In this paper, we address stability and robustness properties of hybrid limit cycles for a class of hybrid systems with multiple jumps in one period. The main results entail equivalent characterizations of stability of hybrid limit cycles for hybrid systems. The hybrid limit cycles may have multiple jumps in one period and the jumps are allowed to occur on sets. Conditions guaranteeing robustne...

In this paper, we study limit cycles in the Liénard equation: ẍ + f(x)ẋ + g(x) = 0 where f(x) is an even polynomial function with degree 2m, while g(x) is a third-degree, odd polynomial function. In phase space, the system has three fixed points, one saddle point at the origin and two linear centers which are symmetric about the origin. It is shown that the system can have 2m small (local) limi...

M. Pineda-Krch ([email protected]), H. J. Blok and M. Doebeli, Dept of Zoology and Dept of Mathematics, Univ. of British Columbia, Vancouver, Canada BC V6T 1Z4. Present address for MP-K: Center for Animal Disease Modeling and Surveillance, Dept of Medicine and Epidemiology, School of Veterinary Medicine, Univ. of California, Davis, CA 95616, USA. U. Dieckmann, Evolution and Ecology Program,...

We consider the class of polynomial differential equation x&= , 2(,)(,)(,)nnmnmPxyPxyPxy++++2(,)(,)(,)nnmnmyQxyQxyQxy++&=++. For where and are homogeneous polynomials of degree i. Inside this class of polynomial differential equation we consider a subclass of Darboux integrable systems. Moreover, under additional conditions we proved such Darboux integrable systems can have at most 1 limit cycle.

Unlike linear ones, nonlinear business cycle models can generate sustained fluctuations even in the absence of shocks (e.g., via limit cycles/chaos). A popular approach to solving is perturbation methods. I show that, as typically implemented, these methods are incapable finding solutions featuring cycles or chaos. Fundamentally, only required not explode, while standard algorithms seek that me...

In this paper, the existence of limit cycles for the specific bilinear systems is explored. Based on the BellmanGronwall inequality approach, not only the exponentially stable limit cycles phenomenon of such systems can be certified but also the oscillation behaviors of such systems can be correctly predicted. Finally, a numerical example is provided to illustrate the feasibility and effectiven...

An industrially applied LCC power converter is modelled as a hybrid system. It is found that the hybrid system given a parameter set up and different initial conditions has three different limit cycles, one unstable and two asymmetric stable limit cycles. Effects of the asymmetric limit cycles are considered.

Using the method of qualitative analysis we show that five perturbed cubic Hamiltonian systems have the same distribution of limit cycles and have 11 limit cycles for some parameters. The accurate location of each limit cycle is given by numerical exploration. In other words, we demonstrate the existence of 11 limit cycles and their distribution in five perturbed systems in two ways, the result...

Bifurcation of limit cycles in a cubic Hamiltonian system with quintic perturbed terms is investigated using both qualitative analysis and numerical exploration. The investigation is based on detection functions which are particularly effective for the perturbed cubic Hamiltonian system. The study reveals firstly that there are at most 15 limit cycles in the cubic Hamiltonian system with pertur...

We give a family of planar polynomial differential systems whose limit cycles can be explicitly described using polar coordinates. Moreover, we characterize the multiplicity of each one of the limit cycles whenever they exist. The given family of planar polynomial differential systems can have at most two limit cycles, counted with multiplicity. As an application of this result we give an examp...