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

Abstract

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.

References in corpus (2)

• Fractional Equations of Kicked Systems and Discrete Maps
• Long-term memory contribution as applied to the motion of discrete dynamical systems

Cited by in corpus (15)

• Review of Some Promising Fractional Physical Models
• Trends, Directions for Further Research, and Some Open Problems of Fractional Calculus
• Logistic map with memory from economic model
• General Fractional Dynamics
• Stability of Fixed Points and Chaos in Fractional Systems
• Caputo Standard $\alpha$-Family of Maps: Fractional Difference vs. Fractional
• Universal Fractional Map and Cascade of Bifurcations Type Attractors
• Economic Accelerator with Memory: Discrete Time Approach
• Asymptotic cycles in fractional maps of arbitrary positive orders
• Fractional Dynamics with Non-Local Scaling
• Predator-prey models with memory and kicks: Exact solution and discrete maps with memory
• Fractional Dynamical Systems
• From Fractional Differential Equations with Hilfer Derivatives: To Discrete Maps with Memory
• Evolution of Systems with Power-Law Memory: Do We Have to Die?
• Characterizing and quantifying weak chaos in fractional dynamics