Birkhoff Periodic Orbits in the Standard-Like Maps (original) (raw)

Non-Symmetric Non-Birkhoff Period-2 Orbits in the Standard Mapping

Progress of Theoretical Physics, 2001

Period-2 badly ordered orbits (non-Birkhoff orbits) are studied in the standard mapping. Points of symmetric non-Birkhoff orbits appear on symmetry axes due to the saddle-node bifurcation, and non-symmetric non-Birkhoff orbits appear due to the equi-period bifurcation of symmetric non-Birkhoff orbits. The braids of symmetric non-Birkhoff orbits are constructed, and the topological entropy is estimated.

Non-Birkhoff Orbits with 2n Turning Points in the Standard Map

Progress of Theoretical Physics, 2003

One of the two fixed points of the standard map gives rise to a period-doubling bifurcation and becomes a saddle with reflection beyond a certain parameter value. In association with this bifurcation, symmetric non-Birkhoff periodic orbits (SNBOs) with 2n (n ≥ 2) turning points appear and exhibit complicated behavior. We first analyze the structure of stable and unstable manifolds of this saddle and then derive dynamical order relations for these SNBOs and show that a period-3 SNBO implies the existence of SNBOs with all possible numbers of turning points.

Non-Birkhoff Periodic Orbits of Farey Type and Dynamical Ordering in the Standard Mapping

Progress of Theoretical Physics, 2007

Using dynamical ordering for symmetric non-Birkhoff periodic orbits of Farey type (NBF), the properties of the limit at which the rotation numbers of NBFs accumulate to a rational number or irrational number are studied in the case of the standard mapping. Next, the relation between the critical value at which the KAM curve disappears and the critical value at which NBF appears is derived. The slow orbit connecting the two resonance chains is found.

Symmetrical Non-Birkhoff Period-3 Orbits in Standard-Like Mappings

Progress of Theoretical Physics, 2000

A necessary condition for the existence of non-Birkhoff period-3 orbits is derived in C 0 standard-like mappings. Symmetrical non-Birkhoff period-3 orbits of the pseudo-Anosov braid-type are found. In this case, the standard-like mapping is a pseudo-Anosov system in a certain parameter range. The braid types of all period-3 orbits are derived. Using these braids, a lower bound of the topological entropy is obtained. These orbits do not exist in a horseshoe.

Periodic Orbits in the Quadruply Reversible Non-Twist Harper Map

Progress of Theoretical Physics, 2004

The Harper map with a particular set of parameters possesses quadruple reversibilities. For this map, it is shown that singly reversible periodic orbits, doubly reversible periodic orbits, and quadruply reversible periodic orbits exist.

On Mather's Connecting Orbits in Standard Mapping

Progress of Theoretical Physics, 2008

Mather's connecting theorem is proved with the elementary geometrical method essentially using the reversibility of the standard mapping instead of the variational method. In the proof of the existence of orbits connecting quasi-periodic orbits of the same rotation number, non-monotonic quasi-periodic orbits (NMQs) are found, in which almost all orbital points are located in the gap of the Aubry-Mather set. The dynamical order relations among NMQs are derived. The existence of orbits connecting orbits of different irrational rotation numbers is proved using also non-Birkhoff periodic orbits of the Farey type. As an application of our method, a partial proof of Greene's criterion is given.

On the stability of some periodic orbits of a new type for twist maps

Nonlinearity, 2002

We study a two-parameter family of twist maps defined on the torus. This family essentially determines the dynamics near saddle-centre loops of four-dimensional real analytic Hamiltonian systems. A saddle-centre loop is an orbit homoclinic to a saddle-centre equilibrium (related to pairs of pure real, ±ν, and pure imaginary, ±ωi, eigenvalues). We prove that given any period n we can find an open set of parameter values such that this family has an attracting n-periodic orbit of a special type. This has interesting consequences on the original Hamiltonian dynamics.

Periodic Orbits of Maps of Y

Transactions of the American Mathematical Society, 1989

We introduce some notions that are useful for studying the behavior of periodic orbits of maps of one-dimensional spaces. We use them to characterize the set of periods of periodic orbits for continuous maps of Y = (z e C: z3 e [0,1]} into itself having zero as a fixed point. We also obtain new proofs of some known results for maps of an interval into itself.

Long periodic orbits of the triangle map

Proceedings of the American Mathematical Society, 1986

Let τ : [ 0 , 1 ] → [ 0 , 1 ] \tau :[0,1] \to [0,1] be defined by τ ( x ) = 2 x \tau (x) = 2x on [ 0 , 1 / 2 ] [0,1/2] and τ ( x ) = 2 ( 1 − x ) \tau (x) = 2(1 - x) on [ 1 / 2 , 1 ] [1/2,1] . We consider τ \tau restricted to the domain D N = { 2 a / p N , N ⩾ 1 , 0 ⩽ 2 a ⩽ p N , ( a , p ) = 1 } {D_N} = \{ 2a/{p^N},N \geqslant 1,0 \leqslant 2a \leqslant {p^N},(a,p) = 1\} where p p is any odd prime. Let k ⩾ 1 k \geqslant 1 be the minimum integer such that p N | 2 k ± 1 {p^N}|{2^k} \pm 1 . Then there are ( ( p − 1 ) ⋅ p N − 1 ) / 2 k (({\text {p}} - 1) \cdot {p^{N - 1}})/2k periodic orbits of τ | D N \tau {|_{{D_N}}} , having equal length, and there are k k points in each orbit. Furthermore, the proportion of points in any of these periodic orbits which lie in an interval ( c , d ) (c,d) approaches d − c d - c as p N − 1 → ∞ {p^{N - 1}} \to \infty . An application to the irreducibility of certain nonnegative matrices is given.

