ANALYTIC INVARIANT CURVES OF A NONLINEAR SECOND ORDER DIFFERENCE EQUATION

This article studies the existence of analytic invariant curves for a nonlinear second order difference equation which was modeled from macroeconomics of the business cycle. The author not only discusses the case of the eigenvalue off the unit circle S^1 and the case on S^1 with the Diophantine condition but also considers the case of the eigenvalue at a root of the unity, which obviously violates the Diophantine condition.

BIFURCATIONS OF INVARIANT CURVES OF A DIFFERENCE EQUATION

Bifurcation of the invariant curves of a difference equation is studied. The system defined by the difference equation is integrable, so the study of the invariant curves of the difference system can become the study of topological classification of the planar phase portraits defined by a planar Hamiltonian system. By strict qualitative analysis, the classification of the invariant curves in parameter space can be obtained.

GL（n, R）-Equivalence of a Pair of Curves in Terms of Invariants

In this paper, the generator set of R 〈 x1,x2 〉G is obtained in according to the group G = Gl（n,R）. The conditions of G = Gl（n, R） -equivalence of a pair of curves are found in terms of G = Gl（n, R）-invariants. And the independence of GL（n, R） -invariants is shown.

A New Kind of Pre-Mean-Type Mappings and Its Gauss Iteration

A function which is reflexive is called by pre-mean, a more generalized definition of a mean. In this paper, we define a new pre-mean and study its properties, and then using the given invariant curve we consider the problem of convergence of Gauss iteration of a kind of pre-mean type mappings generated by the exponential and logarithmic functions.

LIMIT CYCLES AND INVARIANT PARABOLA IN A KUKLES SYSTEM OF DEGREE THREE

In this article,the authors consider a class of Kukles planar polynomial differential system of degree three having an invariant parabola.For this class of second-order differential systems,it is shown that for certain values of the parameters the invariant parabola coexists with a center.For other values it can coexist with one,two or three small amplitude limit cycles which are constructed by Hopf bifurcation.This result gives an answer for the question given in[4],about the existence of limit cycles for such class of system.

Invariant Curves of Reversible Mappings with Small Twist

In this paper, we are concerned with the existence of invariant curves of reversible mappings. A variant of the classical small twist theorem is given.

INTEGRABILITY VIA INVARIANT ALGEBRAIC CURVES FOR PLANAR POLYNOMIALDIFFERENTIAL SYSTEMS

We present an introduction to the Darboux integrability theory of planar complex and real polynomial differential systems containing some improvements to the classical theory.

Analytic Invariant Curves for a Planar Mapping

This paper is concerned with the existence of analytic invariant curves for a planar mapping. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. In this paper, we discuss not only the general case, but also the critical cases as well, in particular, the case where β is a unit root is discussed.

Analytic Invariant Curves for an Iterative Equation Related to Ricker-type Second-order Equation

In this paper,we consider the existence of analytic invariant curves of an iterative equation f(f(x))=xea-x-f(x)which arises from Ricker-type second-order equation.By reducing the equation with the Schr?der transformation to an auxiliary equation,the author discusses not only that the parameter at resonance,i.e.,at a root of the unity,but also the parameter near resonance under the Brjuno condition.

Existence of invariant curves with prescribed frequency for degenerate area preserving mappings

We consider small perturbations of analytic non-twist area preserving mappings,and prove the existence of invariant curves with prescribed frequency by KAM iteration.Generally speaking,the frequency of invariant curve may undergo some drift,if the twist condition is not satisfied.But in this paper,we deal with a degenerate situation where the unperturbed rotation angle function r→w+r^(2n+1)is odd order degenerate at r=0,and prove the existence of invariant curve without any drift in its frequency.Furthermore,we give a more general theorem on the existence of invariant curves with prescribed frequency for non-twist area preserving mappings and discuss the case of degeneracy with various orders.

Existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions

We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter.Without imposing on any non-degeneracy assumption,we prove a formal KAM theorem for the mappings,which implies many previous KAM-type results under some non-degeneracy conditions.Moreover,by this formal KAM theorem,we can also obtain some new interesting results under some weaker non-degeneracy conditions.Thus,the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x^2 ＋ y^2 ＋ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincare disc.

