Non-autonomous Discrete Systems Research Articles

A note on non-autonomous discrete dynamical systems

In this paper, we define some qualitative properties of non-autonomous discrete dynamical systems such as orbit shift continuum-wise expansivity, orbit shift persistence and orbit shift α-persistence. Then we discuss the relation between these notions and give necessary examples. Moreover, we prove that every continuum-wise expansive non-autonomous discrete system on a compact metric space is orbit shift continuum-wise expansive.

N-Convergence and Chaotic Properties of Non-autonomous Discrete Systems

N-Convergence and Chaotic Properties of Non-autonomous Discrete Systems

A discrete second-order Hamiltonian system with asymptotically linear conditions

<abstract><p>This paper deals with a non-autonomous discrete second-order Hamiltonian system under asymptotically linear conditions. The existence of a periodic solution is obtained via the saddle point theorem.</p></abstract>

Collective Sensitivity and Collective Accessibility of Non-Autonomous Discrete Dynamical Systems

The concepts of collectively accessible, collectively sensitive, collectively infinitely sensitive, and collectively Li–Yorke sensitive are defined in non-autonomous discrete systems. It is proved that, if the mapping sequence h1,∞=(h1,h2,…) is W-chaotic, then hn,∞=(hn,hn+1,…)(∀n∈N={1,2,…}) would also be W-chaotic. W-chaos represents one of the following five properties: collectively accessible, sensitive, collectively sensitive, collectively infinitely sensitive, and collectively Li–Yorke sensitive. Then, the relationship of chaotic properties between the product system (H1×H2,f1,∞×g1,∞) and factor systems (H1,f1,∞) and (H2,g1,∞) was presented. Furthermore, in this paper, it is also proved that, if the autonomous discrete system (X,h^) induced by the p-periodic discrete system (H,h1,∞) is W-chaotic, then the p-periodic discrete system (H,f1,∞) would also be W-chaotic.

Chaos in g-Non Autonomous discrete systems

In this work, we generalized some chaotic propositions in space more general than non-autonomous discrete systems for new system we call it a g-non autonomous discrete systems.

Sensitivity of Uniformly Convergent Mapping Sequences in Non-Autonomous Discrete Dynamical Systems

Let H be a compact metric space. The metric of H is denoted by d. And let (H,f1,∞) be a non-autonomous discrete system where f1,∞={fn}n=1∞ is a mapping sequence. This paper discusses infinite sensitivity, m-sensitivity, and m-cofinitely sensitivity of f1,∞. It is proved that, if fn(n∈N) are feebly open and uniformly converge to f:H→H, fi∘f=f∘fi for any i∈{1,2,…}, and ∑i=1∞D(fi,f)<∞, then (H,f) has the above sensitive property if and only if (H,f1,∞) has the same property where D(·,·) is the supremum metric.

Some Sufficient Conditions Under Which a Non-Autonomous Discrete System or Semi-Flow is Sensitive

Some Sufficient Conditions Under Which a Non-Autonomous Discrete System or Semi-Flow is Sensitive

Reiterative Distributional Chaos in Non-autonomous Discrete Systems

Reiterative Distributional Chaos in Non-autonomous Discrete Systems

The dynamics of nonautonomous dynamical systems with the large deviations theorem

Let be a nonautonomous dynamical system and μ be a Borel measure on X with a full support. The purpose of this paper is to introduce the concepts of large deviations theorem and expansive measure for nonautonomous discrete systems and investigate the dynamics of nonautonomous dynamical systems with the large deviations theorem. Actually, it is proved that is ergodicaly sensitive if the pair satisfies the large deviations theorem and is topologically strongly ergodic; is topologically ergodic if the pair satisfies the large deviations theorem in a sequence of positive integers; μ is expansive if the pair satisfies the large deviations theorem and is topologically strongly ergodic and each measurable set with positive measure with respect to μ has a nonempty interior.

Chaotic properties of a class of coupled mapping lattice induced by fuzzy mapping in non-autonomous discrete systems

In this paper, fuzzy mapping is selected to study the coupled system in non-autonomous systems. The following conclusions are obtained: (1) The fuzzy system (F(X),{f˜n}n=1∞) is P1-chaos, then the coupled system (1–3) is P1 chaos. Where P1-chaos is one of the followings: (F1,F2)-Chaos, Li–Yorke chaos, distributional chaos, spatio-temporal chaos, densely δ-chaos, densely chaos, Ruelle-Takens chaos, Kato’s chaos. (2) The fuzzy system (F(X),{f˜n}n=1∞) is P2- sensitive, then the coupled system (1–3) is P2-sensitive. Where P2-sensitive is one of the followings: sensitive, Li–Yorke sensitive, densely Li–Yorke sensitive, infinitely sensitive, syndetically sensitive, cofinitely sensitive, F-sensitive, (F1,F2)-sensitive. (3) The fuzzy system (F(X),{f˜n}n=1∞) is transitive, exactness or accessible, then the coupled system (1–3) is transitive, exactness or accessible.

Functional Method of Localization and LaSalle Invariance Principle

A functional method of localization has proved to be good in solving the qualitative analysis problems of dynamic systems. Proposed in the 90s, it was intensively used when studying a number of well-known systems of differential equations, both of autonomous and of non-autonomous discrete systems, including systems that involve control and / or disturbances.The method essence is to construct a set containing all invariant compact sets in the phase space of a dynamical system. A concept of the invariant compact set includes equilibrium positions, limit cycles, attractors, repellers, and other structures in the phase space of a system that play an important role in describing the behavior of a dynamical system. The constructed set is called localizing and represents an external assessment of the appropriate structures in the phase space.Relatively recently, it was found that the functional localization method allows one to analyze a behavior of the dynamical system trajectories. In particular, the localization method can be used to check the stability of the equilibrium positions.Here naturally emerges an issue of the relationship between the functional localization method and the well-known La Salle invariance principle, which can be regarded as a further development of the method of Lyapunov functions for establishing stability. The article discusses this issue.

