About a conjecture for uniformly isochronous polynomial centers
E. P. Volokitin
研究了一类特殊的一致等距多项式系统。我们的结果允许解决这样的系统的中心的问题。
We study a specific family of uniformly isochronous polynomial systems. Our results permit to solve a problem about centers of such systems.
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An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the dynamics may change and the phenomenon of resonance can appear. In this context, resonance means that all solutions are unbounded. The theory of resonance is well known for the harmonic oscillator and we extend it to nonlinear isochronous oscillators.
On the non-existence of isochronous centers in planar discontinuous differential systems
本文研究平面分段不连续微分系统，证明以直线 y=0 和 x=0 为切换线的系统在原点不存在等时中心。
The isochronicity problem is a classical problem in the qualitative theory of planar vector fields which consists in characterizing whether a center is isochronous or not, that is, if all the trajectories in a neighbourhood of the center have the same period. This problem is usually investigated by means of the so-called period function. In this paper, we are interested in exploring the isochronicity problem for tangential centers of planar Filippov vector fields. By computing the period function for planar Filippov vector fields around tangential centers, we show that such centers are never i…
Using the mould formalism introduced by Jean Ecalle, we define and study the geometric complexity of an isochronous center condition. The role played by several Lie ideals is discussed coming from the interplay between the universal mould of the correction and the different Lie algebras generated by the comoulds. This strategy enters in the general program proposed by J. Ecalle and D. Schlomiuk in \cite{es} to study the size and splitting of some Lie ideals for the linearisability problem.
In this paper, we study the integrability and linearization of a class of quadratic quasi-analytic switching systems. We improve an existing method to compute the focus values and periodic constants of quasi-analytic switching systems. In particular, with our method, we demonstrate that the dynamical behaviors of quasi-analytic switching systems are more complex than that of continuous quasi-analytic systems, by showing the existence of six and seven limit cycles in the neighborhood of the origin and infinity, respectively, in a quadratic quasi-analytic switching system. Moreover, explicit con…
These last years an increasing interest appeared for studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. One of the difficulties for understanding the dynamics of these systems is the study their limit cycles. In this paper we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line, and formed by combinations of linear centers (consequently isochronous) and cubic isochronous centers with homogeneous nonlinearities. For these …
We study the conjecture of Jarque and Villadelprat stating that every center of a planar polynomial Hamiltonian system of even degree is nonisochronous. This conjecture is proved for quadratic and quartic systems. Using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields, wa are able to describe a very large class of nonisochronous Hamiltonian system of even degree of degree arbitrary large.
Complete solution to the weak persistent center and isochronous center problems of cubic polynomial systems
本文通过奇点量与拟李雅普诺夫常数方法，完整求解了三次多项式系统弱持久中心及等时中心问题，给出了复与实系统的充要条件。
Trivial Isochronous Centers in Odd Degrees: a Two--Branch Picture
该研究重新考察奇数次平面多项式哈密顿系统的平凡等时中心，形式化了三角族的通用构造及二次剪切族的交替出现规律，完善了相关等时中心的刻画理论。
Let $X$ be a polynomial vector field in $\mathbb{R}^2$ which, after one-point compactification of the plane, has a punctured neighbourhood $\dot U$ of the point at infinity which is foliated by closed orbits of $X$. If the period function of $X$ in $\dot U$ is bounded from below by a positive constant, $X$ is necessarily linear, hence conjugated, up to a nonzero constant factor, to $-y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$. This result answers to a question posed by J. Llibre , proving e.g. that a global isochronous center is linear.
