This paper concerns with the number of limit cycles for a cubic Hamiltonian system under cubic perturbation. The fact that there exist 9–11 limit cycles is proved. The different distributions of limit cycles are given by using methods of bifurcation theory and qualitative analysis, among which two distributions of eleven limit cycles are new.  2005 Elsevier Inc. All rights reserved.

This paper will review the consequences of having limit cycles in the renormalization group. Characteristics such as discrete symmetry invariance and log periodic behavior of observables are expected in systems that exhibit limit cycles in their renormalization group. Some realizations of limit cycles will be explained. Specifically limit cycles in RG flows will be used to explain the Efimov ef...

We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable.

Essentially all models that have been proposed for predator-prey systems are shown to possess either a stable point equilibrium or a stable limit cycle. This stable limit cycle, an explicitly nonlinear feature, is commonly overlooked in conventional analyses of these models. Such a stable limit cycle provides a satisfying explanation for those animal communities in which populations are observe...

In this paper, a Kolmogorov-type model, which includes the Gause-type model (Kuang and Freedman, 1988), the general predator-prey model (Huang 1988, Huang and Merrill 1989), and many other specialized models, is studied. The stability of equilibrium points, the existence and uniqueness of limit cycles in the model are proved.

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This paper deals with the problem of finding upper bounds on the number of periodic solutions of a class of one-dimensional non-autonomous differential equations: those with the right-hand sides being polynomials of degree n and whose coefficients are real smooth 1-periodic functions. The case n = 3 gives the so-called Abel equations which have been thoroughly studied and are quite understood. ...

We consider one–parameter families of 2–dimensional vector fields Xμ having in a convenient region R a semistable limit cycle of multiplicity 2m when μ = 0, no limit cycles if μ / 0, and two limit cycles one stable and the other unstable if μ ' 0. We show, analytically for some particular families and numerically for others, that associated to the semistable limit cycle and for positive integer...

Limit cycles are common in hybrid systems. However the nonsmooth dynamics of such systems makes stability analysis difficult. This paper uses recent extensions of trajectory sensitivity analysis to obtain the characteristic multipliers of nonsmooth limit cycles. The stability of a limit cycle is determined by its characteristic multipliers. The concepts are illustrated using a coupled tank syst...

This paper presents some new results which we obtained recently for the study of limit cycles of nonlinear dynamical systems. Particular attention is given to small limit cycles of generalized Liénard systems in the vicinity of the origin. New results for a number of cases of the Liénard systems are presented with the Hilbert number, b H ði; jÞ 1⁄4 b H ðj; iÞ, for j = 4, i = 10,11,12,13; j = 5,...