Exact number of limit cycles for a family of rigid systems

For a given family of planar differential equations it is a very difficult problem to determine an upper bound for the number of its limit cycles. Even when this upper bound is one it is not always an easy problem to distinguish between the case of zero and one limit cycle. This note mainly deals with this second problem for a family of systems with a homogeneous nonlinear part. While the condition that allows us to separate the existence and the nonexistence of limit cycles can be described, it is very intricate.

     

极限环的表达式和计算

本文提出一种解析法和数值法相结合的方法，用来计算多项式微分系统的极限环，极限环表示为x=∑k≥0(ak cos kφ bk sin kφ),y=∑k≥0(ck cos kφ dk sin kφ)，先用解析法求出小参数时极限环的初始表达式，然后用增量法和迭代法求出任意参数时极限环满足给定精度的表达式，半稳定极限环和分叉值也可以计算。     

One and three limit cycles in a cubic predator–prey system

A cubic differential system is proposed, which can be considered a generalization of the predator–prey models, studied recently by many authors. The properties of the equilibrium points, the existence of a uniqueness limit cycle, and the conditions for three limit cycles are investigated. The criterion is easy to apply in applications. Copyright © 2006 John Wiley & Sons, Ltd.     

Limit cycle bifurcations of planar piecewise differential systems with three zones

This paper considers the limit cycle bifurcation problem of planar piecewise differential systems with three zones. Some computation formulas studied the problem of limit cycle bifurcations are provided by introducing multiple parameters. As an application to the obtained method, the number of limit cycles of a piecewise linear system with three zones studied in Lima et al. (2017) is discussed and some more limit cycles are found.     

A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems

In the qualitative theory of differential equations in the plane one of the most difficult objects to study is the existence of limit cycles. There are many papers dedicated to this subject. Here we will present a survey mainly dedicated to the algebraic and explicit non-algebraic limit cycles of the polynomial differential systems in ${\mathbb{R}}^2$ and of the discontinuous piecewise differential systems in ${\mathbb{R}}^2$ formed by two linear differential systems separated by a straight line. For this class of discontinuous piecewise differential systems the study of their algebraic and explicit non-algebraic limit cycles just is starting. Here we provide the first explicit non-algebraic limit cycle for the discontinuous piecewise linear differential systems. Additionally we recall seven open questions related with these types of limit cycles.     

TARGET PATTERNS AND SPIRAL WAVES IN REACTION-DIFFUSION SYSTEMS WITH LIMIT CYCLE KINETICSAND WEAK DIFFUSION

1.IntroductionTargetpatternsandspiralwavesarecommonlyobservedincertainmodelsofchemicalandbiologicalsystemssuchastheBelousov-ZhabotinskiireactionandthesocialamoebasDictyosteliumdiscoideium(of.11--4]andthereferencestherein).Thesesystemsaregovernedbyachemicalorbiologicalreactionandspatialdiffusion,i.e.reaction-diffusionequations.Generallyspeaking,atargetpatternisasetofconcentricringsofconstantconcentrationeachmovingoutward.Alonganyradialline,thepatternbehavesasymptoticallylikeaperiodictravellin…     

Qualitative Analysis of Crossing Limit Cycles in Discontinuous Liénard-Type Differential Systems

In this paper, we investigate qualitative properties of crossing limit cycles for a class of discontinuous nonlinear Liénard-type differential systems with two zones separated by a straight line. Firstly, by applying left and right Poincaré mappings we provide two criteria on the existence, uniqueness and stability of a crossing limit cycle. Secondly, by geometric analysis we estimate the position of the unique limit cycle. Several lemmas are given to obtain an explicit upper bound for the amplitude of the limit cycle. Finally, a predatorprey model with nonmonotonic functional response is studied, and Matlab simulations are presented to show the agreement between theoretical results and numerical analysis.     

A New Criterion for Controlling the Number of Limit Cycles of Some Generalized Liénard Equations

We consider a class of planar differential equations which include the Liénard differential equations. By applying the Bendixson-Dulac Criterion for ?-connected sets we reduce the study of the number of limit cycles for such equations to the condition that a certain function of just one variable does not change sign. As an application, this method is used to give a sharp upper bound for the number of limit cycles of some Liénard differential equations. In particular, we present a polynomial Liénard system with exactly three limit cycles.     

Exponential integrability of Itô's processes

In this note we prove the exponential integrability of super-norms of general Itô's processes under certain assumptions, and then apply it to the diffusion processes determined by stochastic differential equations. In particular, a conjecture in [Y. Hu, Exponential integrability of diffusion processes, in: Contemp. Math., vol. 234, 1999, pp. 75-84] is solved.     

A continuous dependence result for a nonstandard system of phase field equations

The present note deals with a nonstandard system of differential equations describing a two‐species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors, as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential. Copyright © 2013 John Wiley & Sons, Ltd.     

