Calculation of superdiffusion for the Chirikov-Taylor model

Records and Occupation Time Statistics for Area-Preserving Maps

A relevant problem in dynamics is to characterize how deterministic systems may exhibit features typically associated with stochastic processes. A widely studied example is the study of (normal or anomalous) transport properties for deterministic systems on non-compact phase space. We consider here two examples of area-preserving maps: the Chirikov-Taylor standard map and the Casati-Prosen triangle map, and we investigate transport properties, records statistics, and occupation time statistics. Our results confirm and expand known results for the standard map: when a chaotic sea is present, transport is diffusive, and records statistics and the fraction of occupation time in the positive half-axis reproduce the laws for simple symmetric random walks. In the case of the triangle map, we retrieve the previously observed anomalous transport, and we show that records statistics exhibit similar anomalies. When we investigate occupation time statistics and persistence probabilities, our numerical experiments are compatible with a generalized arcsine law and transient behavior of the dynamics.

Symplectic Gaussian process regression of maps in Hamiltonian systems

We present an approach to construct structure-preserving emulators for Hamiltonian flow maps and Poincaré maps based directly on orbit data. Intended applications are in moderate-dimensional systems, in particular, long-term tracing of fast charged particles in accelerators and magnetic plasma confinement configurations. The method is based on multi-output Gaussian process (GP) regression on scattered training data. To obtain long-term stability, the symplectic property is enforced via the choice of the matrix-valued covariance function. Based on earlier work on spline interpolation, we observe derivatives of the generating function of a canonical transformation. A product kernel produces an accurate implicit method, whereas a sum kernel results in a fast explicit method from this approach. Both are related to symplectic Euler methods in terms of numerical integration but fulfill a complementary purpose. The developed methods are first tested on the pendulum and the Hénon-Heiles system and results compared to spectral regression of the flow map with orthogonal polynomials. Chaotic behavior is studied on the standard map. Finally, the application to magnetic field line tracing in a perturbed tokamak configuration is demonstrated. As an additional feature, in the limit of small mapping times, the Hamiltonian function can be identified with a part of the generating function and thereby learned from observed time-series data of the system's evolution. For implicit GP methods, we demonstrate regression performance comparable to spectral bases and artificial neural networks for symplectic flow maps, applicability to Poincaré maps, and correct representation of chaotic diffusion as well as a substantial increase in performance for learning the Hamiltonian function compared to existing approaches.

Diffusion phenomena in a mixed phase space

We show that, in strongly chaotic dynamical systems, the average particle velocity can be calculated analytically by consideration of Brownian dynamics in a phase space, the method of images, and the use of the classical diffusion equation. The method is demonstrated on the simplified Fermi-Ulam accelerator model, which has a mixed phase space with chaotic seas, invariant tori, and Kolmogorov-Arnold-Moser islands. The calculated average velocities agree well with numerical simulations and with an earlier empirical theory.

Global and local diffusion in the standard map

We study the global and the local transport and diffusion in the case of the standard map, by calculating the diffusion exponent μ. In the global case, we find that the mean diffusion exponent for the whole phase space is either μ=1, denoting normal diffusion, or μ=2 denoting anomalous diffusion (and ballistic motion). The mean diffusion of the whole phase space is normal when no accelerator mode exists and it is anomalous (ballistic) when accelerator mode islands exist even if their area is tiny in the phase space. The local value of the diffusion exponent inside the normal islands of stability is μ=0, while inside the accelerator mode islands it is μ=2. The local value of the diffusion exponent in the chaotic region outside the islands of stability converges always to the value of 1. The time of convergence can be very long, depending on the distance from the accelerator mode islands and the value of the nonlinearity parameter K. For some values of K, the stickiness around the accelerator mode islands is maximum and initial conditions inside the sticky region can be dragged in a ballistic motion for extremely long times of the order of 10^{7} or more but they will finally end up in normal mode diffusion with μ=1. We study, in particular, cases with maximum stickiness and cases where normal and accelerator mode islands coexist. We find general analytical solutions of periodic orbits of accelerator type and we give evidence that they are much more numerous than the normal periodic orbits. Thus, we expect that in every small interval ΔK of the nonlinearity parameter K of the standard map there exist smaller intervals of accelerator mode islands. However, these smaller intervals are in general very small, so that in the majority of the values of K the global diffusion is normal.

