Single Periodic Orbit Research Articles

Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci

<p style='text-indent:20px;'>A class of two-fast, one-slow multiple timescale dynamical systems is considered that contains the system of ordinary differential equations obtained from seeking travelling-wave solutions to the FitzHugh-Nagumo equations in one space dimension. The question addressed is the mechanism by which a small-amplitude periodic orbit, created in a Hopf bifurcation, undergoes rapid amplitude growth in a small parameter interval, akin to a <i>canard explosion</i>. The presence of a saddle-focus structure around the slow manifold implies that a single periodic orbit undergoes a sequence of folds as the amplitude grows. An analysis is performed under some general hypotheses using a combination ideas from the theory of canard explosion and Shilnikov analysis. An asymptotic formula is obtained for the dependence of the parameter location of the folds on the singular parameter and parameters that control the saddle focus eigenvalues. The analysis is shown to agree with numerical results both for a synthetic normal-form example and the FitzHugh-Nagumo system.</p>

Convergence of a periodic orbit family close to asteroids during a continuation

In this work, we study the continuation of a periodic orbit on a relatively large scale and discover the existence of convergence under certain conditions, which has profound significance in research on asteroids and can provide a total geometric perspective to understanding the evolution of the dynamic characteristics from a global perspective. Based on the polyhedron model, convergence is derived via a series of theoretical analyses and derivations, which shows that a periodic orbit will evolve into a nearly circular orbit with a normal periodic ratio (e.g., 2:1, 3:2, and 4:3) and almost zero torsion under proper circumstances. As an application of the results developed here, three asteroids, (216) Kleopatra, (22) Kalliope and (433) Eros, are studied, and several representative periodic orbit families are detected, with convergence in three different cases: bidirectional, increasing-directional and decreasing-directional continuation. At the same time, four commonalities among these numerical examples are concluded. First, a (pseudo) tangent bifurcation arises at the cuspidal points during the variations in the periodic ratios in a single periodic orbit family. In addition, these cuspidal points in the periodic ratio coincide with the turning points during the variations in the average radius, the maximal torsion and the maximal radius of curvature. Furthermore, the periodic ratio increases (or decreases) with an increase (or decrease) in the Jacobian constant overall. We find the relationship between periodic ratio and Jacobian constant. The results implies that the periodic ratios for a fixed resonant status have an infimum and a supremum. Finally, if the periodic orbit converges to a point, it can be an unstable equilibrium point of the asteroid.

Cyclotron dynamics of neutral atoms in optical lattices with additional magnetic field and harmonic trap potential**Project supported by the National Natural Science Foundation of China (Grant Nos. 11764039, 11847304, 11865014, 11475027, 11305132, and 11274255), the Natural Science Foundation of Gansu Province, China (Grant No. 17JR5RA076), and the Scientific Research Project of Gansu Higher Education Department, China (Grant No. 2016A-005).

We analytically and numerically discuss the stability and dynamics of neutral atoms in a two-dimensional optical lattice subjected to an additional harmonic trap potential and artificial magnetic field. The harmonic trap potential plays a key role in modifying the equilibrium state properties of the system and stabilizing the cyclotron orbits of the condensate. Meanwhile, the presence of the harmonic trap potential and lattice potential results in rich cyclotron dynamics of the condensate. The coupling effects of lattice potential, artificial magnetic field, and harmonic trap potential lead to single periodic, multi-periodic or quasi-periodic cyclotron orbits of the condensate. So we can control the cyclotron dynamics of neutral atoms in optical lattice by manipulating the strength of harmonic confinement, artificial magnetic field, and initial conditions. Our results provide a direct theoretical evidence for the cyclotron dynamics of neutral atoms in optical lattices exposed to the artificial gauge magnetic field and harmonic trap potential.

