Existence of chaos and the approximate solution of the Lorenz–Lü–Chen system with the Caputo fractional operator

Abstract

Dynamical systems and fractional differential equations can be modeled using variable-order differential operators. In this study, the dynamics of a variable-order fractional Lorenz–Lü–Chen system with variable-order and constant-order derivatives are examined. We propose a generalized numerical scheme for simulating fractional differential operators with power-law kernels. The numerical scheme is based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. Furthermore, we examine how random parameter variations affect an approximate unified chaotic model with variable order. Specifically, we examine chaos disappearance in systems that rapidly switch from one family to another. In our simulation study, we show that increasing the key parameter and reducing its reciprocal fractional order together can suppress chaotic fluctuations much earlier than if the parameters remain fixed. Different simulation techniques have also been explored for fractional order switched chaotic systems with random parameters. The proposed system can also be solved using power series methods. The novelty of this paper is not to examine weak chaos masked by additive noise, but rather to describe the dynamics of systems that are subjected to such noise as parameter switching. By adjusting the structure, parameters, and order of the system, different chaotic oscillations and special dynamic behaviors of the Lorenz–Lü–Chen family chaos system are discovered and analyzed.