We study returns in dynamical systems: when a set of points, initially populating a prescribed region, swarms around phase space according to a deterministic rule of motion, we say that the return of the set occurs at the earliest moment when one of these points comes back to the original region. We describe the statistical distribution of these "first--return times" in various settings: when phase space is composed of sequences of symbols from a finite alphabet (with application for instance to biological problems) and when phase space is a one and a two-dimensional manifold. Specifically, we consider Bernoulli shifts, expanding maps of the interval and linear automorphisms of the two dimensional torus. We derive relations linking these statistics with Renyi entropies and Lyapunov exponents.
We present a study of the chaotic behavior of the bouncing ball billiard. The work is realised on the purpose of finding at least certain causes of separation of the neighbouring trajectories. Having in view the geometrical construction of the system, we report a clear origin of chaoticity of the bouncing ball billiard. By this we claim that in case when the floor is made of arc of circles - in a certain interval of frequencies - a lower bound for the maximal Ljapunov can be evaluated by sem-ianalitical techniques.
We study dynamics of a ball moving in gravitational field and colliding with a moving table. The motion of the limiter is assumed as periodic with piecewise constant velocity - it is assumed that the table moves up with a constant velocity and then moves down with another constant velocity. The Poincare map, describing evolution from an impact to the next impact, is derived and scenarios of transition to chaotic dynamics are investigated analytically.
In this paper we analyze a recent experiment conducted in an anechoic chamber, where the scattering of microwaves from an array of metallic cylinders was measured. This is a system which displays chaotic scattering in the short wave limit. The analysis of the experimental data is aimed at elucidating the effects of the underlying chaos. We describe a robust numerical method which provides the scattering matrix for any number of non overlapping reflecting cylinders. We use this method to calculate the scattering from one, two and three cylinders, and to compare the results in the numerical simu…
We study the coupling of bouncing-ball modes to chaotic modes in two-dimensional billiards with two parallel boundary segments. Analytically, we predict the corresponding decay rates using the fictitious integrable system approach. Agreement with numerically determined rates is found for the stadium and the cosine billiard. We use this result to predict the asymptotic behavior of the counting function N_bb(E) ~ E^δ. For the stadium billiard we find agreement with the previous result δ= 3/4. For the cosine billiard we derive δ= 5/8, which is confirmed numerically and is well below the previousl…
We investigate the nonequilibrium dynamics of classical particles in a driven quasiperiodic lattice based on the Fibonacci sequence. An intricate transient dynamics of extraordinarily long ballistic flights at distinct velocities is found. We argue how these transients are caused and can be under- stood by a hierarchy of block decompositions of the quasiperiodic lattice. A comparison to the cases of periodic and fully randomized lattices is performed.
Transport properties of open chaotic ballistic systems and their statistics can be expressed in terms of the scattering matrix connecting incoming and outgoing wavefunctions. Here we calculate the dependence of correlation functions of arbitrarily many pairs of scattering matrices at different energies on the Ehrenfest time using trajectory based semiclassical methods. This enables us to verify the prediction from effective random matrix theory that one part of the correlation function obtains an exponential damping depending on the Ehrenfest time, while also allowing us to obtain the addition…
We study the classical and quantum mechanics of a three-dimensional stadium billiard. It consists of two quarter cylinders that are rotated with respect to each other by 90 degrees, and it is classically chaotic. The billiard exhibits only a few families of nongeneric periodic orbits. We introduce an analytic method for their treatment. The length spectrum can be understood in terms of the nongeneric and unstable periodic orbits. For unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory. For equal radii the billiard exhibits an additi…
Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands appear due to homoclinic structures created from unstable 2 - cycles in a corner-type bifurcation.
We analyze the decay of classically chaotic quantum systems in the presence of fast ballistic escape routes on the Ehrenfest time scale. For a continuous excitation process, the form factor of the decay cross section deviates from the universal random-matrix result on the Heisenberg time scale, i.e. for times much larger than the time for ballistic escape. We derive an exact analytical description and compare our results with numerical simulations for a dynamical model.
The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits. I show that this belief is erroneous. The Gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits. Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of t…
Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare map, describing evolution from an impact to the next impact, is described. Displacement of the limiter is assumed as periodic, cubic function of time. Due to simplicity of this function analytical computations are possible. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands are created from fixed points after first period doubling in a corner-type bifu…
A powerful tool is developed for the characterization of chaotic signals. The approach is based on the symbolic encoding of time series (according to their ordinal patterns) combined with the ensuing characterization of the corresponding cylinder sets. Quantitative estimates of the Kolmogorov-Sinai entropy are obtained by introducing a modified permutation entropy which takes into account the average width of the cylinder sets. The method works also in hyperchaotic systems and allows estimating the fractal dimension of the underlying attractors.
