Maximum Number Of Limit Cycles Research Articles

Limit Cycles of the Discontinuous Piecewise Differential Systems Separated by a Nonregular Line and Formed by a Linear Center and a Quadratic One

During the last decades, the study of discontinuous piecewise differential systems has become an interesting subject of research due to the important applications of this kind of systems to model natural phenomena. In the qualitative theory of differential equations, one of the interesting problems is the detection of the number of limit cycles and their configurations which remains open to date, except for very particular families of differential equations. Here, we are inspired to study the maximum number of limit cycles of the discontinuous piecewise differential systems separated by a nonregular line and formed by a linear center and one of the four classes of quadratic centers. The main tool used to prove our main results is based on the first integrals of such systems. All the computations of this paper are verified using the algebraic manipulator Mathematica.

Limit cycles in a rotated family of generalized Liénard systems allowing for finitely many switching lines

Analytic rotated vector fields have four significant properties: as the rotated parameter $\alpha$ changes, the amplitude of each stable (or unstable) limit cycle varies monotonically, each semi-stable limit cycle bifurcates at most two limit cycles, the isolated homoclinic loop (if exists) disappears while a unique limit cycle with the same stability arises or no closed orbits arise oppositely, and a unique limit cycle arises near the weak focus (if exists). In this paper, we prove that the four properties remain true for a rotated family of generalized Liénard systems having finitely many switching lines. Furthermore, we discuss variational exponent and use it to formulate multiplicity of limit cycles. Then we apply our results to give exact number of limit cycles to a continuous piecewise linear system with three zones and answer to a question on the maximum number of limit cycles in an SD oscillator.

Four Limit Cycles of Three-Dimensional Discontinuous Piecewise Differential Systems Having a Sphere as Switching Manifold

Because of their applications, the study of piecewise-linear differential systems has become increasingly important in recent years. This type of system already exists to model many different natural phenomena in physics, biology, economics, etc. As is well known, the study of the qualitative theory of piecewise differential systems focuses mainly on limit cycles. Most papers studying the problem of existence and the maximum number of limit cycles of piecewise differential systems have precisely considered planar systems. However, few papers have examined this problem in [Formula: see text]. In this paper, our main goal is to examine a class of discontinuous piecewise differential systems in [Formula: see text], where we consider the unit sphere as the separation surface that divides the entire space into two regions, each one has a linear vector field analogous to planar center. In general, it is hard to determine an exact upper bound for the number of limit cycles that a class of differential systems can exhibit. We prove that this class of differential systems can have at most four limit cycles. We show that there are examples of such differential systems with exactly 1, 2, 3 and 4 limit cycles.

Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line

In this article, we study the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems separated by a straight line. We consider three cases, when both Hamiltonians systems in each side of the discontinuity line have simultaneously degree one, two or three. We obtain that in these three cases, this maximum number is zero, one and three, respectively. Moreover, we prove that there are discontinuous piecewise differential systems realizing these maximum number of limit cycles. Note that we have solved the extension of the 16th Hilbert problem about the maximum number of limit cycles that these three classes of discontinuous piecewise differential systems separated by one straight line and formed by two Hamiltonian systems with a degree either one, two, or three, which such systems can exhibit.

Limit cycles of septic polynomial differential systems bifurcating from the periodic annulus of cubic center

This paper focuses on investigating the maximum number of limit cycles bifurcating from the periodic orbits adapted to the cubic system given by ẋ=y−yx+a2,ẏ=−x+xx+a2,where a is a positive number with a≠1. The study specifically examines the perturbation of this system within the class of all septic polynomial differential systems. Our main result demonstrates that the first-order averaging theory associated with the perturbed system yields a maximum of twenty-two limit cycles.

An extension of the 16th Hilbert problem for continuous piecewise linear–quadratic centers separated by a non-regular line

In the last few decades, there has been much interest in studying piecewise differential systems. This is mainly due to the fact that these differential systems allow us to modelize many natural phenomena. In order to describe the dynamics of a differential system, we need to control its periodic orbits and, especially, its limit cycles. In particular, providing an upper bound for the maximum number of limit cycles that such differential systems can exhibit would be desirable, that is solving the extended 16th Hilbert problem. In general, this is an unsolved problem. In this paper, we give an upper bound for the maximum number of limit cycles that a class of continuous piecewise differential systems formed by an arbitrary linear center and an arbitrary quadratic center separated by a non-regular line can exhibit. So for this class of continuous piecewise differential systems, we have solved the extended 16th Hilbert problem, and the upper bound found is seven. The question whether this upper bound is sharp remains open.

