Anti-integrable Limit Research Articles

Anti-integrability for Three-Dimensional Quadratic Maps

We study the dynamics of the three-dimensional quadratic diffeomorphism using a concept first introduced 30 years ago for the Frenkel--Kontorova model of condensed matter physics: the anti-integrable (AI) limit. At the traditional AI limit, orbits of a map degenerate to sequences of symbols and the dynamics is reduced to the shift operator, a pure form of chaos. Under nondegeneracy conditions, a contraction mapping argument can show that infinitely many AI states continue to orbits of the deterministic map. For the 3D quadratic map, the AI limit that we study is a quadratic correspondence whose branches, a pair of one-dimensional maps, introduce symbolic dynamics on two symbols. The AI states, however, are nontrivial orbits of this correspondence. The character of these orbits depends on whether the quadratic takes the form of an ellipse, a hyperbola, or a pair of lines. Using contraction arguments, we find parameter domains for each case such that each symbol sequence corresponds to a unique AI state. In some parameter domains, sufficient conditions are then found for each such AI state to continue away from the limit becoming an orbit of the original 3D map. Numerical continuation methods extend these results, allowing computation of bifurcations to obtain orbits with horseshoe-like structures and intriguing self-similarity. We conjecture that pairs of periodic orbits in saddle-node and period-doubling bifurcations have symbol sequences that differ in exactly one position.

Equilibrium Configurations for Generalized Frenkel–Kontorova Models on Quasicrystals

I study classes of generalized Frenkel–Kontorova models whose potentials are given by almost-periodic functions which are closely related to aperiodic Delone sets of finite local complexity. Since such Delone sets serve as models for quasicrystals, this setup presents Frenkel–Kontorova models for the type of aperiodic crystals which have been discovered since Shechtman’s discovery of quasicrystals. Here I consider models with configurations $$u:{\mathbb {Z}}^r \rightarrow {\mathbb {R}}^d$$ , where d is the dimension of the quasicrystal, for any r and d. The almost-periodic functions used for potentials are called pattern-equivariant and I show that if the interactions of the model satisfies a mild $$C^2$$ requirement, and if the potential satisfies a mild non-degeneracy assumption, then there exist equilibrium configurations of any prescribed rotation rotation number/vector/plane. The assumptions are general enough to satisfy the classical Frenkel–Kontorova models and its multidimensional analoges. The proof uses the idea of the anti-integrable limit.

Transitions and anti-integrable limits for multi-hole Sturmian systems and Denjoy counterexamples

Abstract For a Denjoy homeomorphism $f$ of the circle $S$, we call a pair of distinct points of the $\omega$-limit set $\omega (\,f)$ whose forward and backward orbits converge together a gap, and call an orbit of gaps a hole. In this paper, we generalize the Sturmian system of Morse and Hedlund and show that the dynamics of any Denjoy minimal set of finite number of holes is conjugate to a generalized Sturmian system. Moreover, for any Denjoy homeomorphism $f$ having a finite number of holes and for any transitive orientation-preserving homeomorphism $f_1$ of the circle with the same rotation number $\rho (\,f_1)$ as $\rho (\,f)$, we construct a family $f_\varepsilon$ of Denjoy homeomorphisms of rotation number $\rho (\,f)$ containing $f$ such that $(\omega (\,f_\varepsilon ), f_\varepsilon )$ is conjugate to $(\omega (\,f), f)$ for $0<\varepsilon <\tilde{\varepsilon }<1$, but the number of holes changes at $\varepsilon =\tilde{\varepsilon }$, that $(\omega (\,f_\varepsilon ), f_\varepsilon )$ is conjugate to $(\omega (\,f_{\tilde{\varepsilon }}), f_{\tilde{\varepsilon }})$ for $\tilde{\varepsilon }\leqslant \varepsilon <1$ but $\lim _{\varepsilon \nearrow 1}f_\varepsilon (t)=f_1(t)$ for any $t\in S$, and that $f_\varepsilon$ has a singular limit when $\varepsilon \searrow 0$. We show this singular limit is an anti-integrable limit (AI-limit) in the sense of Aubry. That is, the Denjoy minimal system reduces to a symbolic dynamical system. The AI-limit can be degenerate or nondegenerate. All transitions can be precisely described in terms of the generalized Sturmian systems.

Existence of non-smooth bifurcations of uniformly hyperbolic invariant manifolds in skew product systems

In this paper we study the anti-integrable limit scenario of skew product systems. We consider a generalization of such systems based on the Frenkel–Kontorova model, and prove that under certain mild regularity conditions on the potential the structure of the orbits is of Cantor type. From our results we deduce the existence of the non-smooth folding bifurcation (conjectured by Figueras and Haro (2015 Chaos 25 123119)).Lastly we present some results which are useful in determining if a potential satisfies the regularity conditions required for the Cantor sets of orbits to exist and are also of independent interest. We also prove the existence of orbits with any fibered rotation number in systems of both one and two degrees of freedom. In particular, our results also apply to two-dimensional maps with degenerate potentials (vanishing second derivative), so extending the results of existence of Cantori to more general twist maps.

