Sequence Of Periodic Orbits Research Articles

On the hyperbolicity of [formula omitted]-generic homoclinic classes

The works of Liao, Mañé, Franks, Aoki, and Hayashi characterized a lack of hyperbolicity for diffeomorphisms by the existence of weak periodic orbits. In this note, we announce a result that can be seen as a local version of these works: for C1-generic diffeomorphisms, a homoclinic class either is hyperbolic or contains a sequence of periodic orbits that have a Lyapunov exponent arbitrarily close to 0.

A C1 closing lemma for nonuniformly partially hyperbolic diffeomorphisms of class C1+α

We prove a C1 closing lemma for C1+α diffeomorphisms under the existence of dominated splittings associated with the Lyapunov splittings over the supports of ergodic measures, which creates a sequence of periodic orbits having the local strong stable and unstable manifolds with uniform sizes over some compact set whose measure arbitrarily close to 1. This corresponds to Pesin's stable and unstable manifolds theorem for nonuniformly hyperbolic diffeomorphisms of class C1+α.

Periodic orbits and binary collisions in the classical three-body Coulomb problem

In the helium case of the classical three-body Coulomb problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. Focusing our attention on binary collisions with these tools, a sequence of periodic orbits are predicted and are actually found numerically. A family of periodic orbits found has regularity in their actions. For this family of periodic orbits, it is shown that thanks to its regularity, a partial summation of the Gutzwiller trace formula with a daring approximation gives a Rydberg series of energy levels.

Periodic and Quasi-Periodic Orbits�for the Standard Map

We consider both periodic and quasi-periodic solutions for the standard map, and we study the corresponding conjugating functions, i.e. the functions conjugating the motions to trivial rotations. We compare the invariant curves with rotation numbers ω satisfying the Bryuno condition and the sequences of periodic orbits with rotation numbers given by their convergents ω N = p N /q N . We prove the following results for N→ ∞: (1) for rotation numbers ω N N we study the radius of convergence of the conjugating functions and we find lower bounds on them, which tend to a limit which is a lower bound on the corresponding quantity for ω; (2) the periodic orbits consist of points which are more and more close to the invariant curve with rotation number ω; (3) such orbits lie on analytical curves which tend uniformly to the invariant curve.

Renormalization and Periodic Orbits¶for Hamiltonian Flows

We consider a renormalization group transformation \(\) for analytic Hamiltonians in two or more dimensions, and use this transformation to construct invariant tori, as well as sequences of periodic orbits with rotation vectors approaching that of the invariant torus. The construction of periodic and quasiperiodic orbits is limited to near-integrable Hamiltonians. But as a first step toward a non-perturbative analysis, we extend the domain of \(\) to include any Hamiltonian for which a certain non-resonance condition holds.

On the Stability of Double Homoclinic Loops

We consider 2-degrees of freedom Hamiltonian systems with an involutive symmetry and a pair of orbits bi-asymptotic (homoclinic) to a saddle-center equilibrium (related to pairs of pure real, ±ν, and pure imaginary eigenvalues, ±ω i). We show that the stability of this double homoclinic loop is determined by the reflection coefficient of a one-dimensional scattering problem and ω/ν. We also show that the mechanism for losing stability is the creation of an infinite heteroclinic chain connecting a sequence of periodic orbits that accumulates at the double loop.

On the duality between periodic orbit statistics and quantum level statistics

We discuss consequences of a recent observation that the sequence of periodic orbits in a chaotic billiard behaves like a poissonian stochastic process on small scales. This enables the semiclassical form factor $K_{sc}(\tau)$ to agree with predictions of random matrix theories for other than infinitesimal $\tau$ in the semiclassical limit.

