Skew-product Systems Research Articles

Topological synchronisation or a simple attractor?

A few recent papers introduced the concept of topological synchronisation. We refer in particular to (Lahav et al 2022 Sci. Rep. 12 2508), where the theory was illustrated by means of a skew product system, coupling two logistic maps. In this case, we show that the topological synchronisation could be easily explained as the birth of an attractor for increasing values of the coupling strength and the mutual convergence of two marginal empirical measures. Numerical computations based on a careful analysis of the Lyapunov exponents suggest that the attractor supports an absolutely continuous physical measure (acpm). We finally show that for some unimodal maps such acpm exhibit a multifractal structure.

On Katznelson’s Question for skew-product systems

Katznelson’s Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelson’s Question for certain towers of skew-product extensions of equicontinuous systems, including systems of the form ( x , t ) ↦ ( x + α , t + h ( x ) ) (x,t) \mapsto (x + \alpha , t + h(x)) . We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers.

SOME SYSTEMS WITH <i>C</i><sub>1</sub> REGULARITY AND ONLY NEGATIVE LYAPUNOV EXPONENTS

<abstract> In this paper, we prove that for a $C^{1}$ diffeomorphism preserving an ergodic measure $\mu$ with only negative Lyapunov exponents, the support set of $\mu$ is a periodic orbit. For a skew product system preserving an ergodic measure with only negative fiberwise exponents, whose fiber maps are $C^{1}$ diffeomorphisms, we get that for almost all fibers, the disintegration of this measure on fibers is supported on finitely many points. </abstract>

The multi-transitivity of free semigroup actions

In this paper, we introduce the conceptions of multi-transitivity, [Formula: see text]-transitivity and [Formula: see text]-mixing property for free semigroup actions and give some equivalent conditions for a free semigroup action to be multi-transitive, multi-transitive with respect to vectors and strongly multi-transitive, respectively. For instance, we prove that a free semigroup action is multi-transitive or multi-transitive with respect to a vector if and only if its corresponding skew product system is multi-transitive or multi-transitive with respect to the same vector.

Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations

Linear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as~well as on a space of $p$-summable functions. The main result states that in both cases, the Lyapunov exponents are identical, and that the Oseledets decompositions are related by natural embeddings.

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.

Invariant graphs for chaotically driven maps

This paper investigates the geometrical structures of invariant graphs of skew product systems of the form driven by a hyperbolic base map (e.g. a baker map or an Anosov surface diffeomorphism) and with monotone increasing fibre maps having negative Schwarzian derivatives. We recall a classification, with respect to the number and to the Lyapunov exponents of invariant graphs, for this class of systems. Our major goal here is to describe the structure of invariant graphs and study the properties of the pinching set, the set of points where the values of all of the invariant graphs coincide. In Keller and Otani (2018 Stoch. Dyn. 18 1850009), the authors studied skew product systems driven by a generalized baker map with the restrictive assumption that depend on only through the stable coordinate x of θ. Our aim is to relax this assumption and construct a fibre-wise conjugation between the original system and a new system for which the fibre maps depend only on the stable coordinate of the drive. As an application of this construction we prove that, when S is an Anosov diffeomorphism, the pinching set is a union of global unstable fibres for S.

The variation of invariant graphs in forced systems.

In skew-product systems with contractive factors, all orbits asymptotically approach the graph of the so-called sync function; hence, the corresponding regularity properties primarily matter. In the literature, sync function Lipschitz continuity and differentiability have been proved to hold depending on the derivative of the base reciprocal, if not on its Lyapunov exponent. However, forcing topological features can also impact the sync function regularity. Here, we estimate the total variation of sync functions generated by one-dimensional Markov maps. A sharp condition for bounded variation is obtained depending on parameters, which involves the Markov map topological entropy. The results are illustrated with examples.

