scholarly journals Hidden Attractors in Discrete Dynamical Systems

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 616
Author(s):  
Marek Berezowski ◽  
Marcin Lawnik
Keyword(s):  
Dynamical Systems ◽  
Chaos Theory ◽  
Scientific Literature ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Hidden Attractors ◽  
Negative Effects ◽  
Chemical Reactors ◽  
Points Of View

Research using chaos theory allows for a better understanding of many phenomena modeled by means of dynamical systems. The appearance of chaos in a given process can lead to very negative effects, e.g., in the construction of bridges or in systems based on chemical reactors. This problem is important, especially when in a given dynamic process there are so-called hidden attractors. In the scientific literature, we can find many works that deal with this issue from both the theoretical and practical points of view. The vast majority of these works concern multidimensional continuous systems. Our work shows these attractors in discrete systems. They can occur in Newton’s recursion and in numerical integration.

NUMERICAL COMPUTATIONS OF CONNECTING ORBITS IN DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (07) ◽  
pp. 1281-1293 ◽  
Author(s):  
FENGSHAN BAI ◽  
GABRIEL J. LORD ◽  
ALASTAIR SPENCE
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Discrete System ◽  
Numerical Technique ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Connecting Orbits

The aim of this paper is to present a numerical technique for the computation of connections between periodic orbits in nonautonomous and autonomous systems of ordinary differential equations. First, the existence and computation of connecting orbits between fixed points in discrete dynamical systems is discussed; then it is shown that the problem of finding connections between equilibria and periodic solutions in continuous systems may be reduced to finding connections between fixed points in a discrete system. Implementation of the method is considered: the choice of a linear solver is discussed and phase conditions are suggested for the discrete system. The paper concludes with some numerical examples: connections for equilibria and periodic orbits are computed for discrete systems and for nonautonomous and autonomous systems, including systems arising from the discretization of a partial differential equation.

scholarly journals Strong Li–Yorke Chaos for Time-Varying Discrete Dynamical Systems with A-Coupled-Expansion

2015 ◽  
Vol 25 (13) ◽  
pp. 1550186 ◽  
Author(s):  
Hua Shao ◽  
Yuming Shi ◽  
Hao Zhu
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Strong Sense ◽  
Time Varying ◽  
Transition Matrices ◽  
Complete Metric Spaces ◽  
Compact Subsets

This paper is concerned with strong Li–Yorke chaos induced by [Formula: see text]-coupled-expansion for time-varying (i.e. nonautonomous) discrete systems in metric spaces. Some criteria of chaos in the strong sense of Li–Yorke are established via strict coupled-expansions for irreducible transition matrices in bounded and closed subsets of complete metric spaces and in compact subsets of metric spaces, respectively, where their conditions are weaker than those in the existing literature. One example is provided for illustration.

Non-stationary dynamic system characteristics recovery from three test signals

2020 ◽  
pp. 9-15
Author(s):  
Ilia V. Boikov ◽  
Nikolay P. Krivulin
Keyword(s):  
Dynamical Systems ◽  
Impulse Response ◽  
Discrete Systems ◽  
Exact Formula ◽  
Algebraic Expression ◽  
Continuous Systems ◽  
Exact Reconstruction ◽  
Input Signals ◽  
The Laplace Transform ◽  
System Output

Algorithms of exact restoration in an analytical form of dynamic characteristics of non-stationary dynamic systems are constructed. Non-stationary continuous dynamical systems modeled by Volterra integral equations of the first kind and non-stationary discrete dynamical systems modeled by discrete analogues of Volterra integral equations of the first kind are considered.The article consists of an introduction and three sections: 1) The exact restoration of the dynamic characteristics of continuous systems, 2) The restoration of the transition characteristics of discrete systems, 3) Conclusions. The introduction provides a statement of the problem and provides an overview of dynamical systems for which algorithms for exact reconstruction in ananalytical form of the impulse response (in the case of continuous systems) and the transition characteristic (in the case of discrete systems) are constructed. In the first section, the algorithm is constructed for the exact reconstruction of the impulse response of an non-stationary continuous dynamic system from three interconnected input signals. The first signal may be arbitrary, the second and third signals are associated with the first signal by integral operator. The exact formula for the Laplace transform of the impulse response, represented by an algebraic expression from the Laplace transform of the system output signals, is given. A model example illustrating the effectiveness of the algorithm is given. The practical application of the presented algorithm isdiscussed. In the second section, an algorithm is constructed for the exact reconstruction of the transition response of a non-stationary discrete dynamical system from three input signals that are interconnected. The first signal may be arbitrary, the second and third signals are associated with the first summing operator. The exact formula of the Z-transform of the transition characteristic is presented, which is represented by an algebraic expression from the Z-transform of the system output signals. A model example is given. The “Conclusions” section provides a summary of the results presented in the article and describes the dynamic systems to which the proposed algorithms can be extended.

Controlling Hidden Dynamics and Multistability of a Class of Two-Dimensional Maps via Linear Augmentation

2021 ◽  
Vol 31 (03) ◽  
pp. 2150047
Author(s):  
Liping Zhang ◽  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Qinsheng Bi
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Coupling Strength ◽  
Complex Dynamics ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Hidden Attractor ◽  
Hidden Attractors

This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point, are studied. Our numerical simulations show the effectiveness of the linear augmentation method. As the coupling strength of the controller increases or decreases, hidden attractor can be annihilated or altered to be self-excited, and multistability of the map can be controlled to being bistable or monostable.

