Linear Estimation of the Number of Zeros of Abelian Integrals for Some Cubic Isochronous Centers

This paper consists of two parts. In the first part we study the relationship between conic centers (all orbits near a singular point of center type are conics) and isochronous centers of polynomial systems. In the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics, when we perturb such systems inside the class of all polynomial systems of degree n.     

Non-isochronicity of the center at the origin in polynomial Hamiltonian systems with even degree nonlinearities

In 2002 X. Jarque and J. Villadelprat proved that no center in a planar polynomial Hamiltonian system of degree 4 is isochronous and raised a question: Is there a planar polynomial Hamiltonian system of even degree which has an isochronous center? In this paper we give a criterion for non-isochronicity of the center at the origin of planar polynomial Hamiltonian systems. Moreover, the orders of weak centers are determined. Our results answer a weak version of the question, proving that there is no planar polynomial Hamiltonian system with only even degree nonlinearities having an isochronous center at the origin.     

Isochronous Centers and Isochronous Functions

Abstract We study isochronous centers of two classes of planar systems of ordinary differential equations.Forthe first class which is the Linard systems of the form =y-F(x),=-g(x) with a center at the origin, we provethat if g is isochronous(see Definiton 1.1),then the center is isochronous if and only if F≡0.For the secondclass which is the Hamiltonian systems of the form =-g(y),=f(x) with a center at the origin,we prove thatif f or g is isochronous,then the center is isochronous if and only if the other is also isochronous.     

Isochronicity of centers in a switching Bautin system

In this paper isochronicity of centers is discussed for a class of discontinuous differential system, simply called switching system. We give some sufficient conditions for the system to have a regular isochronous center at the origin and, on the other hand, construct a switching system with an irregular isochronous center at the origin. We give a computation method for periods of periodic orbits near the center and use the method to discuss a switching Bautin system for center conditions and isochronous center conditions. We further find all of those systems which have an irregular isochronous center.     

Numerical Continuation of Hamiltonian Relative Periodic Orbits

The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years, there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity. But as yet, there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step toward a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First, we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally, we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our path following algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous figure eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating eight family and compute the rotating choreographies bifurcating from it.      

一类五次多项式系统原点等时中心的分析

研究了一类五次系统原点复等时中心的问题.先通过一种最新算法求出了这类五次系统原点的周期常数,从而得到复等时中心的必要条件,并利用一些有效途径证明它们的充分性.这实际上解决了这类五次系统的伴随系统原点等时中心问题与其自身为实系统时鞍点可线性化的问题.     

On Periodic Solutions of One Class of Systems of Differential Equations

We study the problem of the existence of periodic solutions of two-dimensional linear inhomogeneous periodic systems of differential equations for which the corresponding homogeneous system is Hamiltonian. We propose a new numerical-analytic algorithm for the investigation of the problem of the existence of periodic solutions of two-dimensional nonlinear differential systems with Hamiltonian linear part and their construction. The results obtained are generalized to systems of higher orders. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 483–495, April, 2005.     

Closed characteristics on non-degenerate star-shaped hypersurfaces in R2n

In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula of C. Viterbo for non-degenerate star-shaped Hamiltonian systems to the general case. Finally we prove that there exist at least two geometrically distinct closed characteristics on any non-degenerate star-shaped compact smooth hypersurface on R2n with n ＞ 1. Here we call a hypersurface non-degenerate, if all the closed characteristics on the given hypersurface together with all of their iterations are non-degenerate as periodic solutions of the corresponding Hamiltonian system. We also study the ellipticity of closed characteristics when n=2.     

Periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces

This paper discusses the existence and multiplicity of periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces. We prove that each such energy hypersurface carries at least one symmetric periodic orbit. Under some suitable pinching conditions, we also get an existence result of multiple symmetric periodic orbits.     

On nonlinear motions of Hamiltonian system in case of fourth order resonance

We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin, which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3: 1. We study nonlinear conditionally periodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the so-called truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionally periodic. By using the KAM theory methods we show that most of the conditionally periodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that are not conditionally periodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession.     

Gutzwiller’s semiclassical trace formula and Maslov-type index theory for symplectic paths

Gutzwiller’s famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with a view toward the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley–Zehnder index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley–Zehnder’s index theory in which we will recall all essential ingredients is the Maslov-type index theory for symplectic paths developed by Long and his collaborators. It would shed new light on the computations and understandings of the semiclassical trace formula. The insights in Gutzwiller’s work also seems plausible for the studies of Hamiltonian systems.     