Synchronization in Minimal Iterated Function Systems on Compact Manifolds

We treat synchronization for iterated function systems generated by diffeomorphisms on compact manifolds. Synchronization here means the convergence of orbits starting at different initial conditions when iterated by the same sequence of diffeomorphisms. The iterated function systems admit a description as skew product systems of diffeomorphisms on compact manifolds driven by shift operators. Under open conditions including transitivity and negative fiber Lyapunov exponents, we prove the existence of a unique attracting invariant graph for the skew product system. This explains the occurrence of synchronization. The result extends previous results for iterated function systems by diffeomorphisms on the circle, to arbitrary compact manifolds.

Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus

In this paper we address the existence and ergodicity of non-uniformly hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems have formulation as a skew product system defined by planar diffeomorphisms, with average contraction condition, forced by any expanding circle map. These attractors are invariant graphs of upper semicontinuous maps which support exactly one physical measure. In our approach, these skew product systems arising from iterated function systems which are generated by finitely many weak contractive diffeomorphisms. Under some conditions including negative fiber Lyapunov exponents, we prove the existence of unique non-uniformly hyperbolic attracting invariant graphs for these systems which attract positive orbits of almost all initial points. Also, we prove that these systems are Bernoulli and therefore they are mixing. Moreover, these properties remain true under small perturbations in the space of endomorphisms on the solid torus.

Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors

Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins. To quantify the degree of intermingledness the uncertainty exponent and the stability index were suggested by various authors and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.

Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems

We consider families of fast-slow skew product maps of the form \begin{align*} x_{n+1} = x_n+\epsilon a(x_n,y_n,\epsilon), \quad y_{n+1} = T_\epsilon y_n, \end{align*} where $T_\epsilon$ is a family of nonuniformly expanding maps, and prove averaging and rates of averaging for the slow variables $x$ as $\epsilon\to0$. Similar results are obtained also for continuous time systems \begin{align*} \dot x = \epsilon a(x,y,\epsilon), \quad \dot y = g_\epsilon(y). \end{align*} Our results include cases where the family of fast dynamical systems consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.

Random interval diffeomorphisms

We consider a class of step skew product systems of interval diffeo-morphisms over shift operators, as a means to study random compositions of interval diffeomorphisms. The class is chosen to present in a simplified setting intriguing phenomena of intermingled basins, master-slave synchronization and on-off intermittency. We provide a self-contained discussion of these phenomena.

Usc/fibred entropy structure and applications

ABSTRACTFor a given metrizable space X, we study continuity properties of the entropy as function not only of the measure but also of the dynamical system on X. We introduce the notion of robust tail entropy, which implies upper semicontinuity of the topological entropy but also stability of measures of maximal entropy (when the topological entropy is continuous). This gives, inter alia, simple proofs of results of Misiurewicz and Raith for multimodal interval maps. We also consider fibred entropy structures which allow us to investigate the symbolic extensions of (smooth) skew-product systems.

Amorphic complexity

We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For instance, it gives positive value to Denjoy examples on the circle and Sturmian subshifts, while being zero for all isometries and Morse–Smale systems.After discussing basic properties and examples, we show that amorphic complexity and the underlying asymptotic separation numbers can be used to distinguish almost automorphic minimal systems from equicontinuous ones. For symbolic systems, amorphic complexity equals the box dimension of the associated Besicovitch space. In this context, we concentrate on regular Toeplitz flows and give a detailed description of the relation to the scaling behaviour of the densities of the p-skeletons. Finally, we take a look at strange non-chaotic attractors appearing in so-called pinched skew product systems. Continuous-time systems, more general group actions and the application to cut and project quasicrystals will be treated in subsequent work.

A model for the nonautonomous Hopf bifurcation

Inspired by an example of Grebogi et al (1984 Physica D 13 261–8), we study a class of model systems which exhibit the full two-step scenario for the nonautonomous Hopf bifurcation, as proposed by Arnold (1998 Random Dynamical Systems (Berlin: Springer)). The specific structure of these models allows a rigorous and thorough analysis of the bifurcation pattern. In particular, we show the existence of an invariant ‘generalised torus’ splitting off a previously stable central manifold after the second bifurcation point.The scenario is described in two different settings. First, we consider deterministically forced models, which can be treated as continuous skew product systems on a compact product space. Secondly, we treat randomly forced systems, which lead to skew products over a measure-preserving base transformation. In the random case, a semiuniform ergodic theorem for random dynamical systems is required, to make up for the lack of compactness.

