Bursting Dynamics in a Singular Vector Field with Codimension Three Triple Zero Bifurcation

As a kind of dynamical system with a particular nonlinear structure, a multi-time scale nonlinear system is one of the essential directions of the current development of nonlinear dynamics theory. Multi-time scale nonlinear systems in practical applications are often complex forms of coupling of high-dimensional and high codimension characteristics, leading to various complex bursting oscillation behaviors and bifurcation characteristics in the system. For exploring the complex bursting dynamics caused by high codimension bifurcation, this paper considers the normal form of the vector field with triple zero bifurcation. Two kinds of codimension-2 bifurcation that may lead to complex bursting oscillations are discussed in the two-parameter plane. Based on the fast-slow analysis method, by introducing the slow variable W=Asin(?t), the evolution process of the motion trajectory of the system changing with W was investigated, and the dynamical mechanism of several types of bursting oscillations was revealed. Finally, by varying the frequency of the slow variable, a class of chaotic bursting phenomena caused by the period-doubling cascade is deduced. Developing related work has played a positive role in deeply understanding the nature of various complex bursting phenomena and strengthening the application of basic disciplines such as mechanics and mathematics in engineering practice.

Automated summary

This paper discusses two types of codimension-2 bifurcation that may lead to complex bursting oscillations by considering the normal form of the vector field with triple zero bifurcation. By using the fast-slow analysis method and introducing the slow variable W=Asin(?t), the evolution process of the system's motion trajectory changing with W was investigated, and the dynamical mechanism of several types of bursting oscillations was revealed. Furthermore, a class of chaotic bursting phenomena caused by the period-doubling cascade is deduced by varying the frequency of the slow variable. These studies have played a positive role in deepening the understanding of the nature of various complex bursting phenomena and strengthening the application of basic disciplines such as mechanics and mathematics in engineering practice.