Section 2 for an example.
Remark 6.7. In the case of LDS with n spatial dimensions, i.e. those parameterized by multiindices k ∈ Z n instead of k ∈ Z, the result of Theorem 6.1, obviously, remains true under the properly modified definition of the space-time topological entropy. In fact, this case is just formally reduced to k ∈ Z by an appropriate reparameterization of the grid Z n by the points from Z.