Principal Lyapunov Exponents and Principal Floquet Spaces of Positive Random Dynamical Systems. III. Parabolic Equations and Delay Systems

This is the third part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors’ paper Mierczyński and Shen (Trans Am Math Soc 365(10):5329–5365, 2013), to positive continuous-time random dynamical systems on infinite dimensional ordered Banach spaces arising from random parabolic equations and random delay systems. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by random parabolic equations and by cooperative systems of linear delay differential equations admit measurable families of generalized principal Floquet subspaces, and generalized principal Lyapunov exponents.

Stability index for chaotically driven concave maps

We study skew product systems driven by a hyperbolic base map S ^ : Θ → Θ (for example, a baker map or an Anosov surface diffeomorphism) and with simple concave fibre maps on R + like x ↦ g ^ ( θ ) arctan ( x ) , where θ ∈ Θ is a parameter driven by the base map. The fibrewise attractor is the graph of an upper semicontinuous function θ ↦ φ ^ ∞ ( θ ) ∈ R + . For many choices of g ^ , φ ^ ∞ has a residual set of zeros, but φ ^ ∞ > 0 μ SRB -almost surely, where μ SRB is the Sinai–Ruelle–Bowen measure of S ^ - 1 . In such situations, we evaluate the stability index of the global attractor of the system, which is the subgraph { ( θ , x ) ∈ Θ × R + : 0 ⩽ x ⩽ φ ^ ∞ ( θ ) } of φ ^ ∞ , at all regular points ( θ , 0 ) in terms of the local exponents Γ ^ ( θ ) : = lim n → ∞ ( 1 / n ) log g ^ n ( θ ) and Λ ^ ( θ ) : = lim n → ∞ ( 1 / n ) log | D u S ^ - n ( θ ) | and of the positive zero s * of a certain thermodynamic pressure function associated with S ^ and g ^ . (In queuing theory, an analogue of s * is known as Loynes’ exponent [R. M. Loynes, ‘The stability of a queue with non-independent inter-arrival and service times’, Proc. Cambridge Philos. Soc. 58 (1962) 497–520].) The stability index was introduced by Podvigina and Ashwin [‘On local attraction properties and a stability index for heteroclinic connections’, Nonlinearity 24 (2011) 887–929.] to quantify the local scaling of basins of attraction.

On the invariant density of the random β-transformation

We construct a Lebesgue measure preserving natural extension of a skew product system related to the random β-transformation K β . This allows us to give a formula for the density of the absolutely continuous invariant probability measure of K β , answering a question of Dajani and de Vries, and also to evaluate some estimates on the typical branching rate of the set of β-expansions of a real number.

Coupled skinny baker's maps and the Kaplan–Yorke conjecture

The Kaplan–Yorke conjecture states that for ‘typical’ dynamical systems with a physical measure, the information dimension and the Lyapunov dimension coincide. We explore this conjecture in a neighborhood of a system for which the two dimensions do not coincide because the system consists of two uncoupled subsystems. We are interested in whether coupling ‘typically’ restores the equality of the dimensions. The particular subsystems we consider are skinny baker's maps, and we consider uni-directional coupling. For coupling in one of the possible directions, we prove that the dimensions coincide for a prevalent set of coupling functions, but for coupling in the other direction we show that the dimensions remain unequal for all coupling functions. We conjecture that the dimensions prevalently coincide for bi-directional coupling. On the other hand, we conjecture that the phenomenon we observe for a particular class of systems with uni-directional coupling, where the information and Lyapunov dimensions differ robustly, occurs more generally for many classes of uni-directionally coupled systems (also called skew-product systems) in higher dimensions.