Limit cycle

In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854-1912).

We consider a two-dimensional dynamical system of the form

where

is a smooth function. A trajectory of this system is some smooth function ${\displaystyle x(t)}$ with values in ${\displaystyle \mathbb {R} ^{2}}$ 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 ${\displaystyle t_{0}>0}$ such that ${\displaystyle x(t+t_{0})=x(t)}$ for all ${\displaystyle t\in \mathbb {R} }$. An orbit is the image of a trajectory, a subset of ${\displaystyle \mathbb {R} ^{2}}$. 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.

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