Planar Polynomial Systems Research Articles

Limit cycles and homoclinic networks in two-dimensional polynomial systems.

In this paper, the properties of equilibriums in planar polynomial dynamical systems are studied. The homoclinic networks of sources, sinks, and saddles in self-univariate polynomial systems are discussed, and the numbers of sources, sinks, and saddles are determined through a theorem, and the first integral manifolds are determined. The corresponding proof of the theorem is completed, and a few illustrations of networks for source, sinks, and saddles are presented for a better understanding of the homoclinic networks. Such homoclinic networks are without any centers even if the networks are separated by the homoclinic orbits. The homoclinic networks of positive and negative saddles with clockwise and counterclockwise limit cycles in crossing-univariate polynomial systems are studied secondly, and the numbers of saddles and centers are determined through a theorem, and the first integral manifolds are determined through polynomial functions. The corresponding proof of the theorem is given, and a few illustrations of networks of saddles and centers are given to show the corresponding geometric structures. Such homoclinic networks of saddles and centers are without any sources and sinks. Since the maximum equilibriums for such two types of planar polynomial systems with the same degrees are discussed, the maximum centers and saddles should be obtained, and maximum sinks, sources, and saddles should be achieved. This paper may provide a different way to determine limit cycles in the Hilbert 16th problem.

Critical periods in planar polynomial centers near a maximum number of cusps

We provide the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. This lower bound doubles the previous one from the literature, and we end up with at least n2−2 (resp. n2−2n−1) critical periods for planar polynomial systems of odd (resp. even) degree n. Key idea is the perturbation of a vector field with many cusp equilibria, whose construction is by itself a nontrivial construction that uses elements of catastrophe theory.

The number of limit cycles for regularized piecewise polynomial systems is unbounded

In this paper, we extend the slow divergence-integral from slow-fast systems, due to De Maesschalck, Dumortier and Roussarie, to smooth systems that limit onto piecewise smooth ones as ϵ→0. In slow-fast systems, the slow divergence-integral is an integral of the divergence along a canard cycle with respect to the slow time and it has proven very useful in obtaining good lower and upper bounds of limit cycles in planar polynomial systems. In this paper, our slow divergence-integral is based upon integration along a generalized canard cycle for a piecewise smooth two-fold bifurcation (of type visible-invisible called VI3). We use this framework to show that the number of limit cycles in regularized piecewise smooth polynomial systems is unbounded.

The Songling system has exactly four limit cycles

Determining how many limit cycles a planar polynomial system of differential equations can have is a remarkably hard problem. One of the main difficulties is that the limit cycles can reside within areas of vastly different scales. This makes numerical explorations very hard to perform, requiring high precision computations, where the necessary precision is not known in advance. Using rigorous computations, we can dynamically determine the required precision, and localize all limit cycles of a given system. We prove that the Songling system of planar, quadratic polynomial differential equations has exactly four limit cycles. Furthermore, we give precise bounds for the positions of these limit cycles using rigorous computational methods based on interval arithmetic. The techniques presented here are applicable to the much wider class of real-analytic planar differential equations.

On the dynamic of the isotropic star

In this paper, we investigate the dynamical behavior of the planar polynomial system which describes the evolution of the isotropic star. It is shown that the system has no global [Formula: see text] and no global analytic first integrals. It has an invariant algebraic curve with algebraic multiplicity [Formula: see text] and an exponential factor that comes from the multiplicity of the infinite invariant straight line. It is proved that the system can be changed into the Liénard system. By using a dominant balance analysis, we prove the system has a general solution which eventuates to a finite-time singularity. Finally, we prove the trajectories of the vector field associated with the planar system of the isotropic star do not create a trajectory manifold which means there is no pseudo-Riemannian metric [Formula: see text] in the sense of trajectory metric such that the trajectories of the isotropic star system be geodesic.

Local cyclicity in low degree planar piecewise polynomial vector fields

In this work, we are interested in isolated crossing periodic orbits in planar piecewise polynomial vector fields defined in two zones separated by a straight line. In particular, in the number of limit cycles of small amplitude. They are all nested and surrounding one equilibrium point or a sliding segment. We provide lower bounds for the local cyclicity for planar piecewise polynomial systems, Mpc(n), with degrees 2, 3, 4, and 5. More concretely, Mpc(2)≥13,Mpc(3)≥26,Mpc(4)≥40, and Mpc(5)≥58. The computations use parallelization algorithms.

