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Licensed Unlicensed Requires Authentication Published by De Gruyter 2019

Dynamics of nonlinear systems with power-law memory

Mark Edelman

Abstract

Dynamics of fractional (with power-law memory) nonlinear systems may demonstrate features, which are fundamentally different from behavior of regular (memoryless) nonlinear systems. The new features of fractional dynamics include self-intersection of trajectories in dimensions α, less than two, and overlapping of chaotic attractors; existence of periodic points only in an asymptotic sense, and a cascade of bifurcations-type attracting trajectories. Fractional nonlinear systems demonstrate the standard scenario of the transition to chaos, through a cascade of period-doubling bifurcations, which depends on the fractional dimension (memory parameter) α. Two-dimensional bifurcation diagrams, limiting the value of the coordinate xlim as a function of the nonlinearity parameter K and the memory parameter α, possess some universal properties. Bifurcations with the change in α, (α, x)-bifurcation diagrams, allow control of fractional systems by changing the memory parameter.

© 2019 Walter de Gruyter GmbH, Berlin/Munich/Boston
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