Abstract
In this paper, we first estimate the boundedness of a new proposed 4-dimensional (4D) hyper-chaotic system with complex dynamical behaviors. For this system, the ultimate bound set Ω1 and globally exponentially attractive set Ω2 are derived based on the optimization method, Lyapunov stability theory and comparison principle. Numerical simulations are presented to show the effectiveness of the method and the boundary regions. Then, to prove the existence of hyper-chaos, the hyper-chaotic dynamics of the 4D nonlinear system is investigated by means of topological horseshoe theory and numerical computation. Based on the algorithm for finding horseshoes in three-dimensional hyper-chaotic maps, we finally find a horseshoe with two-directional expansions in the 4D hyper-chaotic system, which can rigorously prove the existence of the hyper-chaos in theory.
Highlights
A 4D hyper-chaotic system is usually considered as a chaotic system with two positive Lyapunov exponents, which can enhance the randomness and unpredictability of the nonlinear system
Constructing a new hyper-chaotic system with complex dynamical behaviors may be more useful in some research fields, such as communication [1, 37], encryption [5, 20], and synchronization [3, 22, 31, 32]
If one can prove that a hyper-chaotic system has a globally exponentially attractive set, one concludes that the system cannot have the equilibrium points, periodic or quasi-periodic solutions, or other chaotic or hyper-chaotic attractors existing outside the attractive set, which greatly simplifies the dynamics analysis of a chaotic or hyper-chaotic system
Summary
A 4D hyper-chaotic system is usually considered as a chaotic system with two positive Lyapunov exponents, which can enhance the randomness and unpredictability of the nonlinear system. It is meaningful to investigate the boundedness of the new proposed hyper-chaotic system based on the optimization method, Lyapunov stability theory, and comparison principle. A hyper-chaotic system usually has a large negative LE, its attractor is often https://www.mii.vu.lt/NA contracted closely to a certain surface Based on this feature, we can use a remarkable algorithm proposed by Li and Tang [16] to detect a horseshoe with two-directional expansions effectively by deducting the dimension along the direction of contraction.
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