Generic Stability Properties of Periodic Points

Meyer, K. R.

doi:10.2307/1995443

Cited by 3 publications

(3 citation statements)

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“…Sometimes we shall make explicit the dependence of these constants with respect to F and write ;=;(F),... Next result shows that when F is sufficiently smooth the normal form of degree 3 decides in most cases the stability of the origin. [11] when (2.2) holds. These works assume that F is very smooth because their proofs are based on the Theorems of the invariant curve in [15,25].…”

Section: The Normal Form Of Degree Three Of An Elliptic Fixed Pointmentioning

confidence: 99%

Periodic Solutions of a Newtonian Equation: Stability by the Third Approximation

Ortega

1996

Journal of Differential Equations

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Section: The Normal Form Of Degree Three Of An Elliptic Fixed Pointmentioning

confidence: 99%

Periodic Solutions of a Newtonian Equation: Stability by the Third Approximation

Ortega

1996

Journal of Differential Equations

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“…In particular, there is a one parameter family p 3 (t, h) of hyperbolic periodic solution of period 3τ 0 + · · · whose limit is p 0 (t) as h −→ h 0 (see [Mey71,MH92] for details).…”

Section: Bifurcating Periodic Solutions Of the 3-body Problemmentioning

confidence: 99%

High-Dimensional Chaotic and Attractor Systems

Ivancevic¹

2007

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“…Let M be a 4-dimensional symplectic manifold and let k be an integer, greater than 2. Then there is a residual set R k C Ck(M) such that for each H ~ Rk and each closed orbit 7 of Xn, for which the differential of the Poincark map, restricted to the H level of ~, is a rotation over n_~ 2re with m < k, there is a m generic bifurcation of periodic points for the Poincar~ map as described in [5] and hence in M a generic bifurcation of closed orbits.…”

Section: Be a Countable Set Of Normal Subsets Of Jk S(2n --1) Then mentioning

confidence: 99%

Hamiltonian systems: Generic properties of closed orbits and local perturbations

Takens

1970

Math. Ann.

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Introduction, Definitions and TheoremThe aim of this paper is to extend parts of R. C. Robinsons work on generic properties of Hamiltonian systems [7]. We first give some definitions.Let M be a 2n-dimensional C°-manifold with symplectic form co (i.e. co is a closed 2-form and co A---A co (n-times) is a nowhere zero volume form). A Hamiltonian system on M is a vectorfield X such that the 1-form Zxco is exact (i.e. txCO is the differential of a function). In general a Hamiltonian system is given by a function H : M ~IIL the Hamiltonian; the correspondin 9 vectorfield X n on M is then determined by the equation txnC0 = dH. We say that the Hamiltonian system, defined by H :The flow ~n : Un ~ M, U u an open subset of M × IlL of the Hamiltonian H is the map defined by: (i) ~H(m, 0) = m for each m ~ m (m × {0} CUn). (ii) t~.~n(m, t) is the integral curve of Xn through m. (iii) U H is the largest connected open subset of M × IR on which ~n can be defined.IfH is C k+l, ~n is Ck; for more details see S. Lang [4]. A closed orbit ? of a Hamiltonian system is a subset of M which is of the form ~n (m, [0, t]) where ~n(m, t): ~n(m, 0)=m and @n(m, t')~m for all t'e (0, t); t is called the period of ?. In order to study the properties of the flow @H near a closed orbit y, we introduce a flowbox with local section and a Poincar6 map for y.For m ~ ? (dH),. 4= O. Then, by the flowbox theorem [2], there is a neighbourhood U ofm and a map A: U~IR 2" such that: Xl ..... x,, Yl ..... y. are the coordinate funci fions on IR2". HJU=y~oA.A is called a flowbox at m. The local section Z,,,,a is defined by X,,,a=A-l({xl =x~(A(m))}). Z,. is called a local section at m if it can be obtained in the above way from some flowbox at m. Notice that A.(Xn) = O/Ox~ ; so Xu is nowhere tangent to Z,,.

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Generic Stability Properties of Periodic Points

Abstract: Abstract. A classification of the periodic points of a generic area-perserving diffeomorphism which depends on a parameter is given. The stability properties of each periodic point in the classification is decided.

Cited by 3 publications

References 1 publication

Periodic Solutions of a Newtonian Equation: Stability by the Third Approximation

Periodic Solutions of a Newtonian Equation: Stability by the Third Approximation

High-Dimensional Chaotic and Attractor Systems

Hamiltonian systems: Generic properties of closed orbits and local perturbations

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