📂Dynamics

Poincaré Map

Definition ¹

Let us consider a vector field given as a differential equation in the Euclidean space $\mathbb{R}^{n}$ and an open set $U \subset \mathbb{R}^{n}$ where the function $f : U \to \mathbb{R}^{n}$ is differentiable up to $r$ times. $$ \dot{x} = f(x) $$ Let’s denote that flow as $\phi_t \left( \cdot \right)$ and consider a $n-1$-dimensional surface $\Sigma$ crossing the vector field. For an open set $V \subset \Sigma$, the following map $P$ is called the Poincaré Map. $$ \begin{align*} P : V &\to \Sigma \\ x &\mapsto \phi_{\tau (x)} (x) \end{align*} $$ Here, $\tau (x)$ represents the time it takes to start from $x$ and return to $\Sigma$.

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• If $f(x) \cdot n (x) \ne 0$ at every point of $\Sigma$, it is said that $\Sigma$ crosses the vector field.

Explanation

Through $P$, the flow $\phi$ on $\Sigma$ will wander off $\Sigma$, skipping the intermediaries.

This is similar to the plane constraint that blocks long-range attacks in Denma. The Poincaré Map is a map that reduces the system represented by the vector field to $\Sigma$ by one dimension. Although much information might be lost since one dimension disappears, it is still worth using in any problem if one is interested in the overall behavior.