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 ?
Ge, "Amplitude control of limit cycle from Hopf bifurcation in the Langford system," Acta Physica Sinica, vol.
0], and a limit cycle (periodic solution) will appear near the equilibrium point [E.
Let us now analytically study the amplitude of the limit cycle by using the average method [13].
b) There are equations (1) under condition (ii) having exactly one limit cycle surrounding either 1, 7 or 13 critical points, and equations (1) under condition (i) having exactly one hyperbolic limit cycle surrounding either 7 critical points if [p.
The procedure is repeated till we get a limit cycle or a fixed point.
Guzan, "Boundary surface and limit cycles of ternary memory by using forward and backward integration", in Proc.
There are 2 + ([eta] - 2) +1 state transitions, and we have a limit cycle of length c = [eta] + 1.
A significant suppression of the limit cycle magnitude is achieved using this controller.
Time delay induced death in coupled limit cycle oscillators.
They explore the phenomenon of complexity at the lowest level of interacting heterogeneous agents; a middle level where processes for endogenous reasons fail to converge to a point, a limit cycle, or a simple expansion or contraction; and a high level named here meta-complexity that is moving into new perspectives and techniques.
Principles of limit cycle and phase plane approach [16] is adopted to investigate the stability of the system and evaluate the associated hunting critical velocity.