Number Of Small-amplitude Limit Cycles Research Articles

An upper bound for the number of small-amplitude limit cycles in non-smooth Liénard system

The non-smooth Liénard system, a well-known nonlinear model, appears in a natural way in physics, chemistry, biology, and so on, in which periodic phenomena play a relevant role. In this paper, we investigate the small-amplitude limit cycles generated by the Hopf bifurcation of the non-smooth Liénard system. By utilizing the Picard–Fuchs equation, we gain the upper bounds of the number of small-amplitude limit cycles for a generic non-smooth Liénard system. Finally, an example is given to illustrate the efficiency of the theoretical results.

Bifurcation of Limit Cycles from a Focus-Parabolic-Type Critical Point in Piecewise Smooth Cubic Systems

In this paper, we investigate the maximum number of small-amplitude limit cycles bifurcated from a planar piecewise smooth focus-parabolic type cubic system that has one switching line given by the x-axis. By applying the generalized polar coordinates to the parabolic subsystem and computing the Lyapunov constants, we obtain 11 weak center conditions and 9 weak focus conditions at (0,0). Under these conditions, we prove that a planar piecewise smooth cubic system with a focus-parabolic-type critical point can bifurcate at least nine limit cycles. So far, our result is a new lower bound of the cyclicity of the piecewise smooth focus-parabolic type cubic system.

Limit cycles in a switching Liénard system

<p style='text-indent:20px;'>In this paper, we consider a class of quadratic switching Liénard systems with three switching lines. We give an algorithm for computing the Lyapunov constants of this system. Based on this method, we obtain a center condition and three limit cycles bifurcating from the focus <inline-formula><tex-math id="M1">\begin{document}$ (0,0) $\end{document}</tex-math></inline-formula>. Further, an example of quadratic switching systems is constructed to show the existence of six limit cycles bifurcating from the center. This is a new low bound on the maximal number of small-amplitude limit cycles obtained in such quadratic switching systems.</p>

Limit Cycles from Hopf Bifurcation in Nongeneric Quadratic Reversible Systems with Piecewise Perturbations

In this paper, we consider a nongeneric quadratic reversible system with piecewise polynomial perturbations. We use the expansion of the first order Melnikov function to obtain the maximal number of small-amplitude limit cycles produced by Hopf bifurcation in the perturbed systems.

Maximum Number of Small Limit Cycles in Some Rational Liénard Systems with Cubic Restoring Terms

In this paper, we obtain an upper bound for the number of small-amplitude limit cycles produced by Hopf bifurcation in one particular type of rational Liénard systems in the form of [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomials in [Formula: see text] with degrees [Formula: see text] and [Formula: see text], respectively. Furthermore, we show that the upper bound presented here is sharp in the case of [Formula: see text].

Small-Amplitude Limit Cycles of Certain Planar Differential Systems

In this work we consider the polynomial differential system $$\dot{x} = -y + x f(y)$$ , $$\dot{y} =x + y f(x)$$ , where f is a polynomial. This system is a certain generalization of the classical Lienard system. For that system, we solve the center problem for such family and compute the order of degeneracy of a weak focus for obtaining the maximum number of small-amplitude limit cycles.

Bifurcations of small limit cycles in Liénard systems with cubic restoring terms

In this paper, we study bifurcations of small-amplitude limit cycles of Liénard systems of the form x˙=y−F(x), y˙=−g(x), where g(x) is a cubic polynomial, and F(x) is a smooth or piecewise smooth polynomial of degree n. By using involutions, we obtain sharp upper bounds of the number of small-amplitude limit cycles produced around a singular point for some systems of this type.

Bifurcation analysis on a class of Z2-equivariant cubic switching systems showing eighteen limit cycles

In this paper, the method developed for computing the Lyapunov constants of planar switching systems associated with an elementary singular point is applied to study bifurcation of limit cycles in a cubic switching system. A complete classification on the center conditions and 16 limit cycles of this system are obtained around the two foci (1,0) and (−1,0). Further, with the method, an example of cubic switching systems is constructed to show the existence of 18 small-amplitude limit cycles bifurcating from centers. This is a new lower bound on the maximal number of small-amplitude limit cycles obtained in such cubic switching systems. Finally, a method is present to show the realization of the 18 limit cycles.

Hopf Bifurcation of Z2-Equivariant Generalized Liénard Systems

In this paper, we consider a class of Liénard systems, described by [Formula: see text], with [Formula: see text] symmetry. Particular attention is given to the existence of small-amplitude limit cycles around fine foci when [Formula: see text] is an odd polynomial function and [Formula: see text] is an even function. Using the methods of normal form theory, we found some new and better lower bounds of the maximal number of small-amplitude limit cycles in these systems. Moreover, a complete classification of the center conditions is obtained for such systems.

Liénard Equation and Its Generalizations

In this paper, we first present a survey of the known results on limit cycles and center conditions for Liénard differential systems. Next we propose a generalization of such systems and we study their center conditions and the number of small-amplitude limit cycles that can bifurcate from the origin. Computing the focal values and using Gröbner bases we find the center conditions for such systems up to a certain degree. We also establish a conjecture about the center conditions for such systems when they have arbitrary degree.

Limit Cycles for a Class of Piecewise Smooth Quadratic Differential Systems with Multiple Parameters

This paper considers a class of quadratic differential systems with an isochronous center under small piecewise smooth perturbations. Two perturbation parameters at different scales are included in the system. By using the first order Melnikov function, we obtain some new results on the number of small-amplitude limit cycles bifurcating around an isochronous center.

Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations

This paper is the continuation of our previous papers [16] and [17] where we studied small-amplitude limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations. We find optimal upper bounds for the number of small-amplitude limit cycles in these slow-fast codimension 3 bifurcations. We use techniques from geometric singular perturbation theory.

Center conditions in a switching Bautin system

A new method with an efficient algorithm is developed for computing the Lyapunov constants of planar switching systems, and then applied to study bifurcation of limit cycles in a switching Bautin system. A complete classification on the conditions of a singular point being a center in this Bautin system is obtained. Further, an example of switching systems is constructed to show the existence of 10 small-amplitude limit cycles bifurcating from a center. This is a new lower bound of the maximal number of small-amplitude limit cycles obtained in quadratic switching systems near a singular point.

HOPF BIFURCATION OF LIÉNARD SYSTEMS BY PERTURBING A NILPOTENT CENTER

As we know, Liénard system is an important model of nonlinear oscillators, which has been widely studied. In this paper, we study the Hopf bifurcation of an analytic Liénard system by perturbing a nilpotent center. We develop an efficient method to compute the coefficients bl appearing in the expansion of the first order Melnikov function by finding a set of equivalent quantities B2l+1 which are able to calculate directly and can be used to study the number of small-amplitude limit cycles of the system. As an application, we investigate some polynomial Liénard systems, obtaining a lower bound of the maximal number of limit cycles near a nilpotent center.

Symbolic computation of limit cycles associated with Hilbert’s 16th problem

This paper is concerned with the practical complexity of the symbolic computation of limit cycles associated with Hilbert’s 16th problem. In particular, in determining the number of small-amplitude limit cycles of a non-linear dynamical system, one often faces computing the focus values of Hopf-type critical points and solving lengthy coupled polynomial equations. These computations must be carried out through symbolic computation with the aid of a computer algebra system such as Maple or Mathematica, and thus usually gives rise to very large algebraic expressions. In this paper, efficient computations for the focus values and polynomial equations are discussed, showing how to deal with the complexity in the computation of non-linear dynamical systems.

Coexistence of limit cycles and invariant algebraic curves for a Kukles system

We consider a Kukles system of the form x ˙ = - y , y ˙ = f ( x , y ) where f ( x , y ) is a polynomial with real coefficients of degree d without y as a divisor. We study the maximum number of small-amplitude limit cycles for these kind of systems which can coexist with invariant algebraic curves. We give all the possible distributions of invariant straight lines for a Kukles system and we give some bounds for the number of limit cycles. We also give some necessary conditions for the existence of an invariant algebraic curve of degree ⩾ 2 and we study the possible coexistence of this curve and a limit cycle. Finally, we give two examples of cubic Kukles systems both with an invariant hyperbola. In the first example the hyperbola coexists with a center and in the second one it coexists with two small-amplitude limit cycles. These two examples contradict a previous result given in Ann. Differential Equations 7(3) (1991) 323.

Generalized cubic Liénard equations

This article presents a new method for determining the Liapunov quantities of Liénard systems with either cubic damping or restoring terms. The first eleven quantities have been computed on a PC, whereas the algorithm used previously requires the use of high powered computers with lots of memory. The reduction part of the algorithm is simplified by expressing the Liapunov quantities in a special form. The maximum number of small-amplitude limit cycles which may be bifurcated from the origin is given for certain systems.

Quadratic Systems with Center and Their Perturbations

We study the quadratic systems on a plane ż = (i + λ)z + Az2 + B |z|2 + Cz̄2, z = x + iy. We give a simple algebraic proof of the center conditions, λ = 0 and B = 0 or A = −12, B = 1 or A = Ā, B = 1, C = C̄ or A = 2, B = 1, |C| = 1, and of the theorem of Bautin that the number of small-amplitude limit cycles bifurcating from a center or a focus is not greater than 3. We present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case. Here one mistake in Bautin′s work is revealed and some questions concerning focus numbers are answered. We study also global (not small) limit cycles for a small perturbation of the center case with two invariant lines (the Lotka-Volterra system) using Abelian integrals. We show that the number of zeroes of the corresponding Abelian integral is 0, 1, or 2. Some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two-fold center (when two center conditions hold). The problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients.

The number of limit cycles of certain polynomial differential equations

SynopsisTwo-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.