Generic bifurcation of periodic points

Abstract

Abstract. This paper discusses the bifurcation of periodic points of a generic symplectic diffeomorphism of a two manifold that depends on a parameter. A complete classification of the types of bifurcation that can occur in the generic case is given.Introduction. This paper discusses the bifurcation of periodic points of an area preserving diffeomorphism that depends on a parameter. In the spirit of Thorn and Smale only the generic case is considered. As in Smale's work on discrete dynamical systems the present problem was suggested by problems in ordinary differential equations (see [1]). Whereas Smale modeled his theory on differential equations with dissipation the present problem was suggested by conservative differential equations.Throughout this paper smooth will always mean C°°. Let M be a smooth, compact, two dimensional, symplectic manifold, S=SX the circle considered as a smooth manifold and F the space of all smooth mappings <p: MxS-> M with the property that for each s e S the map <ps=<p(-, s): M-> M is a symplectic diffeomorphism. Let F have the usual topology of C°° maps. The space F is complete and therefore is a Baire space. A point (x, s) e M x S is called a periodic point of least period m if <p™(x)=x and m is the smallest positive integer for which this holds. Any multiple of m will be called a period of (x, s). If (x, s) is a periodic point of <p of least period m then each point (<p\ix), s) e M x S is a distinct periodic point of least period m for i = 1,..., m and the set U™= i (<pl(x), s) is called the orbit of <p through (x, s).

Universal quantum signature of mixed dynamics in antidot lattices

“…They have been classified by Meyer and Bruno. 32,33 A list of the normal forms can be obtained from Table I by setting x 2 =1 ͑except for r = 1 where it is P 2 + x 1 Q + Q 3 ͒. These normal forms are simplified versions in which all constants and terms that are irrelevant for the determination of ␤ and ␥ have been removed.…”

Geometry and topology of escape. I. Epistrophes

“…A fixed point of the map corresponds to a stationary point of G(q, p), where ∂G/∂p = ∂G/∂q = 0. The fixed point is stable or unstable according to whether the Hessian determinant D = (∂ 2 G/∂p 2 )∂ 2 G/∂q 2 − (∂ 2 G/∂q∂p) 2 is positive or negative [53,54]. Accordingly, the above map has exactly one stable and one unstable fixed point, located at the local minimum and the local maximum of V (q).…”

Generic Stability Properties of Periodic Points

“…Further discussion on the origin of the present problem in the theory of Hamiltonian differential equations can be found in [3].…”

“…Introduction. This paper is a sequel to [3]. In the previous paper a classification of the periodic points of a generic area-preserving diffeomorphism which depends on a parameter was given.…”

“…In [3] a residual set '¡S^F was defined such that if (x, s) is a periodic point of cpeG then either it is (1) an extremal or (2) a transitional or (3) a hyperbolic or [February (4) an elliptic periodic point. If A, A-1 are the multipliers of (x, s) then roughly speaking the classification is (1) A= +1, (2) A= -1, (3) A^ + 1, A real, (4) A^ ± 1, A complex and |A| = 1.…”