On the dynamics of fractional maps with power-law, exponential decay and Mittag-Leffler memory. (English) Zbl 1448.34086
Summary: 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.
MSC:
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
| 34A08 | Fractional ordinary differential equations |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 37M05 | Simulation of dynamical systems |
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