Stability of the discrete Caputo-like fractional hyper-Rössler system

Chaotic behavior is a common feature of nonlinear dynamics.

The subject of our discussion today is the stability of hyperchaos in high-dimensional systems. This study serves as an introductory guide to a discrete fractional four-dimensional hyperchaotic Rössler system with a Caputo-like operator is a complex system that can be used to study the chaos of discrete fractional nonlinear dynamics. Our results demonstrate the existence of a hyperchaotic invariant set in these systems, which leads to extended hyperchaotic transient behaviors. The coexistence of chaos and hyperchaos is evident in the numerical results, which are presented in phase plots and bifurcation diagrams for various fractional orders and different parameters and specific initial conditions. These diagrams provide a comprehensive explanation of the dynamics of the proposed discrete system. This research substantiates the presence of chaos in discrete fractional hyperchaotic Rössler systems that are reminiscent of Caputo-like discrete systems. Low control is offered to display synchronization of coupled Caputo-like discrete fractional hyperchaotic Rössler systems and to force the states of the proposed system to converge asymptotically to zero. The findings of the study are demonstrated through the following implementation of numerical simulations, which has been a significant development in our research.