Semi Stable Limit Cycles in Dynamical Systems Via Algebraic Topological Approach

Most phenomena in the natural and engineering fields are represented through dynamical systems composed of differential equations that incorporate parameters. A bifurcation occurs when a parameter in these systems varies, resulting in a significant impact on the solution. One of these bifurcations is a semi-stable limit cycle. In this paper, we investigate the structure of the dynamical system exhibiting a semi-stable limit cycle bifurcation using concepts from algebraic topology, and subsequently, we present new findings regarding the homology Conley index for all components of these bifurcations. This technique is vital for enhancing the topological approach to examining this type of bifurcation.