Differential equations with fractional derivative and universal map with memory

Discrete maps with long-term memory are obtained from nonlinear differential equations with Riemann–Liouville and Caputo fractional derivatives. These maps are generalizations of the well-known universal map. The memory means that their present state is determined by all past states with special forms of weights. To obtain discrete maps from fractional differential equations, we use the equivalence of the Cauchy-type problems and to the nonlinear Volterra integral equations of the second kind. General forms of the universal maps with memory, which take into account general initial conditions for the cases of the Riemann–Liouville and Caputo fractional derivative, are suggested.

[1]  A. Kleczkowski,et al.  Nonlinear Maps with Memory , 1987 .

[2]  M Giona,et al.  Dynamics and relaxation properties of complex systems with memory , 1991 .

[3]  V. E. Tarasov.  Universal electromagnetic waves in dielectric , 2008, 0907.2163.

[4]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[5]  D. A. Usikov,et al.  Nonlinear Physics: From the Pendulum to Turbulence and Chaos , 1988 .

[6]  A. Kilbas,et al.  The cauchy problem for differential equations with fractional caputo derivative , 2004 .

[7]  V. E. Tarasov.  Fractional equations of Curie–von Schweidler and Gauss laws , 2008, 0907.1837.

[8]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[9]  V. E. Tarasov.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[10]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[11]  J. Trujillo,et al.  EXISTENCE AND UNIQUENESS THEOREMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS , 2000 .

[12]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[13]  Fick,et al.  Logistic equation with memory. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[14]  A. Jonscher.  Dielectric relaxation in solids , 1983 .

[15]  M. Caputo,et al.  A new dissipation model based on memory mechanism , 1971 .

[16]  A. Stanislavsky,et al.  Long-term memory contribution as applied to the motion of discrete dynamical systems. , 2006, Chaos.

[17]  B. Chirikov.  A universal instability of many-dimensional oscillator systems , 1979 .

[18]  M. Caputo.  Linear models of dissipation whose Q is almost frequency independent , 1966 .

[19]  I. Podlubny.  Fractional differential equations , 1998 .

[20]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[21]  K. Hartwich,et al.  Hopf bifurcations in the logistic map with oscillating memory , 1993 .

[22]  Jason A. C. Gallas,et al.  Simulating memory effects with discrete dynamical systems , 1993 .

[23]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[24]  George M. Zaslavsky.  Hamiltonian Chaos and Fractional Dynamics , 2005 .

[25]  A. Kilbas,et al.  Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions , 2005 .

[26]  Fractional equations of kicked systems and discrete maps , 2008, 1107.3953.

[27]  J. Trujillo,et al.  Existence and uniqueness theorem for nonlinear fractional differential equations , 2000 .