We provide an effective uniform upper bond for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order $k$ of the Melnikov function. The generic case $k=1$ was considered by Binyamini, Novikov and Yakovenko (\cite{BNY-Inf16}). The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.
We study the problem of subharmonic bifurcations for analytic systems in the plane with perturbations depending periodically on time, in the case in which we only assume that the subharmonic Melnikov function has at least one zero. If the order of zero is odd, then there is always at least one subharmonic solution, whereas if the order is even in general other conditions have to be assumed to guarantee the existence of subharmonic solutions. Even when such solutions exist, in general they are not analytic in the perturbation parameter. We show that they are analytic in a fractional power of th…
This paper is devoted to the study of the maximum number of limit cycles, $H(m,n)$, of a planar piecewise linear differential system with two zones separated by the curve $y^n-x^m=0$, with $n,m$ being positive integers. More precisely, we provide a lower estimate of $ H(m,n)$. for all $m,n\in \mathbb{N}$, for piecewise linear perturbations of the linear center using some recent results about Chebyshev systems with positive accuracy and Melnikov Theory
We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection. We show how our definition reproduces the classical methods of Poincaré and Melnikov and specializes to methods previously used for exact symplectic and volume-preserving maps. We use the method to detect …
Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds
State Key Lab of Processors, Institute of Computing TechnologyChinese Academy of Sciences（处理器国家重点实验室，中国科学院计算技术研究所）；School of Computer Science and TechnologyUniversity of Chinese Academy of Sciences（中国科学技术大学计算机科学与技术学院）
In this paper we study polynomial Hamiltonian systems $dF=0$ in the plane and their small perturbations: $dF+εω=0$. The first nonzero Melnikov function $M_μ=M_μ(F,γ,ω)$ of the Poincaré map along a loop $γ$ of $dF=0$ is given by an iterated integral. In a previous work (see arXiv 1703.03837), we bounded the length of the iterated integral $M_μ$ by a geometric number $k=k(F,γ)$ which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system $F$ and its orbit $γ$ having infinite orbit depth. If our conjecture is true, for this example th…
We consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant with the frequency vector of the perturbation. We assume that the unperturbed system is locally integrable and anisochronous, and that the frequency vector of the perturbation satisfies the Bryuno condition. Existence of resonant solutions is related to the zeroes of a suitable function, called the Melnikov function - by analogy with the periodic case. We show that, if the Melnikov function has a zero …
We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is diffe…
In this paper, we are interested in providing lower estimations for the maximum number of limit cycles $H(n)$ that planar piecewise linear differential systems with two zones separated by the curve $y=x^n$ can have, where $n$ is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that $H(2)\geq 4,$ $H(3)\geq 8,$ $H(n)\geq7,$ for $n\geq 4$ even, and $H(n)\geq 9,$ for $n\geq 5$ odd. This improves all the previous results for $n\geq2.$ Our analysis is mainly based on some recent results about C…
We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter $\varepsilon>0$ and give several necessary conditions for their persistence. Here we treat homoclinic orbits not only to equilibria but also to periodic orbits. We also discuss some relationships of these results with the standard subharmonic and homoclinic Melnikov methods for time-periodic perturbations of single-degree-of-freedom Hamiltonian systems, and with another version of the homoclinic Melnikov method for autonomous perturbations of…
We compute the cyclicity of open period annuli of the following generalized Rayleigh-Liénard equation $$\ddot{x}+ax+bx^3-(λ_1+λ_2 x^2+λ_3\dot{x}^2+λ_4 x^4+λ_5\dot{x}^4+λ_6 x^6)\dot{x}=0$$ and the equivalent planar system $X_λ$, where the coefficients of the perturbation $λ_j$ are independent small parameters and $a, b$ are fixed nonzero constants. Our main tool is the machinery of the so called higher-order Poincaré-Pontryagin-Melnikov functions (Melnikov functions $M_n$ for short), combined with the explicit computation of center conditions and the corresponding Bautin ideal. We consider firs…
In this paper, the general perturbation problem of piecewise smooth integrable differential systems with two switching planes is considered. Firstly, when the unperturbed system has a family of periodic orbits, the first order Melnikov vector function is derived which can be used to study the number of periodic orbits bifurcated from the period annuli. Then, by using the obtained Melnikov vector function, we get an upper bound of the number of periodic orbits of a concrete $n$-dimensional piecewise smooth differential system.
We propose a program for finding the cyclicity of period annuli of quadratic systems with centers of genus one. As a first step, we classify all such systems and determine the essential one-parameter quadratic perturbations which produce the maximal number of limit cycles. We compute the associated Poincare-Pontryagin-Melnikov functions whose zeros control the number of limit cycles. To illustrate our approach, we determine the cyclicity of the annuli of two particular reversible systems.
