Adaptive memory switching for chaos synchronization in variable-order fractional dynamical systems

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Abstract

We study the synchronization of piecewise-constant-order Caputo systems whose lower terminal is reset at prescribed times. Under linear feedback, a computable dissipativity margin yields an error bound consisting of one Mittag-Leffler factor per completed reset interval. The argument combines standard local Caputo inequalities with continuous endpoint matching; periodic schedules give an explicit cycle factor and a physical-time decay envelope. We also quantify the difference between resetting and retaining memory: A total-variation estimate controls the integrated omitted history and the resulting trajectory discrepancy. An exact scalar example shows that refining the reset mesh need not approximate a fixed-origin equation. A conditional bounded-region Lorenz result gives explicit sufficient gains. Matched-grid computations with identical gains and order schedules compare reset and fixed-origin models and include step refinement and discarded-history diagnostics. In the alternating 0.94/0.86 benchmark with unit dwell, the reset model reaches a 10⁻⁵ error near t = 2.04, whereas the fixed-origin error is approximately 2.15 × 10⁻³ at t = 8. The reset master approaches an equilibrium over this horizon. These results concern synchronization under prescribed memory reinitialization; they establish neither chaotic dynamics nor a general performance advantage for a fixed-memory plant.

Keywords

Caputo memory reset / piecewise fractional order / synchronization / Mittag-Leffler bounds

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