Limit cycles appearing from the perturbation of a cubic isochronous center
本文研究三次等时中心 $S^*_2$ 在任意多项式扰动下的极限环分支问题，并利用阿贝尔积分方法给出了极限环数量的上界。
From an iteration formula to Poincaré's Isochronous Center Theorem for holomorphic vector fields
首先将一元全纯映射的经典迭代公式推广到高维全纯映射的迭代公式。然后，作为应用，我们给出了一个经典定理的简短而直观的证明，由于H。Poincaré，对于全纯向量场的奇点是等距中心的条件。
Topological properties on isochronous centers of polynomial Hamiltonian differential systems
本文研究n次复多项式哈密顿微分系统等时中心的拓扑性质，证明消没圈平凡性的条件、给出等时性的简单必要条件，并建立Gavrilov问题与偶次实哈密顿系统非等时性猜想的关联。
Emergence of Multiple Crossing Limit Cycles in Planar Piecewise Systems with Isochronous Centers and Nonsmooth Switching Manifolds
研究平面分段系统中多重交叉极限环的产生机制，通过非光滑切换流形增强动力学复杂性，推导出交叉极限环数量的上界并构造实例。
Isochronous Centers of Lienard Type Equations and Applications
本文研究了中心为0的方程$(E) \ddot x + f(x) \dot x^2 + g(x) = 0$，并研究了它的等时性条件。当$f$和$g$是解析的（不一定是奇的）时，给出了0的等时性的一个充要条件.这种方法使我们能够提出一种算法，获得的条件（E）的一个点是一个等值线中心。特别地，我们又用另一种方法找到了二次Loud系统$(L_{D,F})$的等时线.也考虑了一些Kukles类。此外，我们还对一类具有等熵中心的5参数可逆三次系统进行了分类。关键词：周期函数，单调性，等时性，中心，多项式系统。
The cyclicity of period annulus of cubic isochronous Hamiltonian systems
本文研究了具有原点等时中心的三次哈密顿系统中的弱希尔伯特第16问题，证明极限环的最大数目为n-1并达到该数目。
Center conditions for a simple class of quintic systems
我们获得了原点是一致等距奇点的5次$O$对称系统的中心条件。在修订后的论文中，参考文献列表中的一些印刷错误得到了纠正。
Dynamics of Kahan-Hirota-Kimura maps with rational invariant fibrations
本文提出一种研究平面Kahan-Hirota-Kimura映射动力学的方法，展示可积KHK映射可能表现出复杂动力学，即使其来源于行为简单的向量场。通过研究二次平面向量场的KHK映射，证明其保持原第一积分并具有向量场作为李对称性，且在密集的积分步长下全局周期性。
The Briot-Bouquet systems and the center families for holomorphic dynamical systems
给出了全纯动力系统等熵中心族存在性的完全解。对n维全纯动力系统的中心族的研究自然会导致对相空间中的（n-1）维Briot-Bouquet系统的研究。我们首先给出了一个详细的研究Briot-Bouquet系统。然后基于二维Briot-Bouquet理论证明了三维系统在平衡点附近存在等距中心族。同样的方法适用于任意维度。
Isochronicity and Commutation of Polynomial Vector Fields
研究了多项式向量场中心的等时性与多项式交换系统存在性之间的联系。我们证明了一个不与任何多项式系统交换的4次等值线系统。证明了在Newton多项式系统中，只有Lienard和Abel系统可以与横截多项式域交换。我们给出了Abel多项式系统的中心化子的一个完整的和建设性的描述。我们给出了新的例子的isoorthoscopic系统。
On the period function of Newtonian systems
本文研究了平面自治系统$$(S) \quad \dot x=y,\qquad \dot y = -h(x) - g(x)y - f(x)y^2.$$中心的存在性。我们感兴趣的是围绕中心0的周期函数$T$。给出了（S）在0点等时性的一个充分条件.当$f,g,h$是解析函数时，这样的条件也是必要的。在这种情况下，给出了系统（S）的等距中心的一个特征。一些应用程序将被导出。特别地，将描述等熵中心的新家族
In this paper we investigate the isochronicity and linearizability problem for a cubic polynomial differential system which can be considered as a generalization of the Riccati system. Conditions for isochronicity and linearizability are found. The global structure of systems of the family with an isochronous center is determined. Furthermore, we find the order of weak center and study the problem of local bifurcation of critical periods in a neighborhood of the center.