Heteroclinic loop bifurcations by perturbing a class of Z2-equivariant quadratic switching Hamiltonian systems with nilpotent singular points

This paper investigates the problem of heteroclinic loop bifurcation by perturbing a class of Z2-equivariant quadratic switching systems with nilpotent singular points. Firstly, it provides sufficient and necessary conditions for the occurrence of a heteroclinic loop. Next, the system is perturbed by piecewise polynomial systems of degree n≥1. The paper then considers the lower bound for the maximum number of limit cycles that bifurcate from the generalized heteroclinic loop, finding at least n+[n2] limit cycles.

New lower bound for the Hilbert number in low degree Kolmogorov systems

Our main goal in this paper is to study the number of small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point a class of polynomial Kolmogorov systems. We denote by MK(n) the maximum number of limit cycles bifurcating from the equilibrium point via a degenerate Hopf bifurcation for a polynomial Kolmogorov vector field of degree n. In this work, we obtain another example such that MK(3)≥6. In addition, we obtain new lower bounds for MK(n) proving that MK(4)≥13 and MK(5)≥22.

Some families of quadratic systems with at most one limit cycle

The work of Chicone and Shafer published in 1982 together with the work of Bamon published in 1986 proved that any polynomial differential system of degree two has finitely many limit cycles. But the problem remains open of providing a uniform upper bound for the maximum number of limit cycles that a polynomial differential system of degree two can have, i.e. the second part of the 16th Hilbert problem restricted to the polynomial differential systems of degree two remains open. Here we present six subclasses of polynomial differential systems of degree two for which we can prove that an upper bound for their maximum number of limit cycles is one.

Limit cycles obtained by perturbing a degenerate center

This paper deals with the maximum number of limit cycles bifurcating from the degenerate centre \[ \dot{x}=-y(3x^2+y^2),\: \dot{y}=x(x^2-y^2), \] when we perturb it inside a class of all homogeneous polynomial differential systems of degree \(5\). Using averaging theory of second order, we show that, at most, five limit cycles are produced from the periodic orbits surrounding the degenerate centre under quintic perturbation. In addition, we provide six examples that give rise to exactly \(5, 4, 3, 2, 1\) and \(0\) limit cycles.

Piecewise Smooth Perturbations to a Class of Planar Cubic Centers

This paper studies the limit cycle bifurcations of a class of planar cubic isochronous centers. For different values of two key parameters, we give an estimate of the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems under arbitrarily small piecewise smooth polynomial perturbation. The main method and technique are based on the first order averaging theory for discontinuous systems and the Argument Principle in complex analysis.

Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region

In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line except for at most one point. In the research literature, many papers deal with the problem of determining the maximum number of limit cycles that these differential systems can have. This problem has been usually approached via large case-by-case analyses which distinguish the many different possibilities for the spectra of the matrices of the differential systems. Here, by using a novel integral characterization of Poincaré half-maps, we prove, without unnecessary distinctions of matrix spectra, that the optimal uniform upper bound for the number of limit cycles of these differential systems is one. In addition, it is proven that this limit cycle, if it exists, is hyperbolic and its stability is determined by a simple condition in terms of the parameters of the system. As a byproduct of our analysis, a condition for the existence of the limit cycle is also derived.

Global dynamics of a degenerate planar piecewise linear differential system with three zones

We study the dynamics of a continuous planar piecewise linear differential system with three zones separated by two parallel lines. We provide a complete study on the maximum number of limit cycles by geometric techniques when the system is degenerate in the left and central zones. The bifurcation diagrams and global phase portraits in the Poincaré disc are given. It is shown that a degenerate continuous planar piecewise linear differential system can have scabbard bifurcation, grazing bifurcation and double limit cycle bifurcation. Moreover, we detect three new scabbard bifurcations, such as both the limit cycle with two zones and the limit cycle with three zones can be bifurcated from a singular continuum with infinite length.