In the study of the number of limit cycles of near-Hamiltonian systems, the first order Melnikov function plays an important role. This paper aims to establish a development of a known method to estimate the upper bound of the number of zeros of the function.
This paper studies the family of piecewise linear differential systems in the plane with two pieces separated by a switching curve $y=x^{m}$, where $m>1$ is an arbitrary positive. By analysing the first order Melnikov function, we give an upper bound and an lower bound of the maximum number of limit cycles which bifurcate from the period annulus around the origin under polynomial perturbations of degree $n$. The results shows that the degree of switching curves affect the number of limit cycles.
We study bifurcation behavior in periodic perturbations of two-dimensional symmetric systems exhibiting codimension-two bifurcations with a double eigenvalue when the frequencies of the perturbation terms are small. We transform the periodically perturbed system to a simpler one which is a periodic perturbation of the normal form for codimension-two bifurcations with a double zero eigenvalue and symmetry, and apply the subharmonic and homoclinic Melnikov methods to analyze bifurcations occurring in the system. In particular, we show that there exist transverse homoclinic or heteroclinic orbits…
We consider perturbed pendulum-like equations on the cylinder of the form $ \ddot x+\sin(x)= \varepsilon \sum_{s=0}^{m}{Q_{n,s} (x)\, \dot x^{s}}$ where $Q_{n,s}$ are trigonometric polynomials of degree $n$, and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case $\varepsilon=0$ in terms of $m$ and $n$. Our first result gives upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of pol…
In this paper a generalized Rayleigh-Liénard oscillator is consider and lower bounds for the number of limit cycles bifurcating from weak focus equilibria and saddle connections are provided. By assuming some open conditions on the parameters of the considered system the existence of up to twelve limit cycles is provided. More precisely, the approach consists in perform suitable changes in the sign of some specific parameters and apply Poincaré-Bendixson Theorem for assure the existence of limit cycles. In particular, the method for obtaining the limit cycles through the referred approach is e…
In this paper, we study the bifurcate of limit cycles for Bogdanov-Takens system($\dot{x}=y$, $\dot{y}=-x+x^{2}$) under perturbations of piecewise smooth polynomials of degree $2$ and $n$ respectively. We bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles with switching curve $x=y^{2m}$($m$ is a positive integral) are $(39m+36)n+77m+21(m\geq 2)$ and $50n+52(m=1)$ (taking into account the multiplicity). The upper bounds number of limit cycles with switc…
This paper deals with the problem of limit cycle bifurcations for a piecewise near-Hamilton system with four regions separated by algebraic curves $y=\pm x^2$. By analyzing the obtained first order Melnikov function, we give an upper bound of the number of limit cycles which bifurcate from the period annulus around the origin under $n$-th degree polynomial perturbations. In the case $n=1$, we obtain that at least 4 (resp. 3) limit cycles can bifurcate from the period annulus if the switching curves are $y=\pm x^2$ (resp. $y=x^2$ or $y=-x^2$). The results also show that the number of switching …
In this paper, we consider the bifurcation of limit cycles for generic L-V system ($\dot{x}=y+x^2-y^2\pm\frac{4}{\sqrt{3}}xy,~\dot{y}=-x+2xy$) and B-T system ($\dot{x}=y,~\dot{y}=-x+x^2$) under perturbations of piecewise smooth polynomials with degree $n$. Here the switching line is $y=0$. By using Picard-Fuchs equations, we bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles for generic L-V system and B-T system are respectively $36n-65~(n\geq4),~37,57…
In this paper, by using Picard-Fuchs equations and Chebyshev criterion, we study the bifurcate of limit cycles for quadratic Hamilton system $S^{(2)}$ and $S^{(3)}$: $\dot{x}= y+2axy+by^2$, $\dot{y}=-x+x^2-ay^2$ with $a\in(-\frac{1}{2},1)$, $b=(1-a)(1+2a)^{1/2}$ and $a=1$, $b=0$ respectively, under perturbations of piecewise smooth polynomials with degree $n$. The discontinuity is the line $y=0$. We bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles fo…
We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a fast frequency vector $ω/\sqrtε$, with $ω=(1,Ω)$ where $Ω$ is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincaré-Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show tha…
The second part of the Hilbert's sixteenth problem consists in determining the upper bound $\mathcal{H}(n)$ for the number of limit cycles that planar polynomial vector fields of degree $n$ can have. For $n\geq2$, it is still unknown whether $\mathcal{H}(n)$ is finite or not. The main achievements obtained so far establish lower bounds for $\mathcal{H}(n)$. Regarding asymptotic behavior, the best result says that $\mathcal{H}(n)$ grows as fast as $n^2\log(n)$. Better lower bounds for small values of $n$ are known in the research literature. In the recent paper "Some open problems in low dimens…
