The hyperbolic invariant tori of symplectic mappings

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings. Under a Rüssmann-type non-degenerate condition, by introducing a modified KAM iteration scheme, we proved that nearly integrable twist symplectic mappings admit a family of lower dimensional hyperbolic invariant tori as long as the symplectic perturbation is small enough.     

An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

Summary A snapback repeller of an analytic mapping is defined as a full orbit which tends to an unstable fixed point backwards in time and snaps back to the same fixed point. This note gives a rather elementary proof that unstable periodic orbits accumulate near snapback repellers. The proof is entirely selfcontained and uses only standard elementary tools. We exploit that the global semiconjugacy of the entire analytic map to a linear map is itself an entire analytic function and apply the Theorem of Rouché to its zeros. We also generalize Marotto's result about the chaotic motion near a snapback repeller to include the degenerate case.     

Rigidity of critical circle mappings I

We prove that two C ³ critical circle maps with the same rotation number in a special set ? are C ^1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C ⁰ sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ^∞ critical circle maps with the same rotation number that are not C ^1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers. Received November 1, 1998 / final version received July 7, 1999     

Configurations of limit cycles and planar polynomial vector fields

We show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field. Moreover, the realization can be made by algebraic limit cycles, and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles.     

Finite polynomial orbits in finitely generated domains

LetR be a commutative domain of zero characteristic and letf(X) be a polynomial with coefficients inR. It is shown that all finite orbits inR. under the mapping induced byR have their cardinalities bounded by a constant depending onR but not onf. In the case whenR is the ring of all integers in an algebraic number field this constant is effectively determined.     

Non-existence of first integrals in a Laurent polynomial ring for general semi-quasihomogeneous systems

In this paper, we give some simple criteria of non-integrability and partial integrability in a Laurent polynomial ring C[u₁ ^± , ..., u_n ^± ] for general semi-quasihomogeneous systems. Supported by NSFC grant 10401013 and 985 project of Jilin University. (Received: September 17, 2003; revised: March 18/September 7, 2004)     

Limit cycles, invariant meridians and parallels for polynomial vector fields on the torus

We study the polynomial vector fields of arbitrary degree in R³ having the 2-dimensional torus invariant by their flow. We characterize all the possible configurations of invariant meridians and parallels that these vector fields can exhibit. Furthermore we analyze when these invariant either meridians or parallels can be limit cycles.     

Expansive canonical coordinates are hyperbolic

Theorem. Let ?:X→X be an expansive homeomorphism of a compact metric space onto itself and let ? have canonical coordinates. Then there exists a metric compatible with the topology of X with respect to which the canonical coordinates are hyperbolic.     

On the dynamics near infinity of some polynomial mappings in

We construct the Green current for a random iteration of horizontal-like mappings in . This is applied to the study of a polynomial map with the following properties: i. infinity is f-attracting; ii. f contracts the line at infinity to a point not in the indeterminacy set. We study for such mappings the escape rates near infinity, i.e. the set of possible values of the function We show in particular that the set of possible values can contain an interval. On the other hand the Green current T of f can be decomposed into pieces associated to an itinerary defined by the indeterminacy points. This allows us to prove that exists ||T||-a.e. and we give its value in terms of explicit quantities depending on f.     

Reduction and Equivariant Branching Lemma: Dynamical systems,evolution PDEs,and gauge theories

We review and recast the Equivariant Branching Lemma-which has proved a remarkable tool in linearly equivariant bifurcation theory-and consider its extension to the case of nonlinear (Lie-point) symmetries. This is then applied to gauge theories and gauge theoretic problems, and to nonlinear evolution PDE's; the paper also contains an original setting of Lie-point symmetries for evolution PDEs, modelled on the dynamical systems setting.     

Conjugation for polynomial mappings

We consider Keller's functions, namely polynomial functionsf:C ⁿ C ⁿ with detf(x)=1 at allx C ⁿ. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open.Without loss of generality assumef(0)=0 andf'(0)=I. We study the existence of certain mappingsh , > 1, defined by power series in a ball with center at the origin, such thath(0)=I andh (f(x))=h (x). So eachh conjugates f to its linear part I in a ball where it is injective.We conjecture that for Keller's functionsf of the homogeneous formf(x)=x +g(x),g(sx)=s ^dg(x),g(x)ⁿ=0,xC ⁿ,sC the conjugationh for f is anentire function.     

On cubic-linear polynomial mappings

In the field of the Jacobian conjecture it is well-known after Dru?kowski that from a polynomial ‘cubic-homogeneous’ mapping we can build a higher-dimensional ‘cubic-linear’ mapping and the other way round, so that one of them is invertible if and only if the other one is. We make this point clearer through the concept of ‘pairing’ and apply it to the related conjugability problem: one of the two maps is conjugable if and only if the other one is; moreover, we find simple formulas expressing the inverse or the conjugations of one in terms of the inverse or conjugations of the other. Two nontrivial examples of conjugable cubic-linear mappings are provided as an application.     

Generic polynomial vector fields are not integrable

We study some generic aspects of polynomial vector fields or polynomial derivations with respect to their integration. In particular, using a well-suited presentation of Darboux polynomials at some Darboux point as power series in local Darboux coordinates, it is possible to show, by algebraic means only, that the Jouanolou derivation in four variables has no polynomial first integral for any integer value s ≥ 2 of the parameter.Using direct sums of derivations together with our previous results we show that, for all n ≥ 3 and s ≥ 2, the absence of polynomial first integrals, or even of Darboux polynomials, is generic for homogeneous polynomial vector fields of degree s in n variables.     

Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree

In this paper we classify the centers localized at the origin of coordinates, and their isochronicity for the polynomial differential systems in R² of degree d that in complex notation z=x+iy can be written as where j is either 0 or 1. If j=0 then d?5 is an odd integer and n is an even integer satisfying 2?n?(d+1)/2. If j=1 then d?3 is an integer and n is an integer with converse parity with d and satisfying 0<n?[(d+1)/3] where [⋅] denotes the integer part function. Furthermore λ∈R and A,B,C,D∈C. Note that if d=3 and j=0, we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in K.E. Malkin (1964) [17], N.I. Vulpe and K.S. Sibirskii (1988) [25], J. Llibre and C. Valls (2009) [15], and if d=2, j=1 and C=0, we are also obtaining as a particular case the quadratic polynomial differential systems studied in N.N. Bautin (1952) [2], H. Zoladek (1994) [26]. So the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations.     

Local instability and localization of attractors. From stochastic generator to Chua's systems

This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chua's circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.     

Stability of orbit spaces of endomorphisms

In this paper it is first proved that, for a hyperbolic set of aC ¹ (non-invertible) endomorphism of a compact manifold, the dynamical structure of its orbit space (inverse limit space) is stable underC ¹-small perturbations and is semi-stable underC ⁰-small perturbations. It is then proved that if an Axiom A endomorphism satisfies no-cycle condition then its orbit space is Θ-stable andR-stable underC ¹-small perturbations and is semi-Θ-stable and semi-R-stable underC ⁰-small perturbations. This research is supported by the National Natural Science Foundation of China