Modelling discrete time fractional Rucklidge system with complex state variables and its synchronization

This article proposes a discrete time fractional order three dimensional Rucklidge system with complex state variables. The dynamical nature and chaotic behavior exhibited by the Rucklidge system with real state variables and higher dimensional system derived from complex state variables are compared. The stability of the proposed system is analyzed at their equilibrium states by obtaining eigenvalues numerically. Chaotic dynamics exhibited by the system with real and complex variables are investigated employing different meth-ods like bifurcation analysis and maximum Lyapunov exponents via the Jacobian matrix method. Nonlinear controllers are introduced for chaos synchronization of the subsystems of the proposed discrete time Rucklidge system with fractional order. The article further discusses the coexisting behavior of the attractors for the Rucklidge system of real state variables with coexisting bifurcation diagrams. The impact of the parameters on the system dynamics is demonstrated with a sequence of bifurcation diagrams for the simultaneous variation of two parameters.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This article introduces a discrete time fractional order three-dimensional Rucklidge system with complex state variables and compares its dynamical properties and chaotic behavior with the Rucklidge system with real state variables and higher dimensional system derived from complex state variables. The stability of the proposed system is analyzed at equilibrium states by obtaining eigenvalues numerically. Chaotic dynamics exhibited by the system with real and complex variables are studied using bifurcation analysis, maximum Lyapunov exponents, and the Jacobian matrix method. Nonlinear controllers are implemented for chaos synchronization of the subsystems of the proposed system. The article also discusses the coexisting behavior of the attractors in the Rucklidge system with real state variables and coexisting bifurcation diagrams.