MAP DYNAMICS OF REPRODUCTION

One-dimensional return maps characterized by one critical point produce a universal sequence of periodic orbits, the Metropolis, Stein & Stein (MSS) sequence. Iterates of such maps can be regarded as an object whose structure is defined by the MSS sequence. The two critical point mapping [Formula: see text] as the control parameter b varies between [Formula: see text] and 1 is shown to display evolution from a single MSS structure to two identical MSS structures, suggestive of reproduction. The process of reproduction connotes a causal relationship whereby an original structure provides the blueprint for a copy. Here causality is replaced by an omniscient instruction, the mapping, and reproduction is achieved by progressively changing the instruction through variation of the control parameter. Bistability plays a crucial role in map dynamics of reproduction, which is viewed as a holistic process providing an alternative mechanism to digital replication.

Period-multiplying cascades for diffeomorphisms of the disc

It is a well-known result in one-dimensional dynamics that if a continuous map of the interval has positive topological entropy, then it has a periodic orbit of period 2i for each integer i ≥ 0 [15] (see also [12]). In fact, one can say rather more: such a map has a sequence of periodic orbits (P)i ≥ 0 with per (Pi) = 2i which form a period-doubling cascade (that is, whose points are ordered and permuted in the way which would occur had the orbits been created in a sequence of period-doubling bifurcations starting from a single fixed point). This result reflects the central role played by period-doubling in transitions to positive entropy in a one-dimensional setting. In this paper we prove an analogous result for positive-entropy orientation-preserving diffeomorphisms of the disc. Using the notion [9] of a two-dimensional cascade, we shall show that such diffeomorphisms always have infinitely many ‘zero-entropy’ cascades of periodic orbits (including a period-doubling cascade, though this need not begin from a fixed point).

Tupling in three-dimensional reversible mappings

Feigenbaum's parameter scaling exponents S are calculated for l.3m,l.5m,l.7m,l.9m sequences of periodic orbits in a class of three-dimensional non-measure-preserving reversible mappings. Some preliminary results on orbit scaling exponents for l.3m sequences are also included. Scaling exponents found so far are the same as those found for two-dimensional reversible mappings.

Measures on periodic orbits for continuous transformations on the interval

Let be a continuous transformation which is eventually onto, i.e., for any interval I in [o,l] there exists- n such that . Let μ be a [ILM0003]-ergodic measure. Then there exists a sequence of periodic orbits

Stable arcs of diffeomorphisms

We consider here arcs of diffeomorphisms which, being initially structurally stable, go through brusque changes (or bifurcations) in their phase portrait (space of orbits) as the parameter changes. We are mainly interested in stable arcs in the sense that nearby arcs have the same changes in their phase portraits. As we shall see, there are various natural ways of defining stability for arcs. For a of arcs starting as Morse-Smale diffeomorphisms we characterize the arcs that are stable according to the various definitions of stability. We conjecture that this large class really contains all stable arcs of diffeomorphisms starting as Morse-Smale diffeomorphisms. We present below the precise statements of the results, beginning with some definitions and known facts. The proofs will appear elsewhere. denotes a compact C°° manifold without boundary. Diff(A/) is the space of C°° diffeomorphisms of and (M) the usual C°°-topologies. For g G Diff(M), the orbit 0(x) of a point x G is defined by 0{x) = i8^(x)\n G Z}; aj pointy G is called a limit point of g if for some x G and sequence np i = 1, 2, . . . , nG Z, \nt —• °°, lim g'(x) = y. The closure of the limit points of g is called the limit set of g and denoted by L(g). A point x G is a periodic point of g with period n if g(x) = x and £(x) -=h x for all 0 } and {y G M dist(/(y), f{x)) —» 0 for n —• -°°} respectively. If x is a hyperbolic periodic point of g, W(x, g) and W(x, g) are smoothly immersed submanifolds of We say that a diffeomorphism g G Diff(M), with finite limit set, has an «-cycle if there is a sequence of periodic orbits 0(px), . . . , 0(pn) with O(p0) = 0(pn) and 0(pi+1) C closure (W(0(>.)) ~ °(Pi)) f o r 0 1 if no other periodic orbit can be added to the previous sequence.