QUADRATIC SYSTEM WITH HOMOCLINIC CYCLE DESCRIBED BY SEXTIC CURVE

In [2-5], cubic, quartic or quintic homoclinic cycles are found. In this paper, we present a quadratic system with homoclinic cycle which is described by a sextic curve.

RATIONAL GENERAL SOLUTIONS OF HIGHER ORDER ALGEBRAIC ODES

This paper generalizes the method of Ng6 and Winkler （2010, 2011） for finding rational general solutions of a first order non-autonomous algebraic ordinary differential equation （AODE） to the case of a higher order AODE, provided a proper parametrization of its solution hypersurface. The authors reduce the problem of finding the rational general solution of a higher order AODE to finding the rational general solution of an associated system. The rational general solutions of the original AODE and its associated system are in computable 1-1 correspondence. The authors give necessary and sufficient conditions for the associated system to have a rational solution based on proper reparametrization of invariant algebraic space curves. The authors also relate invariant space curves to first integrals and characterize rationally solvable systems by rational first integrals.

A Result on the Quasi-periodic Solutions of Forced Isochronous Oscillators at Resonance

In this paper, the authors are concerned with the forced isochronous oscillators with a repulsive singularity and a bounded nonlinearity x'' + V'(x) + g(x) = e(t, x, x'),where the assumptions on V, g and e are regular, described precisely in the introduction.Using a variant of Moser's twist theorem of invariant curves, the authors show the existence of quasi-periodic solutions and boundedness of all solutions. This extends the result of Liu to the case of the above system where e depends on the velocity.

Boundedness of Solutions of a Quasi-periodic Sublinear Duffing Equation

The authors study the Lagrangian stability for the sublinear Duffing equationsx+e(t)|x|^(α-1) x=p(t)with 0<α<1,where e and p are real analytic quasi-periodic functions with frequencyω.It is proved that if the mean value of e is positive and the frequencyωsatisfies Diophantine condition,then every solution of the equation is bounded.

Boundedness of Solutions of Quasi-periodic p-Laplacian Equations with Jumping Nonlinearity

In this article,we prove the existence of quasi-periodic solutions and the boundedness of all solutions of the p-Laplacian equation(φ_(p)(x’))’+aφ_(p)(x+)-bφ_(p)(x-)=g(x,t)+f(t),where g(x,t)and f(t)are quasi-periodic in t with Diophantine frequency.A new method is presented to obtain the generating function to construct canonical transformation by solving a quasi-periodic homological equation.

Quasi-periodic solutions for the general semilinear Duffing equations with asymmetric nonlinearity and oscillating potential

In this paper,we prove the existence of quasi-periodic solutions and the boundedness of all the solutions of the general semilinear quasi-periodic differential equation x′′+ax^(+)-bx^(-)=G_x(x,t)+f (t),where x^(+)=max{x,0},x^(-)=max{-x,0},a and b are two different positive constants,f(t) is C^(39) smooth in t,G(x,t)is C^(35) smooth in x and t,f (t) and G(x,t) are quasi-periodic in t with the Diophantine frequency ω=(ω_(1),ω_(2)),and D_(x)^(i)D_(t)^(j)G(x,t) is bounded for 0≤i+j≤35.

A CLASS OF DIFFERENTIAL SYSTEM WITH AT MOST FOUR ALGEBRAIC LIMIT CYCLES

A class of quartic and quintic differential system is introduced. We show that under suitable assumptions, one, two or four algebraic limit cycles can occur. These limit cycles are analytically given.

THE PROBLEM OF THE CENTRE FOR CUBIC DIFFERENTIAL SYSTEMS WITH TWO HOMOGENEOUS INVARIANT STRAIGHT LINES AND ONE INVARIANT CONIC

For cubic differential systems with two homogeneous invariant straight lines and one invariant conic, it is proved that a singular point with pure imaginary eigenvalues (a weak focus) is a centre if and only if the first two Lyapunov quantities Lj , j = 1, 2 vanish.