Effective lag synchronisation and control in piecewise linear fractional-order Rössler and Rikitake systems

This paper studies the lag synchronisation and control in piecewise linear fractional-order Rössler and Rikitake systems. We replace certain nonlinear terms in the two systems with piecewise linear approximations using signum functions for improved analytical tractability and enhanced numerical stability. This approach simplifies the mathematical structure while preserving essential system dynamics. We analyse the lag synchronisation dynamics between coupled master-slave configurations of these systems for both integer and fractional orders. Numerical simulations demonstrate that the incorporation of memory effects through fractional derivatives decelerates the process of synchronisation, with far more iterations needed for the system states to become synchronised. Additionally, we demonstrate that an increase in the time-delay in the coupling term negatively impacts synchronisation quality. Active control methods are applied to both systems to control their trajectories. Our findings indicate that integer-order systems are fast in their responses to control inputs but take longer to converge to equilibrium in fractional-order equivalents, again owing to the memory that comes with fractional operators. Our findings bring out the subtle interaction among fractional-order dynamics, piecewise linearity, and synchronisation-control mechanisms to provide insights for designing effective control strategies in complex systems.