# Limit Cycle

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## limit cycle

[′lim·ət ‚sīk·əl]*C*in the plane (corresponding to a periodic solution of the equation) where every point of

*C*has a neighborhood so that every trajectory through it spirals toward

*C*.

## Limit Cycle

The limit cycle of a system of second-order differential equations

is a closed trajectory in the *xy*-phase space which has the property that all trajectories starting in a sufficiently narrow annular neighborhood of this trajectory approach it, as *t* → + ∞ (stable limit cycle) and as *t* → –∞ (unstable limit cycle), or some approach it as *t* → + ∞ and the rest as *t* → —∞ (semistable limit cycle). For example, the system

(*r* and ϕ are polar coordinates), whose general solution is *r* = 1 — (1 — *r*_{0})*e*^{-t}, ϕ = ϕ_{0} + *t* (where *r*_{0} ≥ 0), has the stable limit cycle *r* = 1 (see Figure 1). The concept of limit cycle can be carried over to an *n*th-order system. From a mechanical viewpoint, a stable limit cycle corresponds to a stable periodic motion of the system. Therefore, finding limit cycles is of great importance in the theory of nonlinear oscillations.

### REFERENCES

Pontriagin, L. S.*Obyknovennye differentsial’nye urameniia*, 3rd ed. Moscow, 1970.

Andronov, A. A., A. A. Vitt, and S. E. Khaikin.

*Teoriia kolebanii*, 2nd ed. Moscow, 1959.