Chaotic Dynamics and Chaos Control in a Fractional-Order Satellite Model and Its Time-Delay Counterpart

Abstract

Fractional analysis provides useful tools to describe natural phenomena, and therefore, it is more convenient to describe models of satellites. This work illustrates rich chaotic behaviors that exist in a fractional-order model for satellite with and without time-delay. The proof for existence and uniqueness of the satellite model’s solution with and without time-delay is shown. Chaos control is achieved in this system via a simple linear feedback control criterion. Chaotic attractors and chaos control are also found in a time-delay version of the proposed fractional-order satellite system. Various tools based on numerical simulations such as 2D and 3D attractors and bifurcation diagrams are used to illustrate the variety of rich chaotic dynamics in the satellite models.

Self‐Excited and Hidden Chaotic Attractors in Matouk’s Hyperchaotic Systems

“…where the parameter set 4) is simple in comparison with similar types of 4D systems and reports studies of important problems [3,24,25]. It has the equilibria Eq (1) � (0, 0, 0, 0),…”

“…Assume that ς i is an eigenvalue of the Jacobian matrix J computed after the linearization of Matouk's system (4) and J is evaluated at any of the equilibrium points described in equations (5). For α � 1, Eq (1) has the following eigenvalues:…”

“…For α ∈ (0, 1] and according to eorem 2 in Ref. [25], the original equilibrium point E (1) q satisfies the following cases:…”

“…Theorem 1. Assume that the original equilibrium state Eq (1) satisfies one of conditions (A-C), and assume that Eq (1) is connected to itself by a homoclinic orbit. en, the integerorder form of system ( 4) has Smale horseshoe chaos if eigenvalues (6) of Eq (1) satisfy any of the following conditions:…”

“…Recently, fractional-order calculus (FC) has been utilized to describe complex dynamics arising from nonlinear dynamical systems (NDS) [1][2][3][4][5][6][7][8]. FC also has some interdisciplinary applications in several scientific fields.…”