Can simple renormalization theories describe the trapping of chaotic trajectories in mixed systems?

Stickiness and recurrence plots: An entropy-based approach

The stickiness effect is a fundamental feature of quasi-integrable Hamiltonian systems. We propose the use of an entropy-based measure of the recurrence plots (RPs), namely, the entropy of the distribution of the recurrence times (estimated from the RP), to characterize the dynamics of a typical quasi-integrable Hamiltonian system with coexisting regular and chaotic regions. We show that the recurrence time entropy (RTE) is positively correlated to the largest Lyapunov exponent, with a high correlation coefficient. We obtain a multi-modal distribution of the finite-time RTE and find that each mode corresponds to the motion around islands of different hierarchical levels.

Stickiness in generic low-dimensional Hamiltonian systems: A recurrence-time statistics approach

We analyze the structure and stickiness in the chaotic components of generic Hamiltonian systems with divided phase space. Following the method proposed recently in Lozej and Robnik [Phys. Rev. E 98, 022220 (2018)2470-004510.1103/PhysRevE.98.022220], the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid. We perform extensive numerical studies of three example systems: the Chirikov standard map, the family of Robnik billiards, and the family of lemon billiards. The filling of the cells is compared to the random model of chaotic diffusion, introduced in Robnik et al. [J. Phys. A: Math. Gen. 30, L803 (1997)JPHAC50305-447010.1088/0305-4470/30/23/003] for the description of transport in the phase spaces of ergodic systems. The model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential. In generic systems the stickiness induces correlations in the cell visits. The distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping. We model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times. We introduce the variable S, which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness. We use S to globally assess the distributions of recurrence times. We find that in the bulk of the chaotic sea S=1, while S>1 in areas of stickiness. We present the results in the form of animated grayscale plots of the variable S in the largest chaotic component for the three example systems, included as supplemental material to this paper.

Power-law trapping in the volume-preserving Arnold-Beltrami-Childress map

Understanding stickiness and power-law behavior of Poincaré recurrence statistics is an open problem for higher-dimensional systems, in contrast to the well-understood case of systems with two degrees of freedom. We study such intermittent behavior of chaotic orbits in three-dimensional volume-preserving maps using the example of the Arnold-Beltrami-Childress map. The map has a mixed phase space with a cylindrical regular region surrounded by a chaotic sea for the considered parameters. We observe a characteristic overall power-law decay of the cumulative Poincaré recurrence statistics with significant oscillations superimposed. This slow decay is caused by orbits which spend long times close to the surface of the regular region. Representing such long-trapped orbits in frequency space shows clear signatures of partial barriers and reveals that coupled resonances play an essential role. Using a small number of the most relevant resonances allows for classifying long-trapped orbits. From this the Poincaré recurrence statistics can be divided into different exponentially decaying contributions, which very accurately explains the overall power-law behavior including the oscillations.

Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories

The dynamics in three-dimensional (3D) billiards leads, using a Poincaré section, to a four-dimensional map, which is challenging to visualize. By means of the recently introduced 3D phase-space slices, an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincaré recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories, which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps, we find that (i) trapping takes place close to regular structures outside the Arnold web, (ii) trapping is not due to a generalized island-around-island hierarchy, and (iii) the dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.

Thirty years of turnstiles and transport

To characterize transport in a deterministic dynamical system is to compute exit time distributions from regions or transition time distributions between regions in phase space. This paper surveys the considerable progress on this problem over the past thirty years. Primary measures of transport for volume-preserving maps include the exiting and incoming fluxes to a region. For area-preserving maps, transport is impeded by curves formed from invariant manifolds that form partial barriers, e.g., stable and unstable manifolds bounding a resonance zone or cantori, the remnants of destroyed invariant tori. When the map is exact volume preserving, a Lagrangian differential form can be used to reduce the computation of fluxes to finding a difference between the actions of certain key orbits, such as homoclinic orbits to a saddle or to a cantorus. Given a partition of phase space into regions bounded by partial barriers, a Markov tree model of transport explains key observations, such as the algebraic decay of exit and recurrence distributions.

