# Limit Cycle

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## 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.

## 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.
References in periodicals archive ?
Let us now analytically study the amplitude of the limit cycle by using the average method [13].
There are 2 + ([eta] - 2) +1 state transitions, and we have a limit cycle of length c = [eta] + 1.
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ij]} the manifold on which the corresponding maps f have a center at the origin and to investigate the limit cycles bifurcations of such maps.
Otherwise, a stable limit cycle can make the system to be (non asymptotically) globally Lyapunov's stable with local instability around the equilibrium exhibiting ultimate boundedness.
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According to the theory of dynamic systems, the limit cycle is a stationary trajectory of the real system, which is in the oscillatory regime (Smith 1975, Andronov et al.
Jasinghani and Ray (5) observed the existence of the unstable limit cycle in the MMA and VA polymerization reaction systems.
By comparing the Fort Worth market limit cycle to the hypothetical ideal limit cycle (as defined by these two variables) and by limiting cycles of various other multi-family markets, we can draw inferences concerning market structure and real estate cycle position.
includes vital nonlinear topics such as limit cycle prediction and forced oscillations analysis on the basis of the describing function method and absolute stability analysis by means of the primary classical frequency-domain criteria (e.

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