Limit Cycles In Autonomous Systems Research Articles

On limit cycles of autonomous systems

We consider the problem of the existence of limit cycles for autonomous systems of differential equations. We present quite elementary considerations that can be useful in discussing qualitative issues that arise in the course of ordinary differential equations. We establish that any simple closed curve defined by the equation \(F(x,y)=1\) with a sufficiently general function \(F\) is a limit cycle for the corresponding autonomous system on the plane (and even for an infinite number of systems depending on the real parameter). These systems are written out explicitly. We analyze in detail several specific examples. Graphic illustrations are provided.

Ways for detection of the exact number of limit cycles of autonomous systems on the cylinder

For real autonomous systems of differential equations with continuously differentiable right-hand sides, the problem of detecting the exact number and localization of the second-kind limit cycles on the cylinder is considered. To solve this problem in the absence of equilibria of the system on the cylinder, we have developed our previously proposed ways consisting in a sequential two-step application of the Dulac – Cherkas test or the Dulac test. Additionally, a new way has been worked out using the generalization of the Dulac – Cherkas or Dulac test at the second step, where the requirement of constant sign for divergence is replaced by the transversality condition of the curves on which the divergence vanishes. With the help of the developed ways, closed transversal curves are found that divide the cylinder into subdomains surrounding it, in each of which the system has exactly one second-kind limit cycle.The practical efficiency of the mentioned ways is demonstrated by the example of a pendulum-type system, for which, in the absence of equilibria, the existence of exactly three second-kind limit cycles on the entire phase cylinder is proved.

Dulac-Cherkas Test for Determining the Exact Number of Limit Cycles of Autonomous Systems on the Cylinder

For real autonomous two-dimensional systems of differential equations with continuously differentiable right-hand sides 2π-periodic in one of the variables, we consider the problem of determining the exact number of limit cycles of the second kind on the cylinder. If the system has no equilibria, then we propose to solve this problem by two methods based on a successive two-step application of the Dulac-Cherkas test or one of its modifications, which permits determining closed transversal curves dividing the cylinder into subdomains surrounding it and such that the system has precisely one limit cycle of the second kind in each of them. The efficiency of the proposed methods is illustrated with examples of systems corresponding to the Abel equation, for which the number of limit cycles on the whole phase cylinder is established.

Continuation of limit cycles near saddle homoclinic points using splines in phase space

We present a new algorithm for continuation of limit cycles of autonomous systems as a system parameter is varied. The algorithm works in phase space with an ordered set of points on the limit cycle, along with spline interpolation. Currently popular algorithms in bifurcation analysis packages compute time-domain approximations of limit cycles using either shooting or collocation. The present approach seems useful for continuation near saddle homoclinic points, where it encounters a corner while time-domain methods essentially encounter a discontinuity (a relatively short period of rapid variation). Other phase space-based algorithms use rescaled arclength in place of time, but subsequently resemble the time-domain methods. Compared to these, we introduce additional freedom through a variable stretching of arclength based on local curvature, through the use of an auxiliary index-based variable. Several numerical examples are presented. Comparisons with results from the popular package, MATCONT, are favorable close to saddle homoclinic points.

A Precise Estimate of the Number of Limit Cycles of Autonomous Systems on the Plane

A Precise Estimate of the Number of Limit Cycles of Autonomous Systems on the Plane

Numerical analysis of the flip bifurcation of maps

A numerical procedure for the analysis of the flip bifurcation of maps in R n is described. The procedure is based on normal forms and the center-manifold approach and can be applied to study period doubling of limit cycles in autonomous systems as well as period doubling of periodic solutions of time-periodic systems. The procedure has been programmed in fortran-77, and the code is available on request from the second author.

The energetic/variational interpretation of Poincaré's criterion for the orbital stability of limit cycles of autonomous systems

This paper supplies an energetic interpretation and derivation of Poincaré's criterion for the orbital stability of the limit cycles of the autonomous (quasilinear) equation q ̈ + q = Q(q,q) → εF(q,q) ( q = coordinate , overdot = time derivative, ε = small number, F, Q = msmooth nonlinear functions of q,q), based on the principles of Lagrangean/Hamiltonian dynamics. Specifically, it is shown that the time derivative of the “net work(-ing)” (= a linear differential energetic expression equal to the virtual work of Q, minus the first virtual variation of the system's total energy), equals this net work(-ing) times δQ/δq; a subsequent integration over the limit cycle's solution and period yields the well-known time-integral stability test. The connection of the above with: i) the standard Stoker/McCuskey derivation of the criterion, and ii) the well-known Liouville's theorem of volume change in phase-space are pointed out.