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.
Highlights
Fractional analysis has become a basic topic for research since it presents appropriate mathematical tools to explain a wide variety of engineering, physical and biological phenomena, and some interdisciplinary topics such as neural networks [1,2,3,4,5,6,7,8,9]
Some fractional differential operators were successfully proposed to describe fractional derivatives such as Caputo’s type [10] and Riemann–Liouville’s type [11]. e aforementioned operators are defined by integration so they are sorted as nonlocal operators with singular kernels
Motivated by the aforementioned statements, we further investigate the chaotic dynamics in the fractional-order satellite (FOS) model with and without time delay
Summary
Fractional analysis has become a basic topic for research since it presents appropriate mathematical tools to explain a wide variety of engineering, physical and biological phenomena, and some interdisciplinary topics such as neural networks [1,2,3,4,5,6,7,8,9]. Chaotic attractors and other interesting dynamics are found in a time-delay version of the FOS model. E obtained results show that the time-delay version of the FOS model is stabilized to its origin equilibrium state when using the same FCGs, whose selection is based on the LST of fractional-order systems with time delay. Assume that a nonnegative function P(t) ∈ R is given that it is continuously differentiable and satisfies the following conditions (as α ∈ (0, 1)): DαP(t) ≤ − η1P(t) + η2P(t − τ), P(t) Ψ(t) ≥ 0 and − τ ≤ t ≤ 0,. If η < 1, U Ψ(U) is a contraction mapping, and the following theorem gives a sufficient condition for existence and uniqueness of the solution of FOS model (20).
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