On periodic mapping data of a two-dimensional torus with one saddle orbit

Periodic data of diffeomorphisms with regular dynamics on surfaces were studied using zeta functions in a series of already classical works by such authors as P. Blanchard, J. Franks, S. Narasimhan, S. Batterson and others. The description of periodic data for gradient-like diffeomorphisms of surfaces were given in the work of A. Bezdenezhnykh and V. Grines by means of the classification of periodic surface transformations obtained by J. Nielsen. V. Grines, O. Pochinka, S. Van Strien showed that the topological classification of arbitrary Morse-Smale diffeomorphisms on surfaces is based on the problem of calculating periodic data of diffeomorphisms with a single saddle periodic orbit. Namely, the construction of filtering for Morse-Smale diffeomorphisms makes it possible to reduce the problem of studying periodic surface diffeomorphism data to the problem of calculating periodic diffeomorphism data with a single saddle periodic orbit. T. Medvedev, E. Nozdrinova, O. Pochinka solved this problem in a general formulation, that is, the periods of source orbits are calculated from a known period of the sink and saddle orbits. However, these formulas do not allow to determine the feasibility of the obtained periodic data on the surface of this kind. In an exhaustive way, the realizability problem is solved only on a sphere. In this paper we establish a complete list of periodic data of diffeomorphisms of a two-dimensional torus with one saddle orbit, provided that at least one nodal point of the map is fixed.

Delay embedding of periodic orbits using a fixed observation function

Delay coordinates are a widely used technique to pass from observations of a dynamical system to a representation of the dynamical system as an embedding in Euclidean space. Current proofs show that delay coordinates of a given dynamical system result in embeddings generically with respect to the observation function (Sauer et al., 1991). Motivated by applications of the embedding theory, we consider flow along a single periodic orbit where the observation function is fixed but the dynamics is perturbed. For an observation function that is fixed (as a nonzero linear combination of coordinates) and for the special case of periodic solutions, we prove that delay coordinates result in an embedding generically over the space of vector fields in the Cr−1 topology with r≥2.

Error-feedback control of multistability

We propose the use of error-feedback control for converting a multistable system into a monostable one. The method is tested with a multistable erbium-doped fiber laser which exhibits the coexistence of four periodic orbits. A desired orbit is selected by an external harmonic modulation which frequency is equal to the frequency of one of the coexisting attractors. Additionally, the error signal, proportional to the difference between the external modulation and the laser output intensity, is applied to the pump current of the diode pumped laser. For certain combinations of the amplitude of the external modulation and the error signal, the laser displays monostability with a single periodic orbit at frequency determined by the external modulation. The efficiency of the control is demonstrated by both numerical simulations and experimentally.

Finite-energy pseudoholomorphic planes with multiple asymptotic limits

It’s known from (Ann Inst H Poincaré Anal Non Linéaire 13(3):337–379 [11], Properties of Pseudoholomorphic Curves in Symplectisation. IV. Asymptotics with Degeneracies. In: Contact and symplectic geometry (Cambridge, 1994), vol 8 Publ. Newton Inst., pp 78–117. Cambridge Univ. Press, Cambridge [10], A Morse-Bott Approach to Contact Homology. PhD thesis, Stanford University [2]) that in a contact manifold equipped with either a nondegenerate or Morse-Bott contact form, a finite-energy pseudoholomorphic curve will be asymptotic at each of its nonremovable punctures to a single periodic orbit of the Reeb vector field and that the convergence is exponential. We provide examples here to show that this need not be the case if the contact form is degenerate. More specifically, we show that on any contact manifold $$(M, \xi )$$ with cooriented contact structure one can choose a contact form $$\lambda $$ with $$\ker \lambda =\xi $$ and a compatible complex structure J on $$\xi $$ so that for the associated $$\mathbb {R}$$ -invariant almost complex structure $$\tilde{J}$$ on $$\mathbb {R}\times M$$ there exist families of embedded finite-energy $$\tilde{J}$$ -holomorphic cylinders and planes having embedded tori as limit sets.

Ground states are generically a periodic orbit

We prove that for an expanding transformation the maximizing measures of a generic Lipschitz function are supported on a single periodic orbit.

Outer Billiards with Contraction: Attracting Cantor Sets

We consider the outer billiards map with contraction outside polygons. We construct a 1-parameter family of systems such that each system has an open set in which the dynamics are reduced to that of a piecewise contraction on the interval. Using the theory of rotation numbers, we deduce that every point inside the open set is asymptotic to either a single periodic orbit (rational case) or a Cantor set (irrational case). In particular, we deduce the existence of an attracting Cantor set for certain parameter values. Moreover, for a different choice of 1-parameter family, we prove that the system is uniquely ergodic; in particular, the entire domain is asymptotic to a single attractor.