Complexity reduction of C-algorithm
在[Chouikha 2007]中引入的C-算法被设计用于在一般实解析情况下确定Lienard型微分系统的等值线中心。然而，它有一个大的复杂性，防止计算，即使在四次多项式的情况下。本文的主要成果是一个有效的算法实现的C-算法，称为ReCA（减少C-算法）。此外，在理性的情况下，它提出了一个适应的版本。它被称为RCA（有理C-算法），并广泛用于[BardetBoussaadaChouikhaStrelcyn 2010]和[BoussaadaChouikhaStrelcyn 2010]，以找到Liénard型方程的许多新的等距中心的例子。
Dynamics of polynomial generalized Liénard system near the origin and infinity
本文分类了多项式广义Liénard系统在原点与无穷远处的拓扑相图，刻画了单值性，给出局部中心充要条件，并证明全局中心非等时。
In this paper we investigate the problem of linearizability for a family of cubic complex planar systems of ordinary differential equations. We give a classification of linearizable systems in the family obtaining conditions for linearizability in terms of parameters. We also discuss coexistence of isochronous centers in the systems.
Isochronicity conditions for some planar polynomial systems
本文研究了系统$\dot x=-y+A(x,y), \dot y=x+B(x,y)$在$O\in \mathbb{R}^2$中心的等时性，其中$A, B\in \mathbb{R}[x,y]$，它可以归结为Lienard型方程.使用所谓的C-算法，我们已经找到了27个新的多参数等值线中心。
Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics
我们证明了具有一致紧的$f$不变中心叶理$F^c$的部分双曲$C^1$-单同态$f : M \to M$是动态相干的.进一步地，中心叶理的商空间上的诱导同胚$F : M/F^c \to M/F^c$具有跟踪性质，即对于每个$\varepsilon> 0$，存在$δ> 0$使得中心叶的每个$δ$ -伪轨道被中心叶的轨道$\varepsilon$ -跟踪.虽然跟踪轨道不一定是唯一的，但我们证明了商动力系统的链递归集内周期中心叶的密度。商动力学的其他一些有趣的性质进行了讨论。
Codimension one partially hyperbolic diffeomorphisms with a uniformly compact center foliation
考虑光滑紧致流形M上的部分双曲C1-齐次同态f，其中心叶理为一致紧的f-不变。证明了如果不稳定丛是一维定向丛，则中心叶理的完整性处处为零，中心叶理的商空间是环面，f在其上诱导双曲自同构，特别地，f是中心传递的.我们实际上得到了进一步有趣的结果，而不受限制的不稳定，稳定和中心尺寸：我们证明了一种谱分解的链递归集的商动力学，我们建立了一个完整的不稳定的叶子（Margulis措施）不变的措施家庭的存在。
Isochronicity conditions for some planar polynomial systems II
本文研究了系统$$\dot x=-y+A(x,y),\;\dot y=x+B(x,y),$$在$O\in \mathbb{R}^2$中心的等时性，其中$A,\;B\in \mathbb{R}[x,y]$可化为Liénard型方程.当$deg(A)\leq 4$和$deg(B) \leq 4$，使用所谓的C-算法，我们发现了$36$新的家庭的等离子中心。当Urabe函数为$h=0$时，给出了线性化的显式一般公式.本文是\cite{BoussaadaChouikhaStrelcyn2010}的直接延续，但可以独立阅读。
Let $f:M\to M$ be a dynamically coherent partially hyperbolic diffeomorphism whose center foliation has all its leaves compact. We prove that if the unstable bundle of $f$ is one-dimensional, then the volume of center leaves must be bounded in $M$.
Finiteness of physical measures for diffeomorphisms with multi 1-D centers
该研究针对具有多1维中心的部分双曲$C^2$微分同胚，在Gibbs $u$-态双曲性的条件下，证明了其遍历物理（SRB）测度具有有限性，推进了部分双曲系统的测度理论研究。
Finiteness and Geometric structure of $c$-$cu$-States with Maximal $u$-Entropy
本研究针对$c$-混合系统中具有最大$u$-熵的$c$-$cu$-态，证明其支撑两两不同的最大数量有限，通过改进Smale方法构造了具有混合中心的四维环面微分同胚实例，为物理测度的半连续变化提供了具体例证。
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