An efficient solution procedure for solving higher-codimension Hopf and Bogdanov-Takens bifurcations

TL;DR

This paper introduces an efficient method for analyzing higher-codimension Hopf and Bogdanov-Takens bifurcations, overcoming symbolic computation challenges and parameter restrictions in nonlinear dynamical systems.

Contribution

The paper presents a simplified one-step transformation approach for higher-codimension bifurcations, improving upon classical methods and demonstrating effectiveness with a 2D epidemic model.

Findings

Successfully applied to a 2D epidemic model for bifurcation analysis

Efficient one-step transformation outperforms classical six-step method

Accurately determines the codimension of bifurcations

Abstract

In solving real world systems for higher-codimension bifurcation problems, one often faces the difficulty in computing the normal form or the focus values associated with generalized Hopf bifurcation, and the normal form with unfolding for higher-codimension Bogdanov-Takens bifurcation. The difficulty is not only coming from the teadious symbolic computation of focus values, but also due to the restriction on the system parameters, which frequently leads to failure of the conventional approach used in the computation even for simple $2$-dimensional nonlinear dynamical systems. In this paper, we use a simple 2-dimensional epidemic model, for which the conventional approach fails in analyzing the stability of limit cycles arising from Hopf bifurcation, to illustrate how our method can be efficiently applied to determine the codimension of Hopf bifurcation. Further, we apply the simplest normal form theory to consider codimension-3 Bogdanov-Takens bifurcation and present an efficient one-step transformation approach, compared with the classical six-step transformation approach to demonstrate the advantage of our method.