Existence of discrete solitons in discrete nonlinear Schrödinger equations with non-weak couplings

Discrete solitons are spatially localized periodic solutions in the discrete nonlinear Schrodinger equation. The anti-integrable limit is defined for the discrete nonlinear Schrodinger equation as the limit of vanishing couplings. There are an infinite number of trivial discrete solitons in this limit, each of which consists of a finite number of excited sites. The existence of discrete solitons continued from them has been proved only for sufficiently weak couplings. In the present paper, we focus on the case of non-weak couplings and prove the existence of discrete solitons over an explicitly given range of the coupling constant.

The anti-integrable limit

The anti-integrable limit is one of the convenient and relatively simple methods for the construction of chaotic hyperbolic invariant sets in Lagrangian, Hamiltonian, and other dynamical systems. This survey discusses the most natural context of the method, namely, discrete Lagrangian systems, and then presents examples and applications. Bibliography: 75 titles.

Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps.

We present a computer-assisted numerical study of different bifurcations of saddle invariant tori in quasiperiodic area preserving twist maps. We detect three bifurcation scenarios. In the first scenario, the smooth bifurcation, the regularity of the invariant torus is preserved, and the stable and unstable invariant bundles collide smoothly. In the other scenarios, the spiky and folding breakdowns, the invariant torus looses smoothness, and the invariant bundles collide non-smoothly. In the former, the C(1) seminorm of the torus does not blow up while in the latter it does. Numerics suggest that after the breakdowns non-uniformly hyperbolic invariant objects persist. These are qualitatively different depending on the type of breakdown. Finally, using anti-integrable limit theory, we provide a proof of existence of non-uniformly hyperbolic invariant objects for systems very far from the integrable regime.

Chaotic orbits for differentiable maps near anti-integrable limits

In this work, we study the dynamical system which can be transformed into a difference equation of the form ϵg(xt−n2,…,xt,…,xt+n1;ϵ)+g0(xt)=0 where ϵ∈R is a parameter, g:Rn1+n2+1→R and g0:R→R are both C1 functions and n1,n2 are both nonnegative integers. Suppose that the function g0 has k simple zeros where k≥2. We give criteria of existence of some kinds of chaotic orbits, and construct infinite ones explicitly. As applications of these results, we establish snap-back repellers and heteroclinical repellers for the modified Mira maps and transversal homoclinic orbits or transversal heteroclinic orbits for the Hénon maps, the Arneodo–Coullet–Tresser maps and the quadratic volume preserving maps.

The laminations of a crystal near an anti-continuum limit

The anti-continuum limit of a monotone variational recurrence relation consists of a lattice of uncoupled particles in a periodic background. This limit supports many trivial equilibrium states that persist as solutions of the model with small coupling. We investigate when a persisting solution generates a so-called lamination and prove that near the anti-continuum limit the collection of laminations of solutions is homeomorphic to the (N − 1)-dimensional simplex, with N the number of distinct local minima of the background potential. This generalizes a result by Baesens and MacKay on twist maps near an anti-integrable limit.

Topological horseshoes in travelling waves of discretized nonlinear wave equations

Applying the concept of anti-integrable limit to coupled map lattices originated from space-time discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatio-temporal chaos on the horseshoes.

Abrupt bifurcations in chaotic scattering: view from the anti-integrable limit**Dedicated to the memory of Leonid Pavlovich Shilnikov.

Bleher, Ott and Grebogi found numerically an interesting chaotic phenomenon in 1989 for the scattering of a particle in a plane from a potential field with several peaks of equal height. They claimed that when the energy E of the particle is slightly less than the peak height Ec there is a hyperbolic suspension of a topological Markov chain from which chaotic scattering occurs, whereas for E > Ec there are no bounded orbits. They called the bifurcation at E = Ec an abrupt bifurcation to chaotic scattering.The aim of this paper is to establish a rigorous mathematical explanation for how chaotic orbits occur via the bifurcation, from the viewpoint of the anti-integrable limit, and to do so for a general range of chaotic scattering problems.

Fermi Acceleration in Anti-Integrable Limits of the Standard Map

We consider a dynamical system on the semi-infinite cylinder which models the high energy dynamics of a family of mechanical models. We provide conditions under which we ensure that the set of orbits undergoing Fermi acceleration has measure zero.