Statistics of the island-around-island hierarchy in Hamiltonian phase space

The phase space of a typical Hamiltonian system contains both chaotic and regular orbits, mixed in a complex, fractal pattern. One oft-studied phenomenon is the algebraic decay of correlations and recurrence time distributions. For area-preserving maps, this has been attributed to the stickiness of boundary circles, which separate chaotic and regular components. Though such dynamics has been extensively studied, a full understanding depends on many fine details that typically are beyond experimental and numerical resolution. This calls for a statistical approach, the subject of the present work. We calculate the statistics of the boundary circle winding numbers, contrasting the distribution of the elements of their continued fractions to that for uniformly selected irrationals. Since phase space transport is of great interest for dynamics, we compute the distributions of fluxes through island chains. Analytical fits show that the "level" and "class" distributions are distinct, and evidence for their universality is given.

Fluctuations in the relaxation dynamics of mixed chaotic systems

The relaxation dynamics in mixed chaotic systems are believed to decay algebraically with a universal decay exponent that emerges from the hierarchical structure of the phase space. Numerical studies, however, yield a variety of values for this exponent. In order to reconcile these results, we consider an ensemble of mixed chaotic systems approximated by rate equations and analyze the fluctuations in the distribution of Poincaré recurrence times. Our analysis shows that the behavior of these fluctuations, as a function of time, implies a very slow convergence of the decay exponent of the relaxation.

Fractal Weyl laws for quantum decay in dynamical systems with a mixed phase space

Weyl's law approximates the number of states in a quantum system by partitioning the energetically accessible phase-space volume into Planck cells. Here, we show that resonances in open quantum systems can follow a modified fractal Weyl law, even when their classical dynamics is not globally chaotic but also contains domains of regular motion. Using an appropriate phase-space representation for open quantum systems, we connect this behavior to emerging quantum-to-classical correspondence.

Accumulation of unstable periodic orbits and the stickiness in the two-dimensional piecewise linear map

We investigate the stickiness of the two-dimensional piecewise linear map with a family of marginal unstable periodic orbits (FMUPOs), and show that a series of unstable periodic orbits accumulating to FMUPOs plays a significant role to give rise to the power law correlation of trajectories. We can explicitly specify the sticky zone in which unstable periodic orbits whose stability increases algebraically exist, and find that there exists a hierarchy in accumulating periodic orbits. In particular, the periodic orbits with linearly increasing stability play the role of fundamental cycles as in the hyperbolic systems, which allows us to apply the method of cycle expansion. We also study the recurrence time distribution, especially discussing the position and size of the recurrence region. Following the definition adopted in one-dimensional maps, we show that the recurrence time distribution has an exponential part in the short time regime and an asymptotic power law part. The analysis on the crossover time T(c)(*) between these two regimes implies T(c)(*) approximately -log[micro(R)] where micro(R) denotes the area of the recurrence region.

Universality of algebraic laws in hamiltonian systems

Hamiltonian mixed systems with unbounded phase space are typically characterized by two asymptotic algebraic laws: decay of recurrence time statistics (gamma) and superdiffusion (beta). We conjecture the universal exponents gamma=beta=3/2 for trapping of trajectories to regular islands based on our analytical results for a wide class of area-preserving maps. For Hamiltonian mixed systems with a bounded phase space the interval 3/2< or =gamma_{b}< or =3 is obtained, given that trapping takes place. A number of simulations and experiments by other authors give additional support to our claims.

Poincaré recurrences and transient chaos in systems with leaks

In order to simulate observational and experimental situations, we consider a leak in the phase space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival probability by applying the theory of transient chaos. This expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an equivalence to the classical problem of Poincaré recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle into hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability.

Universality of algebraic decays in Hamiltonian systems

Hamiltonian systems with a mixed phase space typically exhibit an algebraic decay of correlations and of Poincaré recurrences, with numerical experiments over finite times showing system-dependent power-law exponents. We conjecture the existence of a universal asymptotic decay based on results for a Markov tree model with random scaling factors for the transition probabilities. Numerical simulations for different Hamiltonian systems support this conjecture and permit the determination of the universal exponent.

Poincaré recurrences from the perspective of transient chaos

We obtain a description of the Poincaré recurrences of chaotic systems in terms of the ergodic theory of transient chaos. It is based on the equivalence between the recurrence time distribution and an escape time distribution obtained by leaking the system and taking a special initial ensemble. This ensemble is atypical in terms of the natural measure of the leaked system, the conditionally invariant measure. Accordingly, for general initial ensembles, the average recurrence and escape times are different. However, we show that the decay rate of these distributions is always the same. Our results remain valid for Hamiltonian systems with mixed phase space and validate a split of the chaotic saddle in hyperbolic and nonhyperbolic components.

Multipeaked probability distributions of recurrence times

We determine probabilities of recurrence time into finite-sized, physically meaningful subsets of phase space. We consider three different autonomous chaotic systems: (i) scattering in a three-peaked potential, (ii) connected billiards, and (iii) Lorenz equations. We find multipeaked probability distributions, similar to the distributions found in (driven) stochastically resonant systems. In nondriven systems, such as ours, only monotonic decaying distributions (exponentials, stretched exponentials, power laws, and slight variations or combinations of these) have hitherto been reported. Discrete peaks in autonomous systems have as yet escaped attention in autonomous systems and correspond to specific trajectory subsets involving an integer number of loops.

Recurrence time distribution in mushroom billiards with parabolic hat

The recurrence time distribution of mushroom billiards with a parabolic-shaped hat is investigated. Classical dynamics exhibits sharply divided phase space, and the recurrence time distribution obeys the algebraic law like well-known classes of billiards. However, due to the existence of a specific type of marginally unstable periodic orbits that forms a crossing in phase space, the sticky motion occurs not as a simple drift along the straight line. Numerical experiments reveal and also theoretical analyses predict that an exponent for the cumulative recurrence time distribution approaches 2 in the asymptotic regime, but in a relatively small recurrence time scale it significantly deviates from the predicted universality, which is explained by the slowdown behavior around a crossing point of the periodic orbit family.

Survival probability time distribution in dielectric cavities

We study the survival probability time distribution (SPTD) in dielectric cavities. In a circular dielectric cavity the SPTD has an algebraic long time behavior, approximately t(-2) in both the TM and TE cases, but shows different short time behaviors due to the existence of the Brewster angle in the TE case where the short time behavior is exponential. The SPTD for a stadium-shaped cavity decays exponentially, and the exponent shows a relation of gamma approximately n(-2), n is the refractive index, and the proportional coefficient is obtained from a simple model of the steady probability distribution. We also discuss the SPTD for a quadrupolar deformed cavity and show that the long time behavior can be algebraic or exponential depending on the location of islands.

Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space

We investigate the dynamics of chaotic trajectories in simple yet physically important Hamiltonian systems with nonhierarchical borders between regular and chaotic regions with positive measures. We show that the stickiness to the border of the regular regions in systems with such a sharply divided phase space occurs through one-parameter families of marginally unstable periodic orbits and is characterized by an exponent gamma=2 for the asymptotic power-law decay of the distribution of recurrence times. Generic perturbations lead to systems with hierarchical phase space, where the stickiness is apparently enhanced due to the presence of infinitely many regular islands and Cantori. In this case, we show that the distribution of recurrence times can be composed of a sum of exponentials or a sum of power laws, depending on the relative contribution of the primary and secondary structures of the hierarchy. Numerical verification of our main results are provided for area-preserving maps, mushroom billiards, and the newly defined magnetic mushroom billiards.

Stickiness in mushroom billiards

We investigate the dynamical properties of chaotic trajectories in mushroom billiards. These billiards present a well-defined simple border between a single regular region and a single chaotic component. We find that the stickiness of chaotic trajectories near the border of the regular region occurs through an infinite number of marginally unstable periodic orbits. These orbits have zero measure, thus not affecting the ergodicity of the chaotic region. Notwithstanding, they govern the main dynamical properties of the system. In particular, we show that the marginally unstable periodic orbits explain the periodicity and the power-law behavior with exponent gamma=2 observed in the distribution of recurrence times.

Effective dynamics in Hamiltonian systems with mixed phase space

An adequate characterization of the dynamics of Hamiltonian systems at physically relevant scales has been largely lacking. Here we investigate this fundamental problem and we show that the finite-scale Hamiltonian dynamics is governed by effective dynamical invariants, which are significantly different from the dynamical invariants that describe the asymptotic Hamiltonian dynamics. The effective invariants depend both on the scale of resolution and the region of the phase space under consideration, and they are naturally interpreted within a framework in which the nonhyperbolic dynamics of the Hamiltonian system is modeled as a chain of hyperbolic systems.