New Types of Solutions of Non-Linear Fractional Differential Equations

TL;DR

This paper explores various solution types of non-linear fractional differential equations, revealing new phenomena such as overlapping attractors, intersecting trajectories, and cascade bifurcations, using fractional standard maps as examples.

Contribution

It introduces novel solution behaviors in fractional differential equations, including overlapping attractors and intersecting trajectories, expanding understanding of fractional dynamical systems.

Findings

Discovered overlapping attractors

Identified intersecting trajectories

Observed cascade of bifurcations type trajectories

Abstract

Using the Riemann-Liouville and Caputo Fractional Standard Maps (FSM) and the Fractional Dissipative Standard Map (FDSM) as examples, we investigate types of solutions of non-linear fractional differential equations. They include periodic sinks, attracting slow diverging trajectories (ASDT), attracting accelerator mode trajectories (AMT), chaotic attractors, and cascade of bifurcations type trajectories (CBTT). New features discovered include attractors which overlap, trajectories which intersect, and CBTTs.