Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

Abstract

In this paper, the Padé approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The PID controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.

The Shilnikov type homoclinic orbits of perturbed cubic polynomial Moon-Rand systems

“…For [10], the bifurcate parameter value was selected in a reasonable region at the requirement of the Shilnikov theorem, then the analytic expression of the Shilnikov type homoclinic orbit was accomplished by the series expression of manifold. In [4], as the global approximant were analytically expressed in a series, so they can be expanded in power series in the neighborhood of t = 0. And, in [4], the initial conditions of the orbit satisfy x(0) = x 0 , y(0) = y 0 , andż(0) = 0, so, the values of bifurcation parameter and initial value x 0 , y 0 were obtained directly.…”

“…In [4], as the global approximant were analytically expressed in a series, so they can be expanded in power series in the neighborhood of t = 0. And, in [4], the initial conditions of the orbit satisfy x(0) = x 0 , y(0) = y 0 , andż(0) = 0, so, the values of bifurcation parameter and initial value x 0 , y 0 were obtained directly.…”

“…Here, we present a method of how to find homoclinic orbit and bifurcation parameter which is real value in a 3D autonomous nonlinear system, and, the homoclinic orbit is Shilnikov type homoclinic orbit. The initial conditions of the orbit in [4] are modified as x(0) = x 0 , y(0) = y 0 , z(0) = z 0 . The Padé approximation method is introduced and the analytic solution can be expanded in power series in the neighbourhood of t = 0.…”

A new approach of optimal control for a class of continuous-time chaotic systems by an online ADP algorithm

“…Chaotic systems have been widely applied in the areas of biological engineering, communication, chemical processing, and secure information processing. [1][2][3][4] Researchers have studied the control problem of chaotic systems, [5,6] such as impulsive control method [7][8][9][10] and adaptive synchronization control method. [11] It is noted that optimal control is a very important aspect in the control field.…”

A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

“…In the study of chaotic behavior of dynamical system and traveling wave solutions in partial differential equations, homoclinic and heteroclinic orbits play a great role. [1][2][3][4][5][6][7][8] Thus, many methods have been proposed in researching homoclinic and heteroclinic orbits of nonlinear dynamics, such as the Melnikov method, [9] hyperbolic perturbation method, [10] hyperbolic Lindstedt-Poincaré method, [11] perturbation-incremental method, [12] generalized hyperbolic perturbation method, [13] and so on. Besides the perturbationbased method, the rational approximation method can also be used for solving homoclinic and heteroclinic orbits such as Padé approximation.…”