A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system

Dumitru Baleanu,Mehran Namjoo,Amin Jajarmi,Sadegh Zibaei

doi:10.1186/s13662-021-03454-1

Abstract

The aim of this paper is to introduce and analyze a novel fractional chaotic system including quadratic and cubic nonlinearities. We take into account the Caputo derivative for the fractional model and study the stability of the equilibrium points by the fractional Routh–Hurwitz criteria. We also utilize an efficient nonstandard finite difference (NSFD) scheme to implement the new model and investigate its chaotic behavior in both time-domain and phase-plane. According to the obtained results, we find that the new model portrays both chaotic and nonchaotic behaviors for different values of the fractional order, so that the lowest order in which the system remains chaotic is found via the numerical simulations. Afterward, a nonidentical synchronization is applied between the presented model and the fractional Volta equations using an active control technique. The numerical simulations of the master, the slave, and the error dynamics using the NSFD scheme are plotted showing that the synchronization is achieved properly, an outcome which confirms the effectiveness of the proposed active control strategy.

Highlights

1 Introduction One of the most important properties of chaotic systems is the appearance of irregular behavior in their dynamics; the state variables of such systems are highly sensitive to the starting point
Simulation results in time-domain and phase-plane indicate that the new system exhibits both chaotic and nonchaotic behaviors, so we find via the numerical simulations the lowest order in which the system remains chaotic
5 Nonidentical synchronization we study the nonidentical synchronization between the novel fractionalorder system (21) and the fractional-order Volta equations [37] using the active controller developed in [38]

Summary

Introduction

One of the most important properties of chaotic systems is the appearance of irregular behavior in their dynamics; the state variables of such systems are highly sensitive to the starting point. A small change in the initial conditions may cause a wide variation, or even divergence, in the system dynamics. It is very important from both practical and theoretical viewpoints to analyze such complicated systems. The footsteps of chaos are found in various branches of sciences and technologies including economy, biology, physics, secure communications, and so on, and many scientists have devoted growing attention to the analysis, synchronization, and control of complex chaotic systems. In [2] the state-dependent Riccati equations were

Objectives

Conclusion