O(2)-lnvariant Hamiltonians on ℂ4

A normal-form theory and a group-theoretic classification for periodic solutions of O(2)-invariant Hamiltonians on ?⁴ is developed. The theory applies to Hamiltonian systems with an O(2) spatial symmetry that also have a linear-mode interaction. Our motivation is the classic (m, n) mode-interaction problem for capillary-gravity waves. It is well known that the addition of surface-tension effects to irrotational water waves results in a countable infinity of values of the surface-tension coefficient at which two traveling waves of differing wavelength travel at the same speed. However, recognizing the reflectional symmetry in space, the linearized problem is actually spanned by four traveling waves. In other words there is an O(2) symmetry in space. A classification theorem for group-invariant Hamiltonian systems, based on a listing of the isotropy subgroups and their fixed-point spaces, is used to show that there are between seven and fourteen classes of periodic solutions in O(2)-invariant Hamiltonian systems with a mode interaction. The results are used to interpret, from a group-theoretic viewpoint, the classic Wilton ripple.     

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

     

时间可逆系统的等时中心问题

若在中心附近的闭轨线都具有相同的周期,则此中心称为等时中心. 时间可逆多项式系统的等时中心问题是一类公开问题. 为了构造性地解决这一难题,讨论一类范围更广的时间可逆解析动力系统, 给出相应横截交换系统的递推公式,此公式可以用于等时中心条件的推导. 在递推公式的基础上,以吴特征集方法为工具,给出一类时间可逆三次系统具有横截交换系统的充要条件.     

Invariant tori in perturbed three vortex motion

The motion of three point vortices in an ideal fluid in a plane comprises a Hamiltonian dynamical system – one that is completely integrable, so it exhibits numerous periodic orbits, and quasiperiodic orbits on invariant tori. Certain perturbations of three vortex dynamics, such as three vortex motion in a half-plane, are also Hamiltonian, but not completely integrable. Yet these perturbed systems may still have periodic trajectories and invariant tori close to those for the unperturbed dynamics. Extending recent work by the authors [4], invariant 2-tori approximating those for the unperturbed system are located and analyzed using a combination of classical analysis, asymptotics, and Hamiltonian methods. It is shown that the results and approximation methods used are applicable to several perturbations of three vortex dynamics such as three vortices in a half-plane, the restricted four vortex problem in the plane, and three coaxial vortex rings in 3-space. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems

We study some monotonicity and iteration inequality of the Maslov-type index i-1of linear Hamiltonian systems.As an application we prove the existence of symmetric periodic solutions with prescribed minimal period for first order nonlinear autonomous Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity.This result gives a positive answer to Rabinowitz’s minimal period conjecture in this case without strictly convex assumption.We also give a different proof of the existence of symmetric periodic solutions with prescribed minimal period for classical Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity which was proved by Fei,Kim and Wang in 2001.     

Closed characteristics on non-degenerate star-shaped hypersurfaces in R<Subscript>2n</Subscript>

In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula ofC. Viterbo for non-degenerate star-shaped Hamiltonian systems to the general case. Finally we prove that there exist at least two geometrically distinct closed characteristics on any non-degenerate star-shaped compact smooth hypersurface on R²ⁿ with n > 1. Here we call a hypersurface non-degenerate, if all the closed characteristics on the given hypersurface together with all of their iterations are non-degenerate as periodic solutions of the corresponding Hamiltonian system. We also study the ellipticity of closed characteristics whenn = 2     

Periodic solutions of perturbed isochronous hamiltonian systems at resonance

We look for periodic solutions of planar systems obtained by adding an asymptotically positively homogeneous nonlinear term to an isochronous hamiltonian system. Precise computations of the topological degree are obtained by elementary phase-plane analysis.     

On averaging,reduction, and symmetry in hamiltonian systems

The existence of periodic orbits for Hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged Hamiltonian on an associated “reduced space.” Alternatively, in classical (kinetic plus potential energy) Hamiltonians the existence of such orbits can often be established by elementary geometrical arguments. The present paper unifies the two approaches by exploiting discrete symmetries, including reversing diffeomorphisms, that occur in a given system. The symmetries are used to locate the periodic orbits in the averaged Hamiltonian, and thence in the original Hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries. In applications to the Hénon-Heiles Hamiltonian, it is illustrated how “higher order” averaging can sometimes be used to overcome degeneracies encountered at first order.     

Symbolic dynamics and Arnold diffusion

We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolic torus intersect transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometrico-dynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolic periodic orbits near the given torus. The relevance of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolic sets.