Limit cycles in piecewise smooth perturbations of a class of cubic differential systems

In this paper, we study the bifurcation of limit cycles from a class of cubic integrable non-Hamiltonian systems under arbitrarily small piecewise smooth perturbations of degree n . By using the averaging theory and complex method, the lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems are given at first order in ε . It is also shown that in this case, the maximum number of limit cycles produced by piecewise smooth perturbations is almost twice the upper bound of the maximum number of limit cycles produced by smooth perturbations for the considered systems.

Existence of at most two limit cycles for some non-autonomous differential equations

It is know that the non-autonomous differential equations $ {\rm d}x/{\rm d} t = a(t)+b(t)|x| $, where $ a(t) $ and $ b(t) $ are 1-periodic maps of class $ \mathcal{C}^1 $, have no upper bound for their number of limit cycles (isolated solutions satisfying $ x(0) = x(1) $). We prove that if either $ a(t) $ or $ b(t) $ does not change sign, then their maximum number of limit cycles is two, taking into account their multiplicities, and that this upper bound is sharp. We also study all possible configurations of limit cycles. Our result is similar to other ones known for Abel type periodic differential equations although the proofs are quite different.

On the Limit Cycles of a Class of Discontinuous Piecewise Differential Systems Formed by Two Rigid Centers Governed by Odd Degree Polynomials

We provide an upper bound for the maximum number of limit cycles for the class of discontinuous piecewise differential systems formed by two differential systems separated by the straight line [Formula: see text], one of which is a linear rigid center while the other is a rigid center formed of a linear part plus a homogeneous polynomial of odd degree. We solve the extended 16th Hilbert problem for this class of discontinuous piecewise differential systems.

Estimation to the number of limit cycles for generalized Kukles differential system

This article considers the limit cycles of a class of Kukles polynomial differential systems of the form Equation 5. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a linear center \( \dot{x}=y, \dot{y}=-x,\) by using the averaging theory of first and second order.

Limit cycles of perturbed global isochronous center

We apply the averaging method of first order to study the maximum number of limit cycles of the ordinary differential systems of the form ¨x + x = ε (f1(x, y)y + f2 (x, y)) , ¨y + y = ε (g1(x, y)x + g2 (x, y)) , where f1(x, y) and g1(x, y) are real cubic polynomials; f2(x, y) and g2(x, y) are real quadratic polynomials. Furthermore ε is a small parameter.

General study on limit cycle bifurcation near a double homoclinic loop

In this paper, we study the bifurcation problem of limit cycles near a general double homoclinic loop. We establish a general theory to obtain a lower bound of the maximal number of limit cycles near the double homoclinic loop. As an application, we prove that a near-Hamiltonian system of the form x˙=y(y2−1)+ε∑i=0naix2i+1,y˙=−x has at least [52n] limit cycles for 1≤n≤56. This number is maximal that we can find so far for the system.

Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries

This paper aims to study the limit cycles of planar piecewise polynomial Hamiltonian systems of degree n with the switching boundary y=xm, where m and n are positive integers. We answer a version of the 16th Hilbert problem for such systems, providing an upper bound for the maximum number of limit cycles in the function of m and n. We also are devoted to giving a lower bound via perturbing piecewise linear Hamiltonian systems having at the origin a global center. For this, a complete classification of the center conditions is established. In pursuit of a better lower bound, we require that this global center is nonlinear induced by the piecewise linearity, instead of the normal linear differential center considered in most of the existing articles. This renders that the traveling time of the unperturbed periodic orbits in each smooth zone is not calculable explicitly. Thus it is difficult to use the known Melnikov functions and averaged functions to study the bifurcating limit cycles. To overcome this difficulty, we develop an arbitrary order Melnikov-like function, which does not depend on the traveling time, for general d-dimensional piecewise smooth integrable systems allowing nonlinear switching boundaries. Finally, employing the new Melnikov-like function to the considered perturbation problem, we obtain a lower bound and an upper bound for the maximum number of limit cycles bifurcating from the unperturbed periodic orbits up to any order.