Generalizations of Expansiveness in Non-Autonomous Discrete Systems

In this paper, we define and study generalizations of expansiveness, namely n- expansiveness, \(\aleph _0\)-expansiveness, continuum-wise expansive and meagre expansiveness for non-autonomous discrete dynamical systems. We discuss various properties of such non-autonomous systems and give necessary examples. We prove results related to non-existence of \(\aleph _0\)-expansive and meagre expansive non-autonomous system on certain spaces. We also study relation between \(\aleph _0\)-expansive and meagre-expansive non-autonomous systems.

Template Iterations of Quadratic Maps and Hybrid Mandelbrot Sets

As a particular problem within the field of non-autonomous discrete systems, we consider iterations of two quadratic maps \(f_{c_0}=z^2+c_0\) and \(f_{c_1}=z^2+c_1\), according to a prescribed binary sequence, which we call a template. We study the asymptotic behavior of the critical orbits and define the Mandelbrot set in this case as the locus for which these orbits are bounded. However, unlike in the case of single maps, this concept can be understood in several ways. For a fixed template, one may consider this locus as a subset of the parameter space in \((c_0,c_1) \in \mathbb {C}^2\); for fixed quadratic parameters, one may consider the set of templates which produce a bounded critical orbit. In this paper, we consider both situations and hybrid combinations of them, and we study basic topological properties of these sets and interpret them in light of potential applications.

Chaotic Properties in the Sense of Furstenberg Families in Set-Valued Discrete Dynamical Systems

Let be a non-autonomous discrete system and be a set-valued discrete system induced by it. Where, is the space formed by all non-empty compact subsets of X endowed with the Hausdorff metric H, is a set-valued mapping sequence induced by . It is proved that is -chaos, then is -chaos. Where -chaos is denoted to -sensitive, -sensitive, -transitive, -accessible, -weakly mixing, -m-sensitive, infinitely sensitive, or syndetically transitive.

Some Kinds of Distributional Chaos for Non-Autonomous Discrete Systems

Some Kinds of Distributional Chaos for Non-Autonomous Discrete Systems

Specification properties for non-autonomous discrete systems

In this paper notions of strong specification property and quasi-weak specification property for non-autonomous discrete systems are introduced and studied. It is shown that these properties are dynamical properties and are preserved under finite product. It is proved that a k-periodic non-autonomous system on intervals having weak specification is Devaney chaotic whereas if the system has strong specification then the result is true in general. Specification properties of induced systems on hyperspaces and probability measures spaces are also studied. Examples/counter examples are provided wherever necessary to support results obtained.

Multi-transitivity in non-autonomous discrete systems

We introduce and study the notions of multi-transitivity and thick transitivity for non-autonomous discrete dynamical systems. We obtain a sufficient condition under which in minimal non-autonomous discrete dynamical systems, multi-transitivity, thick transitivity and weakly mixing of all orders are equivalent. Counterexamples are constructed justifying that certain results related to above concepts which are true in autonomous discrete dynamical systems need not be true in non-autonomous discrete dynamical systems.

On stability of nonlinear nonautonomous discrete fractional Caputo systems

Conditions to establish Mittag-Leffler stability of solutions for nonlinear nonautonomous discrete Caputo-like fractional systems just from the linear associated system is shown. Mittag-Leffler stability for linear systems is tackled pointing out properties the matrix must satisfy. Additionally features on solutions for linear systems are included in order to establish the equivalence between uniform asymptotical stability and exponential stability. Converse-like-Lyapunov Theorem is developed and a particular result to show Mittag-Leffler stability from a condition on a linear part of the full system is worked.

Some criteria of chaos in non-autonomous discrete dynamical systems

ABSTRACTThis paper establishes several criteria for strong Li–Yorke chaos and distributional chaos in non-autonomous discrete dynamical systems. The main criterion of distributional δ-chaos for some is induced by weak A-coupled-expansion in a sequence of nonempty compact subsets, where A is an irreducible transition matrix with at least one row sum larger than 1. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts. One example of a non-autonomous logistic system is provided for illustration.

Topological Conjugacy Between Induced Non-autonomous Set-Valued Systems and Subshifts of Finite Type

This paper establishes topological (equi-)semiconjugacy and (equi-) conjugacy between induced non-autonomous set-valued systems and subshifts of finite type. First, some necessary and sufficient conditions are given for a non-autonomous discrete system to be topologically semiconjugate or conjugate to a subshift of finite type. Further, several sufficient conditions for it to be topologically equi-semiconjugate or equi-conjugate to a subshift of finite type are obtained. Consequently, estimations of topological entropy and several criteria of Li–Yorke chaos and distributional chaos in a sequence are derived. Second, the relationships of several related dynamical behaviors between the non-autonomous discrete system and its induced set-valued system are investigated. Based on these results, the paper furthermore establishes the topological (equi-)semiconjugacy and (equi-)conjugacy between induced set-valued systems and subshifts of finite type. Consequently, estimations of the topological entropy for the induced set-valued system are obtained, and several criteria of Li–Yorke chaos and distributional chaos in a sequence are established. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts in the literature. Two examples are finally provided for illustration.