DELAYED-FEEDBACK STABILIZATION OF EQUILIBRIA IN THE FRACTIONAL-ORDER RÖSSLER AND LORENZ SYSTEMS: A COMPARATIVE ANALYSIS

Abstract

This study investigates the stabilisation of equilibria in Rössler and Lorenz systems incorporating time-delayed feedback control (TDFC) with delay parameter \(\tau\) and feedback gain \(K\). The research identifies combinations of feedback gain and delay that stabilize the system’s equilibria through two complementary approaches: a local spectral test based on the characteristic equation and Matignon’s criterion, and direct simulation using a fractional forward Euler of Grünwald-Letnikov type. Stabilization in simulations is certified by requiring all trajectories to satisfy a finite-time tolerance \(10^{-2}\), \(\forall\,t\ge200\). The results indicate that TDFC can suppress chaotic dynamics where present and/or enforce convergence to the target equilibrium: The Rössler system exhibits two disjoint stability islands and \(19.5\%\) contraction between theoretical and numerical stability sets, whereas the Lorenz system shows full-domain stability with \(100\%\) agreement; this contrast is traced to a tenfold difference in the oscillation frequency of the unstable eigenvalues. The framework developed here is intended as a template for subsequent studies of other Lorenz-type systems.