Multi-transitivity in non-autonomous discrete systems

Sensitivity of non-autonomous discrete dynamical systems

This paper focuses on chaos induced by weak A-coupled-expansion of non-autonomous discrete systems in compact subsets of metric spaces and in bounded and closed subsets of complete metric spaces, separately. A new concept of weak A-coupled-expansion for non-autonomous discrete systems, whose condition is weaker than that of A-coupled-expansion, is introduced, and several new criteria of chaos induced by weak A-coupled-expansion of non-autonomous discrete systems are established. By applying some close relationships between chaotic dynamical behaviours of the original system and its induced systems, two criteria of chaos are established. One example is provided for illustration.

Let (Y, d) be a nontrivial metric space and (Y, g1,∞) be a nonautonomous discrete dynamical system given by sequences $(g_{l})_{l = 1}^{\infty }$ of continuous maps gl : Y → Y and let $\mathcal {F}$, $\mathcal {F}_{1}$ and $\mathcal {F}_{2}$ be given shift-invariant Furstenberg families. In this paper, we study stronger forms of transitivity and sensitivity for nonautonomous discrete dynamical systems by using Furstenberg family. In particular, we discuss the $\mathcal {F}$-transitivity, $\mathcal {F}$-mixing, $\mathcal {F}$-sensitivity, $\mathcal {F}$-collective sensitivity, $\mathcal {F}$-synchronous sensitivity, $(\mathcal {F}_{1},\mathcal {F}_{2})$-sensitivity and $\mathcal {F}$-multi-sensitivity for the system (Y, g1,∞) and show that under the conditions that gj is semi-open and satisfies gj ∘ g = g ∘ gj for each j ∈ {1, 2, ⋯ } and that $$\sum\limits_{j = 1}^{\infty}D(g_{j},g) $$exists (i.e., $\sum \limits _{j = 1}^{\infty }D(g_{j},g)<+\infty $), the following hold: The above results extend the existing ones.

We introduce the Ellis semigroup of a nonautonomous discrete dynamical system (X, f1,∞) when X is a metric compact space. The underlying set of this semigroup is the pointwise closure of in the space XX. By using the convergence of a sequence of points with respect to an ultrafilter it is possible to give a precise description of the semigroup and its operation. This notion extends the classical Ellis semigroup of a discrete dynamical system. We show several properties that connect this semigroup and the topological properties of the nonautonomous discrete dynamical system.

Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps

In this article, finite nonautonomous discrete event systems are studied in the dioid of formal series. If such a system can be described by a timed event Petri net, a max algebra model of the system can be written. Furthermore, the input-output relation of the nonautonomous system appears in the dioid of formal series. The greatest subsolution of a system of linear equations in the dioid of formal series is found. This subsolution can be used to calculate the latest admissible input of the nonautonomous discrete event system, such that the output of the system is achieved. The application of the theory to an assembly line is given.

The aim of this paper is to present some recent results concerning non-autonomous discrete systems (non-autonomous order one difference equations) and state some problems that arise in this field.

On stability of nonlinear nonautonomous discrete fractional Caputo systems

Weak stability of non-autonomous discrete dynamical systems

We consider homoclinic orbits in non-autonomous discrete time dynamical systems of the form where it is assumed that an n independent fixed point exists. A numerical method for computing finite approximations of transversal homoclinic orbits is introduced and a detailed error analysis is presented. The non-autonomous setup requires special tools. We prove that the analytic condition of transversality of the orbit corresponds to a transversal intersection of the corresponding invariant fiber bundles. The approximation method and the validity of the error estimate is illustrated by an example.

In this chapter, the great sensitivity of nonlinear systems, and especially of chaotic systems, on the initial conditions and on the variation of their parameters, was used to design a novel image encryption scheme. Until now, a great number of chaotic autonomous continuous systems or discrete dynamical systems, have been used in various image encryption processes, as a source of random numbers. However, in this work, a Chaotic Random Bit Generator (CRBG), which is based on a non-autonomous dynamical system, is used. For ridding from the system the influence of the external source and increasing the security of the proposed generator, the Poincar\(\acute{e}\) section for sampling the signal has been used. As a dynamical system, the very well-known Duffing–van der Pol system has been chosen, presenting very good statistical results. The aforementioned CRBG is the “heart” of the proposed image encryption scheme. Finally, the security analysis of the proposed encryption scheme, based on histogram analysis, correlation of two adjacent pixels, differential analysis and information entropy, demonstrate the robustness of the proposed chaotic encryption scheme against all kinds of statistical, cryptanalytic, and brute-force attacks.

Some weak versions of distributional chaos in non-autonomous discrete systems

By using the variant fountain theorem, we study the existence of periodic solutions for a class of superquadratic non-autonomous second-order discrete Hamiltonian systems.

Let be a nonautonomous dynamical system. In this paper, we summarize known definitions of periodic points for general nonautonomous dynamical systems and propose a new definition of asymptotic periodicity. This definition is not only very natural but also resistant to changes of the beginning of the sequence generating the nonautonomous system. We show the relations among these definitions and discuss their properties. We prove that for pointwise convergent nonautonomous systems topological transitivity together with a dense set of asymptotically periodic points imply sensitivity. We also show that even for uniformly convergent systems, the nonautonomous analogue of Sharkovsky's theorem is not valid for most definitions of periodic points.

In this paper we give the complete description of the structure of compact global (forward) attractors for non-autonomous perturbations of discrete autonomous gradient-like dynamical systems under the assumption that the original discrete autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed non-autonomous (in particular -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular the perturbed systems has the same number of -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff solutions). It is shown that the compact global (forward) attractor of non-autonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections.