On the number of limit cycles by perturbing a piecewise smooth Hamilton system with two straight lines of separation

This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than in the case with one straight line of separation and more limit cycles can be bifurcated.     

Lower bounds for the number of limit cycles of trigonometric Abel equations

We consider the Abel equation , where A(t) and B(t) are trigonometric polynomials of degree n and m, respectively, and we give lower bounds for its number of isolated periodic orbits for some values of n and m. These lower bounds are obtained by two different methods: the study of the perturbations of some Abel equations having a continuum of periodic orbits and the Hopf-type bifurcation of periodic orbits from the solution x=0.     

Melnikov functions and limit cycles in piecewise smooth perturbations of a linear center using regularization method

In this article we study limit cycles in piecewise smooth perturbations of a linear center. In this setting it is common to adapt classical results for smooth systems, like Melnikov functions, to non-smooth ones. However, there is little justification for this procedure in the literature. By using the regularization method we give a theoretical proof that supports the use of Melnikov functions directly from the original non-smooth problem.     

Bifurcation of limit cycles by perturbing piecewise non-Hamiltonian systems with nonlinear switching manifold

This paper is devoted to the study of limit cycles that can bifurcate of a perturbation of piecewise non-Hamiltonian systems with nonlinear switching manifold. We derive the first order Melnikov function to these systems. As application, the sharp upper bound of the number of bifurcated limit cycles of two concrete systems, whose switching manifolds are algebraic curves, is presented.     

Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth systems with a cusp

For a piecewise analytical Hamiltonian system with a cusp on a switch line, which has a family of periodic orbits near a generalized homoclinic loop, we study the maximum number of limit cycles bifurcating from the periodic orbits. For doing so, we first obtain the asymptotic expressions of the Melnikov functions near the loop. Finally we present two examples illustrating applications of the main results.     

On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems.New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results.A relevant application to the McKean piecewise linear model of a single neuron activity is included.     

Periodic solutions with nonconstant sign in Abel equations of the second kind

The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation x?x−1, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.     

Estimating the number of limit cycles in polynomials systems

We describe a method based on algorithms of computational algebra for obtaining an upper bound for the number of limit cycles bifurcating from a center or a focus of polynomial vector field. We apply it to a cubic system depending on six parameters and prove that in the generic case at most six limit cycles can bifurcate from any center or focus at the origin of the system.     

On the existence and uniqueness of limit cycles for generalized Liénard systems

In this paper, we consider a generalized Liénard system

     

Method of Lyapunov functions for differential equations with piecewise constant delay

We address differential equations with piecewise constant argument of generalized type [5], [6], [7] and [8] and investigate their stability with the second Lyapunov method. Despite the fact that these equations include delay, stability conditions are merely given in terms of Lyapunov functions; that is, no functionals are used. Several examples, one of which considers the logistic equation, are discussed to illustrate the development of the theory. Some of the results were announced at the 14th International Congress on Computational and Applied Mathematics (ICCAM2009), Antalya, Turkey, in 2009.     

On the reduction principle for differential equations with piecewise constant argument of generalized type

In this paper we introduce a new type of differential equations with piecewise constant argument (EPCAG), more general than EPCA [K.L. Cooke, J. Wiener, Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984) 265-297; J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993]. The Reduction Principle [V.A. Pliss, The reduction principle in the theory of the stability of motion, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1964) 1297-1324 (in Russian); V.A. Pliss, Integral Sets of Periodic Systems of Differential Equations, Nauka, Moskow, 1977 (in Russian)] is proved for EPCAG. The structure of the set of solutions is specified. We establish also the existence of global integral manifolds of quasilinear EPCAG in the so-called critical case and investigate the stability of the zero solution.     

On the limit cycles for a class of generalized Kukles differential systems

In this paper, we consider the limit cycles of a class of polynomial differential systems of the form $\dot{x}=-y, \hspace{0.2cm} \dot{y}=x-f(x)-g(x)y-h(x)y^{2}-l(x)y^{3},$ where $f(x)=\epsilon f_{1}(x)+\epsilon^{2}f_{2}(x),$ $g(x)=\epsilon g_{1}(x)+\epsilon^{2}g_{2}(x),$ $h(x)=\epsilon h_{1}(x)+\epsilon^{2}h_{2}(x)$ and $l(x)=\epsilon l_{1}(x)+\epsilon^{2}l_{2}(x)$ where $f_{k}(x),$ $g_{k}(x),$ $h_{k}(x)$ and $l_{k}(x)$ have degree $n_{1},$ $n_{2},$ $n_{3}$ and $n_{4},$ respectively for each $k=1,2,$ and $\varepsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=-y,$ $\dot{y}=x$ using the averaging theory of first and second order.     

Criteria on the existence of limit cycles in planar polynomial differential systems

We summarize known criteria for the non-existence, existence and on the number of limit cycles of autonomous real planar polynomial differential systems, and also provide new results. We give examples of systems which realize the maximum number of limit cycles provided by each criterion. In particular we consider the class of differential systems of the form $\stackrel{?}{x}=P_n(x,y)+P_m(x,y),\stackrel{?}{y}=Q_n(x,y)+Q_m(x,y),$ where $n,m$ are natural numbers with $m>n\ge 1$ and $(P_i,Q_i)$ for $i=n,m$, are quasi-homogeneous vector fields.     

Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus. The sufficient and necessary conditions for existence and uniqueness of limit cycles are also given.     

Periodic orbits in two classes of piecewise smooth maps with positive nonlinear parts

In this paper we consider two classes of one dimensional piecewise smooth continuous maps that have been derived as normal forms for grazing bifurcations of piecewise smooth dynamical systems. These maps are linear on one side of the phase space and nonlinear on the other side. The case of nonlinear parts with negative coefficients has been studied previously and it is proved that period-adding scenarios are generic in this case. In contrast to this result, in our analytical and numerical results, the period-adding scenarios are not observed when the nonlinear parts have positive coefficients. Furthermore, our results suggest that the typical bifurcation scenario is period doubling cascade leading to chaos in this case, which is similar to that of the smooth logistic map.