Discrete Map with Memory from Fractional Differential Equation of Arbitrary Positive Order

Details Discrete Map with Memory from Fractional Differential Equation of Arbitrary Positive Order Discrete Map with Memory from Fractional Differential Equation of Arbitrary Positive Order

2011-07-22

arxiv.org/abs/1107.4425v1

Journal of Mathematical Physics 50 (2009) 122703

Nonlinear Sciences

Chaotic Dynamics

Scientific paper

Derivatives of fractional order with respect to time describe long-term memory effects. Using nonlinear differential equation with Caputo fractional derivative of arbitrary order $\alpha>0$, we obtain discrete maps with power-law memory. These maps are generalizations of well-known universal map. The memory in these maps means that their present state is determined by all past states with power-law forms of weights. Discrete map equations are obtained by using the equivalence of the Cauchy-type problem for fractional differential equation and the nonlinear Volterra integral equation of the second kind.

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