Limit Cycles

Limit Cycles

An attractor for non-linear dynamic systems which has periodic cycles or orbits in phase space. An example is an undamped pendulum which will have a closed circle orbit equal to the amplitude of the pendulum's swing. See: Attractor, Phase Space.
References in periodicals archive ?
The stationary limit cycles bifurcating from the Hopf points can be obtained by applying the steady-state condition to (48).
Among the topics are basic concepts and linearized problems of systems, an isochronous center in a complex domain, the theory of center-focus and bifurcation of limit cycles for a class of multiple singular points, local and non-local bifurcations of perturbed Zq-equivalent Hamiltonian vector fields, and center-focus problems and bifurcations of limit cycles for three-multiple nilpotent singular points.
Van der Pol found stable oscillations, now known as limit cycles, in electrical circuits employing vacuum tubes.
Other authors [6] have specifically considered this theory when studying the limit cycles and related phenomena in systems with symmetry.
Guzan, "Boundary surface and limit cycles of ternary memory by using forward and backward integration", in Proc.
Our second main result states that when [DELTA] < 2, only fixed points can occur as limit cycles.
The parameter values taken from the background structures consist of chaotic solutions to the tritrophic food chain (modified by McCann and Yodzis, 1994a, which were also employed by McCann and Hastings, 1997) and unstable solutions (two-point limit cycles and top predator extinction to the competition setting).
Phase plane is a direct method refers graphically to determine the existence of limit cycles in place the system behavior over the entire plane can be visualized and limit cycles can be easily identified [16].
Goldwyn and Cox (1965) used Lyapunov theory to generate unstable limit cycles as the boundary of a stable equilibrium.
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.
Langford's fundamental work in the field, the topics here include flow invariant subspaces for lattice dynamical systems, low- to high-dimensional behavior in waves in extended systems, mixed mode oscillations due to the generalized Canard phenomenon, bioremediation of waste in a porous medium, bifurcation of gyroscopic systems near a O:1 resonance, high dimensional data clustering from a dynamical systems point of view, and the computation of limit cycles as the second part of Hilbert's tenth problem.
We have already seen that some specifications of this matrix can lead to limit cycles, but now we focus on cases in which there are no limit cycles.