FAMILY OF INVARIANT CANTOR SETS AS ORBITS OF DIFFERENTIAL EQUATIONS II: JULIA SETS

The Julia set of the quadratic map fμ(z) = μz(1 - z) for μ not belonging to the Mandelbrot set is hyperbolic, thus varies continuously. It follows that a continuous curve in the exterior of the Mandelbrot set induces a continuous family of Julia sets. The focus of this article is to show that this family can be obtained explicitly by solving the initial value problem of a system of infinitely coupled differential equations. A key point is that the required initial values can be obtained from the anti-integrable limit μ → ∞. The system of infinitely coupled differential equations reduces to a finitely coupled one if we are only concerned with some invariant finite subset of the Julia set. Therefore, it can be employed to find periodic orbits as well. We conduct numerical approximations to the Julia sets when parameter μ is located at the Misiurewicz points with external angle 1/2, 1/6, or 5/12. We approximate these Julia sets by their invariant finite subsets that are integrated along the reciprocal of corresponding external rays of the Mandelbrot set starting from the anti-integrable limit μ = ∞. When μ is at the Misiurewicz point of angle 1/128, a 98-period orbit of prescribed itinerary obtained by this method is presented, without having to find a root of a 298-degree polynomial. The Julia sets (or their subsets) obtained are independent of integral curves, but in order to make sure that the integral curves are contained in the exterior of the Mandelbrot set, we use the external rays of the Mandelbrot set as integral curves. Two ways of obtaining the external rays are discussed, one based on the series expansion (the Jungreis–Ewing–Schober algorithm), the other based on Newton's method (the OTIS algorithm). We establish tables comparing the values of some Misiurewicz points of small denominators obtained by these two algorithms with the theoretical values.

Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems

A rigorous numerical proof for establishing existence of a transversal homoclinic orbit for a saddle fixed point with higher dimensional unstable eigenspaces is presented. As the first component of this method, a shadowing theorem that guarantees the existence of such a homoclinic orbit near a suitable pseudo orbit given the invertibility of a certain Jacobian is proved. The second component consists of a refinement procedure for numerically computing a pseudo homoclinic orbit with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component verifies that the homoclinic orbit is transversal. In [6], they proved the existence of transversal homoclinic orbits near anti-integrable limits and near singularities for the Arneodo-Coullet-Tresser maps. In this paper, the existence of transversal homoclinic orbits were proved far away from anti-integrable limits and singularities for these maps.

On topological entropy of billiard tables with small inner scatterers

We present in this paper an approach to studying the topological entropy of a class of billiard systems. In this class, any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers. Combining the concept of anti-integrable limit with the theory of Lyusternik–Shnirel'man, we show that a billiard system in this class generically admits a set of non-degenerate anti-integrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily large topological entropy. The anti-integrable limit is the singular limit when scatterers shrink to points. In order to get around the singular limit and so as to apply the implicit function theorem, on auxiliary circles encircling these scatterers we define a length functional whose critical points are well-defined at the anti-integrable limit and give rise to billiard orbits when the scatterers are not points. Consequently, we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small.

Large Memory Capacity in Chaotic Artificial Neural Networks: A View of the Anti-Integrable Limit

In the literature, it was reported that the chaotic artificial neural network model with sinusoidal activation functions possesses a large memory capacity as well as a remarkable ability of retrieving the stored patterns, better than the conventional chaotic model with only monotonic activation functions such as sigmoidal functions. This paper, from the viewpoint of the anti-integrable limit, elucidates the mechanism inducing the superiority of the model with periodic activation functions that includes sinusoidal functions. Particularly, by virtue of the anti-integrable limit technique, this paper shows that any finite-dimensional neural network model with periodic activation functions and properly selected parameters has much more abundant chaotic dynamics that truly determine the model's memory capacity and pattern-retrieval ability. To some extent, this paper mathematically and numerically demonstrates that an appropriate choice of the activation functions and control scheme can lead to a large memory capacity and better pattern-retrieval ability of the artificial neural network models.

Anti-integrability for the Logistic Maps

The embedding of the Bernoulli shift into the logistic map x → μx(1 − x) for μ > 4 is reinterpreted by the theory of anti-integrability: it is inherited from the anti-integrable limit μ → ∞.

Diffusion speed in the Mather problem

We study the problem of energy growth in a nonautonomous periodic-in-time potential perturbation of a geodesic flow on a 2-dimensional torus. Using the method of the separatrix map and the idea of the anti-integrable limit, we show that there exists a solution such that the energy grows linearly with respect to time.

Topological horseshoes for perturbations of singular difference equations

In this paper, we study solutions of difference equations Φλ(yn, yn+1, …, yn+m) = 0, , of order m with parameter λ, and consider the case when Φλ has a singular limit depending on a single variable as λ → λ0, i.e. , where N is an integer with 0 ⩽ N ⩽ m and φ is a function. We prove that if φ has k simple zeros then for λ close enough to λ0, the difference equation has a k-horseshoe among its solutions, that is, the dynamics is conjugate to the full shift with k symbols. Moreover, we show that these horseshoes change continuously in the uniform topology as λ varies. As applications of these results, we establish the horseshoe structure in families of generalized Hénon-like maps and of Arneodo–Coullet–Tresser maps near their anti-integrable limits as well as in steady states for certain lattice models.

Smale horseshoe via the anti-integrability

Abstract The embedding of the Bernoulli shift into the horseshoe map can be viewed as it is inherited from the anti-integrable limit where the horseshoe has infinite contraction in one direction and infinite expansion in the other. At the limit the map is virtually a subshift of finite type with four symbols or equivalently a full shift with two symbols. We present an algebraic explanation for the equivalence of the two shifts.