Diffusion for ensembles of standard maps

Two types of random evolution processes are studied for ensembles of the standard map with driving parameter K that determines its degree of stochasticity. For one type of process the parameter K is chosen at random from a Gaussian distribution and is then kept fixed, while for the other type it varies from step to step. In addition, noise that can be arbitrarily weak is added. The ensemble average and the average over noise of the diffusion coefficient are calculated for both types of processes. These two types of processes are relevant for two types of experimental situations as explained in the paper. Both types of processes destroy fine details of the dynamics, and the second process is found to be more effective in destroying the fine details. We hope that this work is a step in the efforts for developing a statistical theory for systems with mixed phase space (regular in some parts and chaotic in other parts).

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.

Mechanism for stickiness suppression during extreme events in Hamiltonian systems

In this paper we study how hyperbolic and nonhyperbolic regions in the neighborhood of a resonant island perform an important role allowing or forbidding stickiness phenomenon around islands in conservative systems. The vicinity of the island is composed of nonhyperbolic areas that almost prevent the trajectory to visit the island edge. For some specific parameters tiny channels are embedded in the nonhyperbolic area that are associated to hyperbolic fixed points localized in the neighborhood of the islands. Such channels allow the trajectory to be injected in the inner portion of the vicinity. When the trajectory crosses the barrier imposed by the nonhyperbolic regions, it spends a long time abandoning the vicinity of the island, since the barrier also prevents the trajectory from escaping from the neighborhood of the island. In this scenario the nonhyperbolic structures are responsible for the stickiness phenomena and, more than that, the strength of the sticky effect. We show that those properties of the phase space allow us to manipulate the existence of extreme events (and the transport associated to it) responsible for the nonequilibrium fluctuation of the system. In fact we demonstrate that by monitoring very small portions of the phase space (namely, ≈1×10(-5)% of it) it is possible to generate a completely diffusive system eliminating long-time recurrences that result from the stickiness phenomenon.

Number of first-passage times as a measurement of information for weakly chaotic systems

We consider a general class of maps of the interval having Lyapunov subexponential instability |δxt|∼|δx0|exp[Λt(x0)ζ(t)], where ζ(t) grows sublinearly as t→∞. We outline here a scheme [J. Stat. Phys. 154, 988 (2014)] whereby the choice of a characteristic function automatically defines the map equation and corresponding growth rate ζ(t). This matching approach is based on the infinite measure property of such systems. We show that the average information that is necessary to record without ambiguity a trajectory of the system tends to 〈Λ〉ζ(t), suitably extending the Kolmogorov-Sinai entropy and Pesin's identity. For such systems, information behaves like a random variable for random initial conditions, its statistics obeying a universal Mittag-Leffler law. We show that, for individual trajectories, information can be accurately inferred by the number of first-passage times through a given turbulent phase-space cell. This enables us to calculate far more efficiently Lyapunov exponents for such systems. Lastly, we also show that the usual renewal description of jumps to the turbulent cell, usually employed in the literature, does not provide the real number of entrances there. Our results are supported by exhaustive numerical simulations.

Survey on the role of accelerator modes for anomalous diffusion: the case of the standard map

We perform an extensive and detailed analysis of the generalized diffusion processes in deterministic area preserving maps with noncompact phase space, exemplified by the standard map, with the special emphasis on understanding the anomalous diffusion arising due to the accelerator modes. The accelerator modes and their immediate neighborhood undergo ballistic transport in phase space, and also the greater vicinity of them is still much affected ("dragged") by them, giving rise to the non-Gaussian (accelerated) diffusion. The systematic approach rests upon the following applications: the GALI method to detect the regular and chaotic regions and thus to describe in detail the structure of the phase space, the description of the momentum distribution in terms of the Lévy stable distributions, and the numerical calculation of the diffusion exponent and of the corresponding diffusion constant. We use this approach to analyze in detail and systematically the standard map at all values of the kick parameter K, up to K = 70. All complex features of the anomalous diffusion are well understood in terms of the role of the accelerator modes, mainly of period 1 at large K ≥ 2π, but also of higher periods (2,3,4,...) at smaller values of K ≤ 2π.

