Hyperbolicity of Limit Cycles in Dynamics
Definition
Euclidean space $\mathbb{R}^{n}$ and open set $U \subset \mathbb{R}^{n}$, consider a continuous function $f : U \to \mathbb{R}^{n}$ with the following vector field given by a differential equation. $$ \dot{x} = f(x) $$ Let $P : \Sigma \to \Sigma$ denote the Poincaré map defined on a manifold $\Sigma$ that is traversed by the limit cycle $L_{0}$ of this system. Assume that a point $\xi_{0}$ at the intersection of $L_{0}$ and $\Sigma$ is a fixed point of the map $P$, i.e., it satisfies $P \left( \xi_{0} \right) = \xi_{0}$.
- If $\xi_{0}$ is a hyperbolic fixed point of $P$, then the limit cycle $L_{0}$ is called hyperbolic.
- If $\xi_{0}$ is a saddle of $P$, then the hyperbolic cycle $L_{0}$ is called a saddle cycle.
- If $f(x) \cdot n (x) \ne 0$ at all points on $\Sigma$, then $\Sigma$ is said to be transverse to the vector field.
Explanation
In a continuous system, cycles of interest consist of infinitely many points, making it difficult to grasp directly. However, by reducing the dimension using the Poincaré map, the information of curves can be summarized as points, allowing the discussion on the hyperbolicity of fixed points to be applied directly. The advantage of this definition lies in the intuitive understanding when considering something termed as stability. It is sufficient to just imagine the movement of points rather than the complex notion of invariant sets being stable or unstable.