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\times10^{-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.
Citation: Lateef Ahmad Wani, Sajad Ahmad Sheikh. Adaptive memory switching for chaos synchronization in variable-order fractional dynamical systems[J]. Electronic Research Archive, 2026, 34(12): 9066-9093. doi: 10.3934/era.2026382
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\times10^{-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.
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Lateef Ahmad Wani, Sajad Ahmad Sheikh. Adaptive memory switching for chaos synchronization in variable-order fractional dynamical systems[J]. Electronic Research Archive, 2026, 34(12): 9066-9093. doi: 10.3934/era.2026382
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