limit cycle

[′lim·ət ‚sīk·əl]

(mathematics)

For a differential equation, a closed trajectory 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.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.