Non-Symmetric Periodic Points in Reversible Maps: Examples from the Standard Map

Progress of Theoretical Physics, 2002

The existence of non-symmetric non-Birkhoff periodic points and non-symmetric periodic points of the accelerator mode in standard-like maps is proved. Positions of these points are approximately determined for large parameter values. The number of such points is shown to diverge as the parameter value goes to infinity. We have found two routes for the appearance of non-symmetric periodic points. One is the equi-period bifurcation of a symmetric periodic point and the other is simultaneous saddle-node bifurcations. These two bifurcations seem to disprove the necessity of 'hidden symmetry' introduced by Murakami et al.(2001). We do not know whether or not these are the only routes for the appearance of these points. Numerical examples are considered for the standard map. with involutions H and G, i.e., H • H = G • G = Id and det∇H = det∇G = −1. Following DeVogelaere, 1) we introduce the operators M n = T n G for n ∈ Z (2) and invariant sets M n = {r|r = M n r} for n ∈ Z. (3) M 0 is the set of points invariant under G and M 1 is the set of points invariant under H. Both M 0 and M 1 are called symmetry axes. We have M 2n = T n M 0 and M 2n+1 = T n M 1 ; that is, we obtain M n as iterates of M 0 or M 1. We also have M −n = GM n. For n > m, define M n,m = M n ∩ M m. (4) M n is called a set of symmetric points and M n,m is called a set of doubly symmetric points. A point is doubly symmetric if two points of its orbit are in the set of doubly symmetric points. For more details, see DeVogelaere. 1) Theorem 1 (Ref. 1)). A point is symmetric periodic if and only if it is doubly symmetric.

Quasisymmetric orbit-flexibility of multicritical circle maps

Ergodic Theory and Dynamical Systems, 2021

Two given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Świątek, that the same holds if f is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in (0,1)(0,1)(0,1) , then the number of equivalence classes is uncountable (Theorem 1.1). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this ...

Periodic orbits in quantum standard maps

Physical Review A, 1992

The importance of periodic orbits, both elliptic and hyperbolic, in analyzing driven quantum systems is now well established. We present a detailed analysis of the quantum mechanics in the vicinity of these orbits for both a piecewise-linear and the original standard maps. We begin by constructing effective Hamiltonian operators valid locally near nondegenerate periodic orbits. Our general formula allows for the calculation of the quasienergy spectrum near elliptic orbits as well as of the strength of scarring on hyperbolic orbits. Preliminary results on the dependence of this measure of scarring on the instability of the classical orbit and on 1i are also presented. PACS number(s): 05.45.+b, 03.65.-w

Oscillatory Orbits in the Standard Mapping

Progress of Theoretical Physics, 2004

Oscillatory motion is unbounded but returns infinitely many times to finite positions. The existence of this motion has been predicted and proved to exist in the three-body problem. In the present paper, we prove its existence in the standard mapping, using orbits of the accelerator mode and non-Birkhoff periodic orbits. This motion appears as soon as the last Kolmogorov-Arnold-Moser curve disintegrates.

PHYSICS LETTERS A Stability of minimal periodic orbits

1998

Symplectic twist maps are obtained from a Lagrangian variational principle. It is well known that nondegenerate minima of the action correspond to hyperbolic orbits of the map when the twist is negative definite and the map is two-dimensional. We show that for more than two dimensions, periodic orbits with minimal action in symplectic twist maps with negative definite twist are not necessarily hyperbolic. In the proof we show that in the neighborhood of a minimal periodic orbit of period n, the nth iterate of the map is again a twist map. This is true even though in general the composition of twist maps is not a twist map. @ 1998 Elsevier Science B.V. PAW 03.2O.+i; 05.45.+b

Action and periodic orbits on annulus

2021

We consider the classical problem of area-preserving maps on annulus A = S × [0, 1] . Let Mf be the set of all invariant probability measures of an area-preserving, orientation preserving diffeomorphism f on A. Given any μ1 and μ2 in Mf , Franks [2][3], generalizing Poincaré’s last geometric theorem (Birkhoff [1]), showed that if their rotation numbers are different, then f has infinitely many periodic orbits. In this paper, we show that if μ1 and μ2 have different actions, even if they have the same rotation number, then f has infinitely many periodic orbits. In particular, if the action difference is larger than one, then f has at least two fixed points. The same result is also true for area-preserving diffeomorphisms on unit disk, where no rotation number is available.

Rotation sets for orbits of degree one circle maps

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2002

Let F be the lifting of a circle map of degree one. In [Bamón et al., 1984] a notion of F-rotation interval of a point was given. In this paper we define and study a new notion of a rotation set of point which preserves more of the dynamical information contained in the sequences than the one preserved from [Bamón et al., 1984]. In particular, we characterize dynamically the endpoints of these sets and we obtain an analogous version of the Main Theorem of [Bamón et al., 1984] in our settings.