Fractional-Order Memory Effects and Chaotic Dynamics in Nonlinear Delay Systems: Computational Analysis of Bifurcation and Multi-Scroll Attractors

Introduction:This paper investigates chaos, bifurcation, and control in a nonlinear two-delay differential equation of the formwhere , and are two distinct time delays encoding memory effects.
Main: Definition[Equilibrium Point]:A point is an equilibrium of the system if . It is asymptotically stable if every trajectory starting near converges to it as .
Definition[Lyapunov Exponent]:The maximal Lyapunov exponent quantifies the average exponential rate of divergence of nearbytrajectories. Chaos is confirmed when .
The system is extended to the fractional-order setting via the Caputo derivative
Theorem[Persistence of Chaos]:There exist parameter values and fractional order such that the fractional system exhibits chaotic behavior, even when the integer-order system is stable.
Theorem[Caos Control via Linear Feedback]:Adding the feedback term for sufficiently large stabilizes the system to .Moreover, a delay-independent condition for master--slave synchronization is derived using linear state feedback control.
Conclusion:

Numerical simulations via Lyapunov exponents and phase portraits confirm multi-scroll chaotic attractors and validate all analytical results. The fractional order acts as a pivotal control parameter governing the transition between stable, periodic, and chaotic regimes, demonstrating that memory depth directly modulates the complexity of the system's dynamical behavior. Notably, the interaction between the two time delays and the fractional memory kernel produces richer bifurcation structures than those observed in either pure delay or pure fractional systems individually. The proposed linear feedback control strategy proves robust across a wide range of fractional orders, confirming its practical applicability in engineering and physical systems governed by hereditary dynamics. Furthermore, the derived delay-independent synchronization condition offers a computationally efficient criterion applicable to coupled nonlinear fractional networks. These findings collectively suggest that fractional-order two-delay systems constitute a fertile mathematical framework for modeling complex physical phenomena, with promising future directions in viscoelastic mechanics, neural dynamics, and anomalous diffusion processes in computational physics.