Visualization of Four Limit Cycles in Near-Integrable Quadratic Polynomial Systems

It has been known for almost 40 years that general planar quadratic polynomial systems can have four limit cycles. Recently, four limit cycles were also found in near-integrable quadratic polynomial systems. To help more people to understand limit cycles theory, the visualization of such four numerically simulated limit cycles in quadratic systems has attracted researchers’ attention. However, for near-integral systems, such visualization becomes much more difficult due to limitation on choosing parameter values. In this paper, we start from the simulation of the well-known quadratic systems constructed around the end of 1979, then reconsider the simulation of a recently published quadratic system which exhibits four big size limit cycles, and finally provide a concrete near-integral quadratic polynomial system to show four normal size limit cycles.

Nonexistence of invariant manifolds in fractional-order dynamical systems

Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional order systems.

New lower bound for the number of critical periods for planar polynomial systems

In this paper, we construct two classes of planar polynomial Hamiltonian systems having a center at the origin, and obtain the lower bounds for the number of critical periods for these systems. For polynomial potential systems of degree n, we provide a lower bound of n−2 for the number of critical periods, and for polynomial systems of degree n, we acquire a lower bound of n2/2+n−5/2 when n is odd and n2/2−2 when n is even for the number of critical periods. To the best of our knowledge, these lower bounds are new, moreover the latter one is twice the existing results up to the dominant term.

A new approach for the study of limit cycles

We prove that star-like limit cycles of any planar polynomial system can also be seen either as solutions defined on a given interval of a new associated planar non-autonomous polynomial system or as heteroclinic solutions of a 3-dimensional polynomial system. We illustrate these points of view with several examples. One of the key ideas in our approach is to decompose the periodic solutions as the sum of two suitable functions. As a first application we use these new approaches to prove that all star-like reversible limit cycles are algebraic. As a second application we introduce a function whose zeroes control the periodic orbits that persist as limit cycles when we perturb a star-like reversible center. As far as we know this is the first time that this question is solved in full generality. Somehow, this function plays a similar role that an Abelian integral for studying perturbations of Hamiltonian systems.

Bifurcation of a Kind of 1D Piecewise Differential Equation and Its Application to Piecewise Planar Polynomial Systems

This paper deals with a kind of piecewise smooth equation which is linear in the dependent variable. We study the problem of lower bounds for the maximum number of limit cycles of such equations using Melnikov functions. First of all, using the first order Melnikov function, we prove that these differential equations have a sharp upper bound for the number of the limit cycles which bifurcate from the periodic orbits and cross the separation straight line. Furthermore, in some cases the maximum number of these limit cycles is three, up to any order analysis. In the end, we apply this result on a kind of piecewise smooth planar system which has a separation curve with [Formula: see text] up to homomorphism.

Limit cycle bifurcations in perturbations of planar piecewise smooth systems with multiply lines of critical points

This paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems, whose unperturbed system has a center at the origin and may have invariant multiply lines. By Melnikov function method, this paper provides the upper and lower bounds for the Poincaré cyclicity. In particular, the exact cyclicities of two specific cases are given using analyzing method, one of which has been discussed in the paper [7] .

Reversibility in polynomial systems of ODE’s

For a given family of real planar polynomial systems of ordinary differential equations depending on parameters, we consider the problem of how to find the systems in the family which become time-reversible after some affine transformation. We first propose a general computational approach to solve this problem, and then demonstrate its usage for the case of the family of quadratic systems.

A counterexample to the composition condition conjecture for polynomial Abel differential equations

Polynomial Abel differential equations are considered a model problem for the classical Poincaré center–focus problem for planar polynomial systems of ordinary differential equations. In the last few decades, several works pointed out that all centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.

Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis

In this paper, we present a method of higher-order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations. This method is equivalent to higher-order Melinikov function approach used for studying bifurcation of limit cycles around a center but simpler. Attention is focused on planar cubic polynomial systems and particularly it is shown that the system studied by Żoła̧dek (1995) [24] can indeed have eleven limit cycles under perturbations at least up to 7th order. Moreover, the pattern of numbers of limit cycles produced near the center is discussed up to 39th-order perturbations, and no more than eleven limit cycles are found.

