📂Dynamics

Poincaré Map

Definition ¹

Let us consider a vector field in Euclidean space $\mathbb{R}^{n}$ and open sets $U \subset \mathbb{R}^{n}$ defined in the function continuous $f : U \to \mathbb{R}^{n}$ as given by the following differential equation. $$ \dot{x} = f(x) $$ Represent the flow as $\phi_t \left( \cdot \right)$ and consider a $\left( n-1 \right)$-dimensional surface $\Sigma$ that intersects the vector field. For an open set $V \subset \Sigma$, we define the following map $P$ as the Poincaré map. $$ \begin{align*} P : V &\to \Sigma \\ x &\mapsto \phi_{\tau (x)} (x) \end{align*} $$ Here, $\tau (x)$ denotes the time it takes to return to $\Sigma$ after departing from $x$.

---

• If at every point on $\Sigma$, $f(x) \cdot n (x) \ne 0$ holds, then we call $\Sigma$ the transverse surface to the vector field.

Explanation

Due to $P$, the flow $\phi$ on $\Sigma$ can be represented by wandering on $\Sigma$, skipping the intermediate steps.

This is similar to the long-range attack prevention plane constraint in Denma². The Poincaré map reduces the system represented by a vector field to $\Sigma$ by one dimension. Although much information is lost due to the dimension reduction, it is useful to apply in any problem focusing on the overall dynamics.

See Also

• The Poincaré map is used to define the hyperbolicity of a limit cycle by using the definition of the hyperbolicity of a fixed point.