Limit Cycle

limit cycle [′lim·ət ‚sīk·əl]

(mathematics)

For a differential equation, a closed trajectoryCin the plane (corresponding to a periodic solution of the equation) where every point ofChas a neighborhood so that every trajectory through it spirals towardC.

---

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 — r₀)e^-t, ϕ = ϕ₀ + t (where r₀ ≥ 0), has the stable limit cycle r = 1 (see Figure 1). The concept of limit cycle can be carried over to an nth-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.

Figure 1

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.