A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities

This paper is devoted to study the planar polynomial system:x˙=ax−y+Pn(x,y),y˙=x+ay+Qn(x,y), where a∈R and Pn,Qn are homogeneous polynomials of degree n≥2. Denote ψ(θ)=cos⁡(θ)⋅Qn(cos⁡(θ),sin⁡(θ))−sin⁡(θ)⋅Pn(cos⁡(θ),sin⁡(θ)). We prove that the system has at most 1 limit cycle surrounding the origin provided (n−1)aψ(θ)+ψ˙(θ)≠0. Furthermore, this upper bound is sharp. This is maybe the first uniqueness criterion, which only depends on a (linear) condition of ψ, for the limit cycles of this kind of systems. We show by examples that in many cases, the criterion is applicable while the classical ones are invalid. The tool that we mainly use is a new estimate for the number of limit cycles of Abel equation with coefficients of indefinite signs. Employing this tool, we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin.

Estimate for the number of limit cycles of Abel equation via a geometric criterion on three curves

This paper is devoted to the investigation of Abel equation $$\dot{x}=S(t,x)=\sum ^3_{i=0}a_i(t)x^i$$ , where $$a_i\in \mathrm C^{\infty }([0,1])$$ . A solution x(t) with $$x(0)=x(1)$$ is called a periodic solution. And an orbit $$x=x(t)$$ is called a limit cycle if x(t) is a isolated periodic solution. By means of Lagrange interpolation formula, we give a criterion to estimate the number of limit cycles of the equation. This criterion is only concerned with S(t, x) on three non-intersecting curves. Applying our main result, we prove that the maximum number of limit cycles of the equation is 4 if $$a_2(t)a_0(t)<0$$ . To the best of our knowledge, this is a nontrivial supplement for a classical result which says that the equation has at most 3 limit cycles when $$a_2(t)\ne 0$$ and $$a_0(t)\equiv 0$$ . We also study a planar polynomial system with homogeneous nonlinearities: $$\begin{aligned} \dot{x}=ax-y+P_n(x,y),\ \ \dot{y}=x+ay+Q_n(x,y), \end{aligned}$$ where $$a\in \mathbb R$$ and $$P_n, Q_n$$ are homogeneous polynomials of degree $$n\ge 2$$ . Denote by $$\psi (\theta )=\cos (\theta )\cdot Q_n\big (\cos (\theta ),\sin (\theta )\big )-\sin (\theta )\cdot P_n\big (\cos (\theta ),\sin (\theta )\big )$$ . We prove that if $$(n-1)a\psi (\theta )+\dot{\psi }(\theta )\ne 0$$ , then the polynomial system has at most 1 limit cycle surrounding the origin, and the multiplicity is no more than 2.

Planar quasi-homogeneous polynomial systems with a given weight degree

In this paper, we investigate a class of quasi-homogeneous polynomial systems with a given weight degree. Firstly, by some analytical skills, several properties about this kind of systems are derived and an algorithm can be established to obtain all possible explicit systems for a given weight degree. Then, we focus on center problems for such systems and provide some necessary conditions for the existence of centers. Finally, for a specific quasi-homogeneous polynomial system, we characterize its center and prove that the center is not isochronous.

Center problem for systems with two monomial nonlinearities

We study the center problem for planar systems with a linear center at the origin that in complex coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several centers inside this family. To the best of our knowledge this list includes a new class of Darboux centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the given list is exhaustive. We get several partial results that seem to indicate that this is the case. In particular, we solve the question for several general families with arbitrary high degree and for all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest known order for weak-foci of planar polynomial systems of some given degrees.

The center problem and composition condition for Abel differential equations

The classical Poincaré center-focus problem for planar polynomial systems of ordinary differential equations can be transformed, in certain particular cases, to the center problem of a trigonometric Abel differential equation. Several research papers focused on the study of the center problem for trigonometric Abel differential equations. Polynomial Abel differential equations are also considered in the literature as a model problem. In this work we make a survey of the most important results in this context and we provide the state of the art of several related conjectures. We give two new results on these conjectures.