information:extraction, linguistic:voice_change
Definition

We consider a two-dimensional dynamical system of the form x ′ (t) = V (x (t)) {\displaystyle x'(t)=V(x(t))} where V: R 2 → R 2 {\displaystyle V:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} is a smooth function. A trajectory of this system is some smooth function x (t) {\displaystyle x(t)} with values in R 2 {\displaystyle \mathbb {R} ^{2}} which satisfies this differential equation. Such a trajectory is called closed (or periodic) if it is not constant but returns to its starting point, i.e. if there exists some t 0 > 0 {\displaystyle t_{0}>0} such that x (t + t 0) = x (t) {\displaystyle x(t+t_{0})=x(t)} for all t ∈ R {\displaystyle t\in \mathbb {R} }. An orbit is the image of a trajectory, a subset of R 2 {\displaystyle \mathbb {R} ^{2}}. A closed orbit, or cycle, is the image of a closed trajectory. A limit cycle is a cycle which is the limit set of some other trajectory.