Chaotic behavior of discrete-time linear inclusion dynamical systems

Chaotic behavior is a term that is attributed to dynamical systems whose solutions are highly sensitive to initial conditions. This means that small perturbations in the initial conditions can lead to completely different trajectories in the solution space. These types of chaotic dynamical systems arise in various natural or artificial systems in biology, circuits, engineering, computer science, and more. This chapter reports on some new chaotic discrete time two-dimensional maps that are derived from simple modifications to the well-known Hénon, Lozi, Sine-Sine, and Tinkerbell maps. Numerical simulations are carried out for different parameter values and initial conditions, and it is shown that the mappings either diverge to infinity or converge to attractors of many different shapes. The application to random bit generation is then considered using a collection of the proposed maps by applying a simple rule. The resulting bit generator successfully passes all statistical tests performed.

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Structural stability of linear random dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700009998 ◽

1996 ◽

Vol 16 (6) ◽

pp. 1207-1220 ◽

Cited By ~ 2

Author(s):

Nguyen Dinh Cong

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Structural Stability ◽

Discrete Time ◽

Random Dynamical Systems ◽

Random Dynamical System ◽

Time Linear ◽

Structurally Stable

AbstractIn this paper, structural stability of discrete-time linear random dynamical systems is studied. A random dynamical system is called structurally stable with respect to a random norm if it is topologically conjugate to any random dynamical system which is sufficiently close to it in this norm. We prove that a discrete-time linear random dynamical system is structurally stable with respect to its Lyapunov norms if and only if it is hyperbolic.

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New Discrete Time 2D Chaotic Maps

International Journal of System Dynamics Applications ◽

10.4018/ijsda.2017010105 ◽

2017 ◽

Vol 6 (1) ◽

pp. 77-104 ◽

Cited By ~ 9

Author(s):

Lazaros Moysis ◽

Ahmad Taher Azar

Keyword(s):

Dynamical Systems ◽

Discrete Time ◽

Chaotic Behavior ◽

Initial Conditions ◽

Single Point ◽

Solution Space ◽

Small Perturbations ◽

Highly Sensitive ◽

Different Shapes ◽

Parameter Values

Chaotic behavior is a term that is attributed to dynamical systems whose solutions are highly sensitive to initial conditions. This means that small perturbations in the initial conditions can lead to completely different trajectories in the solution space. These types of chaotic dynamical systems arise in various natural or artificial systems in biology, meteorology, economics, electrical circuits, engineering, computer science and more. Of these innumerable chaotic systems, perhaps the most interesting are those that exhibit attracting behavior. By that, the authors refer to systems whose trajectories converge with time to a set of values, called an attractor. This can be a single point, a curve or a manifold. The attractor is called strange if it is a set with fractal structure. Such systems can be both continuous and discrete. This paper reports on some new chaotic discrete time two dimensional maps that are derived from simple modifications to the well-known Hénon, Lozi, Sine-sine and Tinkerbell maps. Numerical simulations are carried out for different parameter values and initial conditions and it is shown that the mappings either diverge to infinity or converge to attractors of many different shapes.

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