📂Dynamical Systems

Bifurcation of Limit Cycles in Dynamics

Definition ¹

For Euclidean space $\mathbb{R}^{n}$ and an open set $U \subset \mathbb{R}^{n}$, let a continuous function $f : U \to \mathbb{R}^{n}$ be given, and suppose the following vector field is given as a differential equation. $$ \dot{x} = f(x) $$ Let $P : \Sigma \to \Sigma$ be the Poincaré map defined on the $\left( n-1 \right)$-dimensional surface $\Sigma$ that the limit cycle $L_{0}$ of this system crosses, and suppose that a point $\xi_{0}$ in the intersection of $L_{0}$ and $\Sigma$ is a fixed point of the map $P_{\alpha}$, that is, it satisfies $P \left( \xi_{0} \right) = \xi_{0}$. The bifurcation that $\xi_{0}$ undergoes as the bifurcation parameter $\alpha$ varies is called a bifurcation of the limit cycle $L$.

Explanation

In other words, a bifurcation of a limit cycle is originally a bifurcation of a fixed point of the Poincaré map. This is analogous to how the hyperbolicity of a limit cycle is defined, and as a result it can be seen as a natural extension of the notion of bifurcation. Indeed, most bifurcation diagrams are collections of points marked on a Poincaré section, for instance the extrema along one axis, so this can be called a very intuitive definition.

Saddle-Node Bifurcation

As in the saddle-node bifurcation, two stable limit cycles meet and disappear.

Period-Doubling Bifurcation

As in the period-doubling bifurcation, a limit cycle whose period is doubled appears.

Neimark-Sacker Bifurcation

As in the Neimark-Sacker bifurcation, a new limit cycle appears at every point of the limit cycle, so that a manifold homeomorphic to a torus arises. The figure above is specifically a supercritical Neimark-Sacker bifurcation, which is why such a phenomenon occurs.