The equations determining intermediate integrals for Monge-Ampère PDE

In this note we will find the differential equations determining the intermediate integrals for Monge-Ampère equations in an arbitrary number of variables.

     

Fixed point indices and invariant periodic sets of holomorphic systems

This note presents an intuitive method to study center families of periodic orbits of complex holomorphic differential equations near singularities, based on some iteration properties of fixed point indices. As an application of this method, we will prove Needham's theorem in a more general version.

     

Periodic solutions of Abel differential equations

For a class of polynomial non-autonomous differential equations of degree n, we use phase plane analysis to show that each equation in this class has n periodic solutions. The result implies that certain rigid two-dimensional systems have at most one limit cycle which appears through multiple Hopf bifurcation.     

关于微分多项式零点的注记

The value distribution of differential polynomials is studied.The re- suits in this paper improve and generalize some previous theorems given by Yang Chungchun(On deficiencies of differential polynomials,Math.Z.,116(1970),197- 204),H.S.Gopalakrishna and S.S.Bhoosnurmath(On distribution of values of differential polynomials,Indian J.Pure Appl.Math.,17(1986),367-372),I.Lahiri (A note on distribution of nonhomogeneous differential polynomials,Hokkaido Math. J.,31(2002),453-458)and Yi Hongxun(On zeros of differential polynomials,Adv. in Math.,18(1989),335-351)et al.Examples show that the results in this paper are sharp.     

A note on the reflected backward stochastic differential equations driven by a Lévy process with stochastic Lipschitz condition

In this note, we prove the existence and uniqueness of the solution for a class of reflected backward stochastic differential equations (RBSDEs in short) related to the subdifferential operator of a lower semi-continuous convex function, driven by Teugels martingales associated with a Lévy process. Some known results are generalized and improved.     

一类微分系统的极限环分布情况

用定性分析和数值判定方法,对一类微分系统x=y,y=x(l-bx2) (α-cx2)y(其中l＞0,b＞0,c≠0)的极限环分布情况进行了研究,得出该系统有3个极限环,并且给出了该系统所有极限环的分布情况.     

Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Quadratic Differential Games

In this paper, we discuss nonzero-sum linear-quadratic differential games. For this kind of games, the Nash equilibria for different kinds of information structures were first studied by Starr and Ho. Most of the literature on the topic of nonzero-sum linear-quadratic differential games is concerned with games of fixed, finite duration; i.e., games are studied over a finite time horizon t _f. In this paper, we study the behavior of feedback Nash equilibria for t _f.In the case of memoryless perfect-state information, we study the so-called feedback Nash equilibrium. Contrary to the open-loop case, we note that the coupled Riccati equations for the feedback Nash equilibrium are inherently nonlinear. Therefore, we limit the dynamic analysis to the scalar case. For the special case that all parameters are scalar, a detailed dynamical analysis is given for the quadratic system of coupled Riccati equations. We show that the asymptotic behavior of the solutions of the Riccati equations depends strongly on the specified terminal values. Finally, we show that, although the feedback Nash equilibrium over any fixed finite horizon is generically unique, there can exist several different feedback Nash equilibria in stationary strategies for the infinite-horizon problem, even when we restrict our attention to Nash equilibria that are stable in the dynamical sense.     

The number and stability of limit cycles for planar piecewise linear systems of node–saddle type

The objective of this paper is to study the number and stability of limit cycles for planar piecewise linear (PWL) systems of node–saddle type with two linear regions. Firstly, we give a thorough analysis of limit cycles for Liénard PWL systems of this type, proving one is the maximum number of limit cycles and obtaining necessary and sufficient conditions for the existence and stability of a unique limit cycle. These conditions can be easily verified directly according to the parameters in the systems, and play an important role in giving birth to two limit cycles for general PWL systems. In this step, the tool of a Bendixon-like theorem is successfully employed to derive the existence of a limit cycle. Secondly, making use of the results gained in the first step, we obtain parameter regions where the general PWL systems have at least one, at least two and no limit cycles respectively. In addition for the general PWL systems, some sufficient conditions are presented for the existence and stability of a unique one and exactly two limit cycles respectively. Finally, some numerical examples are given to illustrate the results and especially to show the existence and stability of two nested limit cycles.     

Indecomposable cycles on special quartics in ℙ<Superscript>3</Superscript>

 In this note we give an example of an indecomposable higher Chow cycle on a special family of quartics in ℙ³. The example is obtained as an extension of a cycle in the higher Chow group CH ² (K,1) of a singular Kummer surface. Received: 19 June 2002 / Revised version: 17 October 2002 Published online: 24 April 2003 Mathematics Subject Classification (2000): 14C15, 14C25, 14H10, 14H28, 14F42     

Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions.