Single Periodic Orbit Research Articles

Dynamics of delayed relay systems

The paper studies the dynamics near periodic orbits in dynamical systems with relays (switches) that switch only after a fixed delay. As a motivating application, we study the problem of stabilizing an unstable equilibrium by feedback control in the presence of a delay in the control loop. We show that saddle-type equilibria can be stabilized to a periodic orbit by a switch even if this switch is subject to an arbitrarily large delay. This is in contrast to linear static feedback control, which fails when the delay is larger than a problem-dependent critical value. Our analysis is based on the reduction of the return map near a generic periodic orbit to a finite-dimensional map. This map is smooth if the periodic orbit satisfies two genericity conditions. A violation of any of these two conditions causes a discontinuity-induced bifurcation of the periodic orbit. We derive asymptotic expressions for the piecewise smooth return map for each of these two codimension-one bifurcations. This analysis shows that the introduction of a small delay into the switching decision can induce chaos in a relay system that had a single stable periodic orbit without delay. This small-delay behaviour is fundamentally different from smooth dynamical systems.

Determination of Chaotic Dynamical Correlations in Terms of Unstable Periodic Orbits

For low-dimensional chaotic systems, we find that time correlation functions can be accurately approximated by a single unstable periodic orbit. The method of this determination consists of the following two steps. First, the time correlation functions are expanded in terms of static correlation functions by making use of the expansion method recently proposed by one of the authors [H. Fujisaka, Prog. Theor. Phys. 114 (2005), 1]. Second, the static quantities are approximated in terms of an unstable periodic orbit embedded in the strange attractor. Thus the dynamical correlation functions in chaotic systems can be determined in terms of an unstable periodic orbit. Furthermore, applying this method to low-dimensional chaotic models, we prove the usefulness of the present approach.

On the relation between quantum lifetimes and classical stability for the systems with a saddle-type potential

Relations between quantum-mechanical and classical properties of open systems with a saddle-type potential, for which at a given energy only one unstable periodic orbit exists, are studied. By considering the convergence of the Gutzwiller trace formula [J. Math. Phys. 12, 343 (1971)] it is confirmed that both for homogeneous and inhomogeneous potentials the poles of the formula are located below the real energy axis, i.e., these kind of potentials do not support bound states, in general. Within the harmonic approximation the widths of resonant (transition) states are proportional to the values of Lyapunov exponent of the single periodic orbit calculated at the energies which are equal to the resonance positions. The accuracy of the semiclassical relation is discussed and demonstrated for several examples.

A method of moments for calculating dynamic responses beyond linear response theory

A method of moments for calculating the dynamic response of periodically driven overdamped nonlinear stochastic systems in the general response sense is proposed, which is a modification of the method of moments confined within linear response theory. The calculating experience suggests that the proposed technique is simple and efficient in implementation and the comparison with stochastic simulation shows that the first three orders of susceptibilities calculated by the proposed technique have high accuracy. The dependence of the spectral amplification parameters at the first three harmonics on the noise intensity is also investigated and another observed phenomenon of stochastic resonance in the systems induced by the location of a single periodic orbit is disclosed and explained.

Correlation of the photodetachment rate of a scarred resonance state with the classical Lyapunov exponent.

A metastable resonance state of the periodically driven Gaussian potential well is shown to be scarred on a single unstable periodic orbit of the classical motion. The photodetachment rate of this quantum state is strongly correlated with the Lyapunov exponent of the unstable periodic orbit over a wide range of driving field strengths.

Coexisting periodic orbits in vibro-impacting dynamical systems

A method is presented to seek for coexisting periodic orbits which may be stable or unstable in piecewise-linear vibro-impacting systems. The conditions for coexistence of single impact periodic orbits are derived, and in particular, it is investigated in details how to assure that no other impacts will happen in an evolution period of a single impact periodic motion. Furthermore, some criteria for nonexistence of single impact periodic orbits with specific periods are also established. Finally, the stability of coexisting periodic orbits is discussed, and the corresponding computation formula is given. Examples of numerical simulation are in good agreement with the theoretic analysis.

