Attractors of Caputo fractional differential equations with triangular vector fields

TL;DR

This paper demonstrates that the attractors of certain Caputo fractional differential equations with triangular vector fields are essentially the same as those of corresponding ordinary differential equations, and explores bifurcations in scalar fractional systems.

Contribution

It establishes the equivalence of attractors between fractional and ordinary differential equations under specific conditions and analyzes bifurcations in scalar fractional systems.

Findings

Attractors of fractional and ordinary systems coincide under given conditions.

Identifies bifurcations such as saddle-node and pitchfork in fractional equations.

Solutions of Caputo FDEs do not intersect in finite time.

Abstract

It is shown that the attractor of an autonomous Caputo fractional differential equation of order $\alpha\in(0,1)$ in $\mathbb{R}^d$ whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an application, we establish several one-parameter bifurcations for scalar fractional differential equations including the saddle-node and the pichfork bifurcations. The proof uses a result of "N. D. Cong and H.T. Tuan, Generation of nonlocal fractional dynamical systems by fractional differential equations. Journal of Integral Equations and Applications, 29 (2017), 1-24" which shows that no two solutions of such a Caputo FDE can intersect in finite time