On the dynamics of fractional maps with power-law, exponential decay and Mittag–Leffler memory

Abstract In this paper, we propose a fractional form of two-dimensional generalized mythical bird, butterfly wings and paradise bird maps involving the fractional conformable derivative of Khalil’s and Atangana’s type, the Liouville–Caputo and Atangana–Baleanu derivatives with constant and variable-order. We obtain new chaotical behaviors considering numerical schemes based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. Also, the dynamics of the proposed maps are investigated numerically through phase plots considering combinations of these derivatives and mixed integration methods for each map. The numerical simulations show very strange and new behaviors for the first time in this manuscript.

[1]  Ilknur Koca,et al.  Existence and Uniqueness Results for a Novel Complex Chaotic Fractional Order System , 2019, Studies in Systems, Decision and Control.

[2]  M. el-Dessoky,et al.  Application of fractional calculus to combined modified function projective synchronization of different systems. , 2019, Chaos.

[3]  Sen Zhang,et al.  A Novel 3D Fractional-Order Chaotic System with Multifarious Coexisting Attractors , 2019, Int. J. Bifurc. Chaos.

[4]  Fairouz Tchier,et al.  Chaotic Attractors with Fractional Conformable Derivatives in the Liouville–Caputo Sense and Its Dynamical Behaviors , 2018, Entropy.

[5]  Zhao Peichen,et al.  Fractional calculus in abstract space and its application in fractional Dirichlet type problems , 2019, Chaos, Solitons & Fractals.

[6]  Michele Caputo,et al.  On the notion of fractional derivative and applications to the hysteresis phenomena , 2017 .

[7]  E. D. Goufo Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systems , 2020, Discrete & Continuous Dynamical Systems - S.

[8]  Yoshisuke Ueda,et al.  The chaos avant-garde : memories of the early days of chaos theory , 2001 .

[9]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[10]  Martin Bohner,et al.  Dynamic cobweb models with conformable fractional derivatives , 2019, Nonlinear Analysis: Hybrid Systems.

[11]  Abdon Atangana,et al.  Blind in a commutative world: Simple illustrations with functions and chaotic attractors , 2018, Chaos, Solitons & Fractals.

[12]  J. F. Gómez‐Aguilar,et al.  Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena , 2018 .

[13]  Adel Ouannas,et al.  On the dynamics, control and synchronization of fractional-order Ikeda map , 2019, Chaos, Solitons & Fractals.

[14]  M. Jleli,et al.  A derivative concept with respect to an arbitrary kernel and applications to fractional calculus , 2018, Mathematical Methods in the Applied Sciences.

[15]  Dumitru Baleanu,et al.  On a new class of fractional operators , 2017, Advances in Difference Equations.

[16]  Alan D. Freed,et al.  Detailed Error Analysis for a Fractional Adams Method , 2004, Numerical Algorithms.

[17]  A. Sheikhani,et al.  Dynamical analysis of a new three-dimensional fractional chaotic system , 2019, Pramana.

[18]  I. Podlubny Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[19]  J. F. Gómez‐Aguilar,et al.  Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense , 2018 .

[20]  A. Matouk,et al.  Computations of synchronisation conditions in some fractional-order chaotic and hyperchaotic systems , 2019, Pramana.

[21]  Fuli He,et al.  On the Wright hypergeometric matrix functions and their fractional calculus , 2018, Integral Transforms and Special Functions.

[22]  Daniel Cao Labora,et al.  Is It Possible to Construct a Fractional Derivative Such That the Index Law Holds , 2018 .

[23]  Wen Wang,et al.  Fractal dimension analysis and control of Julia set generated by fractional Lotka-Volterra models , 2019, Commun. Nonlinear Sci. Numer. Simul..

[24]  M. Sababheh,et al.  A new definition of fractional derivative , 2014, J. Comput. Appl. Math..

[25]  Stefan Samko,et al.  Fractional integration and differentiation of variable order , 1995 .

[26]  E. D. Goufo Development and Elaboration of a Compound Structure of Chaotic Attractors with Atangana–Baleanu Operator , 2019 .

[27]  Abdon Atangana,et al.  Fractional derivatives with no-index law property: Application to chaos and statistics , 2018, Chaos, Solitons & Fractals.

[28]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and an Introduction to Chaos , 2003 .

[29]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[30]  J. F. Gómez‐Aguilar,et al.  Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws , 2018, Chaos, Solitons & Fractals.

[31]  Abdon Atangana,et al.  The role of power decay, exponential decay and Mittag-Leffler function’s waiting time distribution: Application of cancer spread , 2018, Physica A: Statistical Mechanics and its Applications.

[32]  Abdon Atangana A novel model for the lassa hemorrhagic fever: deathly disease for pregnant women , 2015, Neural Computing and Applications.

[33]  Chengyi Zhou,et al.  Coexisting attractors, crisis route to chaos in a novel 4D fractional-order system and variable-order circuit implementation , 2019, The European Physical Journal Plus.

[34]  Dumitru Baleanu,et al.  Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions , 2018, Commun. Nonlinear Sci. Numer. Simul..

[35]  José Francisco Gómez-Aguilar,et al.  Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations , 2018 .

[36]  Thabet Abdeljawad,et al.  Fractional logistic models in the frame of fractional operators generated by conformable derivatives , 2019, Chaos, Solitons & Fractals.

[37]  Dumitru Baleanu,et al.  On fractional calculus with general analytic kernels , 2019, Appl. Math. Comput..

[38]  Abdon Atangana,et al.  Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties , 2018, Physica A: Statistical Mechanics and its Applications.

[39]  E. K. Lenzi,et al.  The Role of Fractional Time-Derivative Operators on Anomalous Diffusion , 2017, Front. Phys..