Discretized rotation has infinitely many periodic orbits (original) (raw)
Giga-Periodic Orbits for Weakly Coupled Tent and Logistic Discretized Maps
Simple dynamical systems often involve periodic motion. Quasi-periodic or chaotic motion is frequently present in more complicated dynamical systems. However, for the most part, underneath periodic motion models chaotic motion. Chaotic attractors are nearly always present in such dissipative systems. Since their discovery in 1963 by E. Lorenz, they have been extensively studied in order to understand their nature. In the past decade, the aim of the research has been shifted to the applications for industrial mathematics. Their importance in this field is rapidly growing. Chaotic orbits embedded in chaotic attractor can be controlled allowing the possibility to control laser beams or chemical processes and improving techniques of communications. They can also produce very long sequences of numbers which can be used as efficiently as random numbers even if they have not the same nature. However, mathematical results concerning chaotic orbits are often obtained using sets of real numbers (belonging to R or R n) (e.g. the famous theorem of A. N. Sharkovskiǐ which defines which ones periods exist for continuous functions such as logistic or tent maps). O.E. Lanford III reports the results of some computer experiments on the orbit structure of the discrete maps on a finite set which arise when an expanding map of the circle is iterated "naively" on the computer. There is a huge gap between these results and the theorem of Sharkovskiǐ, due to the discrete nature of floating points used by computers. This article introduces new models of very very weakly coupled logistic and tent maps for which orbits of very long period are found. The length of these periods is far greater than one billion. We call giga-periodic orbits such orbits for which the length is greater than 10 9 and less than 10 12. Tera, and peta periodic orbits are the name of the orbits the length of which is one thousand or one million greater. The property of these models relatively to the distribution of the iterates (invariant measure) are described. They are found very useful for industrial mathematics for a variety of purposes such as generation of cryptographic keys, computer games and some classes of scientific experiments.
Families of periodic orbits in resonant reversible systems
Bulletin of the Brazilian Mathematical Society, New Series, 2009
We study the dynamics near an equilibrium point p 0 of a Z 2 (R)-reversible vector field in R 2n with reversing symmetry R satisfying R 2 = I and dim Fi x(R) = n. We deal with one-parameter families of such systems X λ such that X 0 presents at p 0 a degenerate resonance of type 0 : p : q. We are assuming that the linearized system of X 0 (at p 0 ) has as eigenvalues: λ 1 = 0 and λ j = ±iα j , j = 2, . . . n. Our main concern is to find conditions for the existence of one-parameter families of periodic orbits near the equilibrium.
Dynamical Systems on the Circle
2019
In this paper we introduce dynamical systems on the circle. Beginning with elementary notions of dynamical systems, we develop several tools to study these systems, leading to the rotation number, a key invariant of circle homeomorphisms. We then show that any non-periodic circle homeomorphism is semiconjugate to the rotation by its rotation number. Finally, we present a proof of Denjoy’s Theorem, which states that any non-periodic circle diffeomorphism with derivative of bounded variation is conjugate to the rotation by its rotation number.
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.
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.
Stable and unstable periodic orbits in the one-dimensional latticeϕ4theory
Physical Review E, 2016
Periodic orbits for the classical φ 4 theory on the one dimensional lattice are systematically constructed by extending the normal modes of the harmonic theory, for periodic, free and fixed boundary conditions. Through the process, we investigate which normal modes of the linear theory can or can not be extended to the full non-linear theory and why. We then analyze the stability of these orbits, clarifying the link between the stability, parametric resonance and the Lyapunov spectra for these orbits. The construction of the periodic orbits and the stability analysis is applicable to theories governed by Hamiltonians with quadratic inter-site potentials and a general on-site potential. We also apply the analysis to theories with on-site potentials that have qualitatively different behavior from the φ 4 theory, with some concrete examples.
Rotation sets and almost periodic sequences
Mathematische Zeitschrift, 2016
We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segments as well as non-convex and even plane-separating continua. This shows that restrictions which hold for rotation set on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable irregular Toeplitz sequences.
Dynamics close to a non semi-simple 1:-1 resonant periodic orbit
Discrete and Continuous Dynamical Systems - Series B, 2005
In this work, our target is to analyze the dynamics around the 1 : −1 resonance which appears when a family of periodic orbits of a real analytic three-degree of freedom Hamiltonian system changes its stability from elliptic to a complex hyperbolic saddle passing through degenerate elliptic. Our analytical approach consists of computing, up to some given arbitrary order, the normal form around that resonant (or critical) periodic orbit.
Rotation numbers for quasi-periodically forced monotone circle maps
Dynamical Systems, 2002
Rotation numbers have played a central role in the study of (unforced) monotone circle maps. In such a case it is possible to obtain a priori bounds of the form » ¡ 1=n µ …1=n †…y n ¡ y 0 † µ » ‡ 1=n, where …1=n †…y n ¡ y 0 † is an estimate of the rotation number obtained from an orbit of length n with initial condition y 0 , and » is the true rotation number. This allows rotation numbers to be computed reliably and e ciently. Although Herman has proved that quasi-periodically forced circle maps also possess a well-de®ned rotation number, independent of initial condition, the analogous bound does not appear to hold. In particular, two of the authors have recently given numerical evidence that there exist quasi-periodically forced circle maps for which y n ¡ y 0 ¡ »n is not bounded. This renders the estimation of rotation numbers for quasi-periodically forced circle maps much more problematical. In this paper, a new characterization of the rotation number is derived for quasiperiodically forced circle maps based upon integrating iterates of an arbitrary smooth curve. This satis®es analogous bounds to above and permits us to develop improved numerical techniques for computing the rotation number. Additionally, the boundedness of y n ¡ y 0 ¡ »n is considered. It is shown that if this quantity is bounded (both above and below) for one orbit, then it is bounded for all orbits. Conversely, if for any orbit y n ¡ y 0 ¡ »n is unbounded either above or below, then there is a residual set of orbits for which y n ¡ y 0 ¡ »n is unbounded both above and below. In proving these results a min±max characterization of the rotation number is also presented. The performance of an algorithm based on this is evaluated, and on the whole it is found to be inferior to the integral based method.
Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps
Nonlinearity, 2013
We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.