An explicit Hopf bifurcation criterion of fractional-order systems with order 1 < {\alpha} < 2

TL;DR

This paper presents an explicit criterion for Hopf bifurcation in fractional-order systems with order between 1 and 2, simplifying analysis by avoiding root solving and enabling direct computation of bifurcation surfaces.

Contribution

It introduces a novel explicit Hopf bifurcation criterion for fractional-order systems that bypasses complex root calculations, facilitating multi-parameter bifurcation analysis.

Findings

Explicit bifurcation conditions expressed by parameters

Avoids solving characteristic polynomial roots

Enables direct computation of bifurcation hyper-surfaces

Abstract

A Hopf bifurcation criterion of fractional-order systems with order 1 < {\alpha} < 2 is established in this paper, in which all conditions are explicitly expressed by parameters without solving the roots of the relevant characteristic polynomial of Hopf bifurcation conditions. It avoids the problem that existing methods may fail due to the computational complexity in the multi-parameter situation. The bifurcation hyper-surface of multi-parameter can be obtained directly.