Indecomposability and the structure of periodic orbits for interval maps

Suppose f :[a,b]→[a,b] is continuous. Barge and Martin, and Ingram have shown that if the inverse limit of {[ a, b], f} is hereditarily decomposable, then the period of every periodic orbit of f is a power of two. We will elaborate on the structure of these orbits, and, assuming f is a Markov map whose partition is a single periodic orbit, give necessary and sufficient conditions for the inverse limit to be (1) decomposable and (2) hereditarily decomposable.

Periodic magnetoconductance fluctuations in triangular quantum dots in the absence of selective probing

We have studied the magnetoconductance of quantum dots with triangular symmetry and areas down to 0.2 square microns, made in a high mobility two-dimensional electron gas embedded in a GaAs-AlGaAs heterostructure. Semiclassical simulations show that the gross features in the measured magnetoconductance are caused by ballistic effects. Below 1 K we observe a strong periodic oscillation, which may be explained in terms of the Aharanov-Bohm flux quantization through the area of a single classical periodic orbit. From a numerical and analytical analysis of possible trajectories in hard- and soft-walled potentials, we identify this periodic orbit as the enscribed triangle. Contrary to other recent experiments, this orbit is not accessible by classical processes for the incoming collimated beam.

Manifestations of classical chaos in the energy level spectrum of a quantum well.

The electronic energy levels and eigenfunctions of a wide potential well in a double barrier structure with a large tilted magnetic field are investigated in the regime of strong classical chaos. Periodic modulations of the level density are identified and related to distinct unstable closed classical orbits. Single periodic orbits ``scar'' the wave functions corresponding to regular subsets of energy levels determined by quantizing the classical action. Pronounced features in the tunneling rates into the well provide an experimentally accessible means of studying wave-function scarring.

Geometric and diffractive orbits in the scattering from confocal hyperbolas.

We study the scattering resonances between two confocal hyperbolas and show that the spectrum is dominated by the effect of a single periodic orbit. There are two distinct cases, depending on whether the orbit is geometric or diffractive. A generalization of periodic orbit theory allows us to incorporate the second possibility. In both cases, we also perform a Wentzel-Kramers-Brillouin (WKB) analysis. Although it is found that the semiclassical approximations work best for resonances with large energies and narrow widths, there is reasonable agreement even for resonances with large widths---unlike the two disk scatterer. We also find agreement with the next order correction to periodic orbit theory.

Hypothesis: the central oscillator of the circadian clock is a controlled chaotic attractor

Controlled chaos may be important for the generation of rhythmic behaviour in living systems. A model is proposed in which the central circadian oscillator is a chaotic attractor. Whereas a limit cycle mechanism (previously invoked to explain circadian clocks as well as ultradian clocks and cell division cycles) can provide only a single stable periodic orbit, a chaotic attractor can generate rich dynamic behaviour. Control by feedback makes accessible a selected stabilized orbit; this can be chosen so as to optimize system performance. Such a system can accommodate a wide variety of requirements, e.g. that a single clock mutation can affect both period and temperature compensation, and the generation of higher periods from an ultradian oscillator. Simultaneous operation of more than one clock (with differing periods) may require a high-dimension chaotic attractor. Attractive features of such a model include versatility of period selection (e.g. as in the per mutants of Drosophila) and the use of control elements of the type already well known in metabolic circuitry.

Periodic orbit theory of broad resonances in two-dimensional hydrogenic Stark effect

The modified version of periodic orbit theory, developed by Robbins (1989) for systems which possess discrete symmetries is used to calculate parameters of broad resonances in the case of the two-dimensional hydrogenic Stark effect at energies above the zero field ionization threshold. It is shown that the contribution of single periodic orbits of the system give results that coincide with those of the comparison equation method also presented.

Bifocal homoclinic orbits in four dimensions

The authors study the bifurcations which occur as they perturb four-dimensional systems of ordinary differential equations having homoclinic orbits that are bi-asymptotic to a fixed point with a double-focus structure. They give several methods of understanding the geometry of the invariant set that exists close to the homoclinic orbit and introduce a multi-valued one-dimensional map which can be used to predict the behaviour and bifurcation patterns which may occur. They argue that, although local strange behaviour is likely to occur, in a global sense (i.e. for large enough perturbations) the whole sequence of bifurcations produces a single periodic orbit, just as in the three-dimensional saddle-focus case.