Formation and interaction of resonance chains in the open three-disk system

In ballistic open quantum systems, one often observes that the resonances in the complex-energy plane form a clear chain structure. Taking the open three-disk system as a paradigmatic model system, we investigate how this chain structure is reflected in the resonance states and how it is connected to the underlying classical dynamics. Using an efficient scattering approach, we observe that resonance states along one chain are clearly correlated, while resonance states of different chains show an anticorrelation. Studying the phase-space representations of the resonance states, we find that their localization in phase space oscillates between different regions of the classical trapped set as one moves along the chains, and that these oscillations are connected to a modulation of the resonance spacing. A single resonance chain is thus not a WKB quantization of a single periodic orbit, but the structure of several oscillating chains arises from the interaction of several periodic orbits. We illuminate the physical mechanism behind these findings by combining the semiclassical cycle expansion with a quantum graph model.

Global invariant manifolds near a Shilnikov homoclinic bifurcation

We consider a three-dimensional vector field with a Shilnikov homoclinic orbit that converges to a saddle-focus equilibrium in both forward and backward time. The one-parameter unfolding of this global bifurcation depends on the sign of the saddle quantity. When it is negative, breaking the homoclinic orbit produces a single stable periodic orbit; this is known as the simple Shilnikov bifurcation. However, when the saddle quantity is positive, the mere existence of a Shilnikov homoclinic orbit induces complicated dynamics, and one speaks of the chaotic Shilnikov bifurcation; in particular, one finds suspended horseshoes and countably many periodic orbits of saddle type. These well-known and celebrated results on the Shilnikov homoclinic bifurcation have been obtained by the classical approach of reducing a Poincare return map to a one-dimensional map. &nbsp In this paper, we study the implications of the transition through a Shilnikov bifurcation for the overall organization of the three-dimensional phase space of the vector field. To this end, we focus on the role of the two-dimensional global stable manifold of the equilibrium, as well as those of bifurcating saddle periodic orbits. We compute the respective two-dimensional global manifolds, and their intersection curves with a suitable sphere, as families of orbit segments with a two-point boundary-value-problem setup. This allows us to determine how the arrangement of global manifolds changes through the bifurcation and how this influences the topological organization of phase space. For the simple Shilnikov bifurcation, we show how the stable manifold of the saddle focus forms the basin boundary of the bifurcating stable periodic orbit. For the chaotic Shilnikov bifurcation, we find that the stable manifold of the equilibrium is an accessible set of the stable manifold of a chaotic saddle that contains countably many periodic orbits of saddle type. In intersection with a suitably chosen sphere we find that this stable manifold is an indecomposable continuum consisiting of infinitely many closed curves that are locally a Cantor bundle of arcs.

Bifurcation from single periodic orbit in discontinuous autonomous systems

This article is devoted to the study of bifurcations of periodic solutions for discontinuous autonomous systems from single periodic solutions of unperturbed discontinuous equations. In addition, local asymptotic properties of derived perturbed periodic solutions are also investigated. An example of planar discontinuous ordinary differential equations is given to illustrate the theory.

Global Invariant Manifolds Near Homoclinic Orbits to a Real Saddle: (Non)Orientability and Flip Bifurcation

Homoclinic bifurcations are important phenomena that cause global rearrangements of the dynamics in phase space, including changes to basins of attractions and the generation of chaotic dynamics. We consider here a homoclinic (or connecting) orbit that converges in both forward and backward time to a saddle equilibrium of a three-dimensional vector field. We assume that the saddle is such that the eigenvalues of its Jacobian are real. If such a homoclinic orbit is broken by varying a suitable parameter, then, generically, a single periodic orbit $\Gamma$ bifurcates. We consider the case that the saddle quantity of the equilibrium is negative so that $\Gamma$ is an attractor (rather than of saddle type). At the moment of bifurcation the two-dimensional stable manifold of the saddle, when followed along the homoclinic orbit, may form either an orientable or nonorientable surface, and one speaks of an orientable or a nonorientable homoclinic bifurcation. A change of orientability occurs at two kinds of codim...