A FAMILY OF COMPLETELY PERIODIC QUADRATIC DISCRETE DYNAMICAL SYSTEM

2008 ◽  
Vol 18 (05) ◽  
pp. 1425-1433 ◽  
Author(s):  
MILAN KUTNJAK ◽  
MATEJ MENCINGER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Dynamical System ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Nonassociative Algebras ◽  
Rank 2 ◽  
The Basin Of Attraction

There is a one-to-one correspondence between homogeneous quadratic dynamical systems and commutative (possibly nonassociative) algebras. The corresponding theory for continuous systems is well known (c.f. [Markus, 1960; Walcher, 1991; Kinyon & Sagle, 1995]). In this paper the dynamics on the boundary of the basin of attraction of the origin, ∂ B Att (0), in homogeneous quadratic discrete dynamical systems is considered. In particular, we consider the dynamical behavior in a family of systems corresponding to a family of algebras [Formula: see text] which admits nilpotents of rank 2 and idempotents. The complete periodicity of a system (and the corresponding algebra) is defined and it is proven that for every n > 2 there are some systems/algebras from [Formula: see text] which are on ∂ BAtt(0) completely periodic with period n. The dynamics on ∂ B Att (0) is considered via a special class of polynomials Pn, n ∈ ℕ ∪ {0, -1}, recursively defined by Pn(α) = 2αPn-2(α) + Pn-1(α); P-1(α) = 0, P0(α) = 1, n ∈ ℕ.

PARAMETER CHARACTERISTICS FOR STABLE AND UNSTABLE SOLUTIONS IN NONLINEAR DISCRETE DYNAMICAL SYSTEMS

2010 ◽  
Vol 20 (10) ◽  
pp. 3173-3191 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
YU GUO
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Hénon Map ◽  
Henon Map ◽  
Mapping Structures ◽  
Stable Periodic ◽  
Unstable Solutions ◽  
Nonlinear Discrete Systems

This paper studies complete stable and unstable periodic solutions for n-dimensional nonlinear discrete dynamical systems. The positive and negative iterative mappings of discrete systems are used to develop mapping structures of the stable and unstable periodic solutions. The complete bifurcation and stability analysis are presented for the stable and unstable periodic solutions which are based on the positive and negative mapping structures. A comprehensive investigation on the Henon map is carried out for a better understanding of complexity in nonlinear discrete systems. Given is the bifurcation scenario based on positive and negative mappings of the Henon map, and the analytical predictions of the corresponding periodic solutions are achieved. The corresponding eigenvalue analysis of the periodic solutions is presented. The Poincare mapping sections relative to the Neimark bifurcations of periodic solutions are presented. A parameter map for periodic and chaotic solutions is developed. The complete unstable and stable periodic solutions in nonlinear discrete systems are presented for the first time. The results presented in this paper provide a new idea for one to rethink the current existing theories.

TIME-VARYING PERTURBATIONS OF CHAOTIC DISCRETE SYSTEMS

2012 ◽  
Vol 22 (03) ◽  
pp. 1250066 ◽  
Author(s):  
LIJUAN ZHANG ◽  
YUMING SHI
Keyword(s):  
Dynamical Systems ◽  
Banach Spaces ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Small Time ◽  
Strong Sense ◽  
Time Varying ◽  
Perturbed System ◽  
Time Varying Systems

This paper is concerned with time-varying discrete dynamical systems in Banach spaces. A criterion of chaos induced by coupled-expansion for time-varying systems is first established and then the persistence of coupled-expansion is considered for time-varying systems under small time-varying perturbations, under which the perturbed system is shown chaotic in the strong sense of Li–Yorke. By applying this result, a map with a regular and nondegenerate snap-back repeller is shown to be still chaotic in the strong sense of Li–Yorke under small time-varying perturbations.

scholarly journals On problems of Topological Dynamics in non-autonomous discrete systems

2016 ◽  
Vol 1 (2) ◽  
pp. 391-404 ◽  
Author(s):  
Francisco Balibrea
Keyword(s):  
Dynamical Systems ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Topological Dynamics ◽  
Open Questions

AbstractMost of problems in Topological Dynamics in the theory of general autonomous discrete dynamical systems have been addressed in the non-autonomous setting. In this paper we will review some of them, giving references and stating open questions.

STABLE, ALMOST STABLE AND ODD POINTS OF DYNAMICAL SYSTEMS

2017 ◽  
Vol 96 (2) ◽  
pp. 245-255 ◽  
Author(s):  
RYSZARD J. PAWLAK ◽  
ANNA LORANTY
Keyword(s):  
Dynamical Systems ◽  
Chaos Theory ◽  
Additional Assumption ◽  
Discrete Dynamical Systems ◽  
Unit Interval ◽  
Stable Points

We consider stable and almost stable points of autonomous and nonautonomous discrete dynamical systems defined on the closed unit interval. Our considerations are associated with chaos theory by adding an additional assumption that an entropy of a function at a given point is infinite.

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