We consider transport properties of the chaotic (strange) attractor along unfolded trajectories of the dissipative standard map. It is shown that the diffusion process is normal except of the cases when a control parameter is close to some special values that correspond to the ballistic mode dynamics. Diffusion near the related crisises is anomalous and non-uniform in time: there are large time intervals during which the transport is normal or ballistic, or even superballistic. The anomalous superdiffusion seems to be caused by stickiness of trajectories to a non-chaotic and nowhere dense inva…
The crossing of a transition state in a multidimensional reactive system is mediated by invariant geometric objects in phase space: An invariant hyper-sphere that represents the transition state itself and invariant hyper-cylinders that channel the system towards and away from the transition state. The existence of these structures can only be guaranteed if the invariant hyper-sphere is normally hyperbolic, i.e., the dynamics within the transition state is not too strongly chaotic. We study the dynamics within the transition state for the hydrogen exchange reaction in three degrees of freedom.…
When coupling a superconductor to a normal conducting region the physical properties of the system are highly affected by the superconductor. We will investigate the effect of one or two superconductors on the conductance of a ballistic chaotic quantum dot to leading order in the total channel number using trajectory based semiclassics. The results show that the effect of one superconductor on the conductance is of the order of the number of channels and that the sign of the correction from the Drude conductance depends on the particular ratios of the numbers of channels of the superconducting…
In this letter we consider the phase diffusion of a harmonically driven undamped pendulum and show that it is anomalous in the strong sense. The role played by the fractal properties of the phase space is highlighted, providing an illustration of the link between deterministic chaos and anomalous transport. Finally, we build a stochastic model which reproduces most properties of the original Hamiltonian system by alternating ballistic flights and random diffusion.
It is shown that conductance fluctuations due to phase coherent ballistic transport through a chaotic cavity generically are fractals. The graph of conductance vs. externally changed parameter, e.g. magnetic field, is a fractal with dimension D=2-b/2 between 1 and 2. It is governed by the exponent b (<2) of the power law distribution P(t) ~ t^{-b} for a classically chaotic trajectory to stay in the cavity up to time t, which is typical for chaotic systems with a mixed (chaotic and regular) phase space. The phenomenon should be observable in semiconductor nanostructures and microwave billiards.
We study the statistics of wave functions in a ballistic chaotic system. The statistical ensemble is generated by adding weak smooth disorder. The conjecture of Gaussian fluctuations of wave functions put forward by Berry and generalized by Hortikar and Srednicki is proven to hold on sufficiently short distances, while it is found to be strongly violated on larger scales. This also resolves the conflict between the above conjecture and the wave function normalization. The method is further used to study ballistic correlations of wave functions in a random magnetic field.
States supported by chaotic open quantum systems fall into two categories: a majority showing instantaneous ballistic decay, and a set of quantum resonances of classically vanishing support in phase space. We present a theory describing these structures within a unified semiclassical framework. Emphasis is put on the quantum diffraction mechanism which introduces an element of probability and is crucial for the formation of resonances. Our main result are boundary conditions on the semiclassical propagation along system trajectories. Depending on whether the trajectory propagation time is shor…
We explore the influence of external perturbations on the energy levels of a Hamiltonian drawn at random from the Gaussian unitary distribution of Hermitian matrices. By deriving the joint distribution function of eigenvalues, we obtain the (n,m)-point parametric correlation function of the initial and final density of states for perturbations of arbitrary rank and strength. A further generalization of these results allows for the incorporation of short-range spatial correlations in diffusive as well as ballistic chaotic structures.
We investigate the two-point correlations in the band spectra of spatially periodic systems that exhibit chaotic diffusion in the classical limit. By including level pairs pertaining to non-identical quasimomenta, we define form factors with the winding number as a spatial argument. For times smaller than the Heisenberg time, they are related to the full space-time dependence of the classical diffusion propagator. They approach constant asymptotes via a regime, reflecting quantal ballistic motion, where they decay by a factor proportional to the number of unit cells. We derive a universal scal…
We study transport through a two-dimensional billiard attached to two infinite leads by numerically calculating the Landauer conductance and the Wigner time delay. In the generic case of a mixed phase space we find a power law distribution of resonance widths and a power law dependence of conductance increments apparently reflecting the classical dwell time exponent, in striking difference to the case of a fully chaotic phase space. Surprisingly, these power laws appear on energy scales {\em below} the mean level spacing, in contrast to semiclassical expectations.
Both in atomic physics and in mesoscopic physics it is sometimes interesting to consider the energy time-dependence of a parametrically-driven chaotic system. We assume an Hamiltonian ${\cal H}(Q,P;x(t))$ where $x(t)=Vt$. The velocity $V$ is slow in the classical sense but not necessarily in the quantum-mechanical sense. The crossover (in time) from ballistic to diffusive energy-spreading is studied. The associated irreversible growth of the average energy has the meaning of dissipation. It is found that a dimensionless velocity $v_{PR}$ determines the nature of the dynamics, and controls the …