Dynamical Properties of A Cellular Automaton on A Countable Group

Topological structure of the living space is one of the most important parameters for cellular automata (CA), which can be applied to various scientific areas. In order to study CA in larger extent, we define a new type of cellular automata on countable groups and investigate some general dynamical properties of attractors and periodicity. It is shown that if such cellular automaton on a countable group is transitive, it is either sensitive or composed of a single periodic orbit. By using of the form of attractors, classification of this type of cellular automata is given on countable groups. The set of eventually periodic points of a cellular automaton on a countable residually finite group is proved to be dense.

Double impact periodic orbits for an inverted pendulum

There exist many types of possible periodic orbits that impact at the walls for the inverted pendulum impacting between two rigid walls. Previous studies only focused on single impact periodic orbits and symmetric periodic orbits that bounce back and forth between the two walls. They respectively correspond to Types I and II orbits in the Chow, Shaw and Rand classification. In this paper we discuss two types of double impact periodic orbits that have not been studied before. The equations need to be solved for double impact orbits are transcendental and it is very hard to see the structure of the solutions. Consequently the analysis of double impact orbits is much more difficult than that of Types I and II orbits. A combination of analytical and numerical methods is employed to investigate the existence, stability and bifurcations of these orbits. Grazing bifurcations, which do not present for Types I and II orbits, are also observed.

Reply to “Comment on ‘Time-averaged properties of unstable periodic orbits and chaotic orbits in ordinary differential equation systems’ ”

We point out that the original work [Saiki and Yamada, Phys. Rev. E 79, 015201(R) (2009)] was aimed at addressing the question of why even a single periodic orbit of low period can capture averaged properties of physical quantities reasonably well in some dynamical systems including turbulent fluids. This point of view is quite different from that of the Comment [Zaks and Goldobin, Phys. Rev. E 81, 018201 (2010)].

A rigorous lower bound for the stability regions of the quadratic map

We establish a lower bound on the measure of the set of stable parameters a for the quadratic map Q a ( x ) = a x ( 1 − x ) . For these parameters, we prove that Q a either has a single stable periodic orbit or a period-doubling bifurcation. From this result, we also obtain a non-trivial upper bound on the set of stochastic parameters for Q a .

Relative asymptotic behavior of pseudoholomorphic half‐cylinders

AbstractIn this article we study the asymptotic behavior of pseudoholomorphic half‐cylinders that converge exponentially to a periodic orbit of a vector field defined by a framed stable Hamiltonian structure. Such maps are of central interest in symplectic field theory and its variants (symplectic Floer homology, contact homology, and embedded contact homology). We prove a precise formula for the asymptotic behavior of the “difference” of two such maps, generalizing results from [6, 7, 12, 15]. Using this result with a technique from [14], we then show that a finite collection of pseudoholomorphic half‐cylinders asymptotic to coverings of a single periodic orbit is smoothly equivalent to solutions to a linear equation. © 2007 Wiley Periodicals, Inc.

On box models of the North Atlantic thermohaline circulation: Intrinsic and extrinsic millennial timescale variability in response to deterministic and stochastic forcing

Millennial timescale variability in the North Atlantic Ocean circulation is often discussed in terms of concepts that are rooted in the dynamics of simple, low‐dimensional box models. In this study we discuss possible explanatory mechanisms for the millennial timescale oscillations of the North Atlantic Thermohaline Circulation that have been revealed by the Greenland ice core record for the late glacial period. We subject three qualitatively different low‐order models to stochastic and sinusoidal perturbations: (1) a bistable model, (2) a model with a single stable equilibrium point and a single stable periodic orbit (limit cycle) corresponding to a collapse‐flush cycle in the North Atlantic circulation, and (3) a model with a single globally stable equilibrium point which nevertheless exhibits complex behavior when perturbed. We discuss both the physical nature of the model response and its parameter dependence. We conclude that the traditional definition of stochastic resonance should be expanded and that the temporal characteristics of the noise terms should be considered an integral part of model construction, as they profoundly affect the efficacy of a given explanatory mechanism.

Wave functions with localizations on classical periodic orbits in weakly perturbed quantum billiards

We analytically investigate quantum wave functions with localizations on classical periodic orbits (POs) in the circular billiards. We construct the coherent states which are the coherent superpositions of nearly degenerate eigenstates and found to be localized around single classical POs. With the constructed coherent states, we can analytically express the mesoscopic eigenstates in the weakly perturbed systems, which are found to be localized on the ensemble of classical POs. Furthermore, these coherent states can be utilized to study the quantum vortex structures in quantum billiards.