Varieties of periodic attractor in cellular automata

We apply three alternate definitions of "attractor" to cellular automata. Examples are given to show that using the different definitions can give different answers to the question "Does this cellular automaton have a periodic attractor?" The three definitions are the topological notion of attractor as used by C. Conley, a more measure-theoretic version given by J. Milnor, and a variant of Milnor’s definition that is based on the concept of the "center of attraction" of an orbit. Restrictions on the types of periodic orbits that can be periodic attractors for cellular automata are described. With any of these definitions, a cellular automaton has at most one periodic attractor. Additionally, if Conley’s definition is used, then a periodic attractor must be a fixed point. Using Milnor’s definition, each point on a periodic attractor must be fixed by all shifts, so the number of symbols used is an upper bound on the period; whether the actual upper bound is 1 1 is unknown. With the third definition this restriction is removed, and examples are given of onedimensional cellular automata on three symbols that have finite "attractors" of arbitrarily large size (with the third definition, a finite attractor is not necessarily a single periodic orbit).

Classical dynamical analysis of the vibrational spectra for small polyatomic molecules

With the study of three specific molecules, Na3(C2E″), HCN(X1Σ+), and O3(D1B2), each showing a distinctly different dynamical behavior at the energies studied, we show how techniques of nonlinear classical mechanics can be applied to simulate low resolution vibrational spectra and to extract the underlying dynamics from them. Particularly the importance of the single unstable periodic orbits in the chaotic regions of phase space is demonstrated.

Bifurcations into pathology for Hamiltonian systems

This paper presents a geometric analysis of bifurcations leading to chaos for Hamiltonian systems with two degrees of freedom of the form x ̇ = y, y ̇ = −▽V(x) . Two bifurcation parameters are considered. One is the energy level and the other is an angle, Φ, between two homoclinic orbits. Though global non-linearities are necessary, the results are obtained by local analysis of the flow near the origin where it is assumed that D 2 V(0) = − I, the 2 × 2 identity matrix. For a fixed energy level it is shown that as Φ decreases through 90 ° the two homoclinic orbits bifurcate into two homoclinic orbits, a periodic orbit, and connecting orbits. These orbits can then be used to define a compact region in R 4. Now treating the energy as a parameter value the trajectory of orbits passing through this compact region can be described using symbolic dynamics. In this case it is shown that a single periodic orbit bifurcates into three periodic orbits whose stable and unstable manifolds intersect transversely.

Numerical experiments on planetary orbits in double stars

This is a numerical study of orbits in the elliptic restricted three-body problem concerning the dependence of the critical orbits on the eccentricity of the primaries. They are defined as being the separatrix between stable and unstable single periodic orbits. As our results are adapted to the existence of planetary orbits in double stars we concentrated first on the P-orbits (defined to surround both primaries). Due to the complexity of the elliptic problem there is no analytical approach possible. Using the results of some 300 integrated orbits for 103 to 3. 103 periods of the primaries we established lower and upper bounds for the critical orbits for different values of the eccentricity.

On the distribution of periodic points for β-shifts

It is shown that for β-shifts the periodic points are uniformly distributed with respect to the unique measure of maximal entropy, and that the invariant measures with support on a single periodic orbit are dense in the space of all invariant measures.

On Dynamical Systems With the Specification Property

A continuous transformation T of a compact metric space X satisfies the specification property if one can approximate distinct pieces of orbits by single periodic orbits with a certain uniformity. There are many examples of such transformations which have recently been studied in ergodic theory and statistical mechanics. This paper investigates the relation between Tinvariant measures and the frequencies of T-orbits. In particular, it is shown that every invariant measure (and even every closed connected subset of such measures) has generic points, but that the set of all generic points is of first category in X. This generalizes number theoretic results concerning decimal expansions and normal numbers.

On dynamical systems with the specification property

A continuous transformation T of a compact metric space X satisfies the specification property if one can approximate distinct pieces of orbits by single periodic orbits with a certain uniformity. There are many examples of such transformations which have recently been studied in ergodic theory and statistical mechanics. This paper investigates the relation between Tinvariant measures and the frequencies of T-orbits. In particular, it is shown that every invariant measure (and even every closed connected subset of such measures) has generic points, but that the set of all generic points is of first category in X. This generalizes number theoretic results concerning decimal expansions and normal numbers.