Scaling investigation for the dynamics of charged particles in an electric field accelerator

Some dynamical properties of an ensemble of trajectories of individual (non-interacting) classical particles of mass m and charge q interacting with a time-dependent electric field and suffering the action of a constant magnetic field are studied. Depending on both the amplitude of oscillation of the electric field and the intensity of the magnetic field, the phase space of the model can either exhibit: (i) regular behavior or (ii) a mixed structure, with periodic islands of regular motion, chaotic seas characterized by positive Lyapunov exponents, and invariant Kolmogorov-Arnold-Moser curves preventing the particle to reach unbounded energy. We define an escape window in the chaotic sea and study the transport properties for chaotic orbits along the phase space by the use of scaling formalism. Our results show that the escape distribution and the survival probability obey homogeneous functions characterized by critical exponents and present universal behavior under appropriate scaling transformations. We show the survival probability decays exponentially for small iterations changing to a slower power law decay for large time, therefore, characterizing clearly the effects of stickiness of the islands and invariant tori. For the range of parameters used, our results show that the crossover from fast to slow decay obeys a power law and the behavior of survival orbits is scaling invariant.

Anomalous transport induced by nonhyperbolicity

In this paper we study how deterministic features presented by a system can be used to perform direct transport in a quasisymmetric potential and weak dissipative system. We show that the presence of nonhyperbolic regions around acceleration areas of the phase space plays an important role in the acceleration of particles giving rise to direct transport in the system. Such an effect can be observed for a large interval of the weak asymmetric potential parameter allowing the possibility to obtain useful work from unbiased nonequilibrium fluctuation in real systems even in a presence of a quasisymmetric potential.

Distributional response to biases in deterministic superdiffusion

We report on a novel response to biases in deterministic superdiffusion. For its reduced map, we show using infinite ergodic theory that the time-averaged velocity (TAV) is intrinsically random and its distribution obeys the generalized arcsine distribution. A distributional limit theorem indicates that the TAV response to a bias appears in the distribution, which is an example of what we term a distributional response induced by a bias. Although this response in single trajectories is intrinsically random, the ensemble-averaged TAV response is linear.

Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets

We present a computational study of a visualization method for invariant sets based on ergodic partition theory, first proposed by Mezić (Ph.D. thesis, Caltech, 1994) and Mezić and Wiggins [Chaos 9, 213 (1999)]. The algorithms for computation of the time averages of observables on phase space are developed and used to provide an approximation of the ergodic partition of the phase space. We term the graphical representation of this approximation--based on time averages of observables--a mesochronic plot (from Greek: meso--mean, chronos--time). The method is useful for identifying low-dimensional projections (e.g., two-dimensional slices) of invariant structures in phase spaces of dimensionality bigger than two. We also introduce the concept of the ergodic quotient space, obtained by assigning a point to every ergodic set, and provide an embedding method whose graphical representation we call the mesochronic scatter plot. We use the Chirikov standard map as a well-known and dynamically rich example in order to illustrate the implementation of our methods. In addition, we expose applications to other higher dimensional maps such as the Froéschle map for which we utilize our methods to analyze merging of resonances and, the three-dimensional extended standard map for which we study the conjecture on its ergodicity [I. Mezić, Physica D 154, 51 (2001)]. We extend the study in our next paper [Z. Levnajić and I. Mezić, e-print arXiv:0808.2182] by investigating the visualization of periodic sets using harmonic time averages. Both of these methods are related to eigenspace structure of the Koopman operator [I. Mezić and A. Banaszuk, Physica D 197, 101 (2004)].

Chaotic Hamiltonian systems: survival probability

We consider the dynamical system described by the area-preserving standard mapping. It is known for this system that P(t), the normalized number of recurrences staying in some given domain of the phase space at time t (so-called "survival probability") has the power-law asymptotics, P(t) approximately t{-nu}. We present new semiphenomenological arguments which enable us to map the dynamical system near the chaos border onto the effective "ultrametric diffusion" on the boundary of a treelike space with hierarchically organized transition rates. In the framework of our approach we have estimated the exponent nu as nu=ln 2/ln(1+r{g}) approximately 1.44, where rg=([square root] 5-1)/2 is the critical rotation number.

Finding critical exponents for two-dimensional Hamiltonian maps

The transition from integrability to nonintegrability for a set of two-dimensional Hamiltonian mappings exhibiting mixed phase space is considered. The phase space of such mappings show a large chaotic sea surrounding Kolmogorov-Arnold-Moser islands and limited by a set of invariant tori. The description of the phase transition is made by the use of scaling functions for average quantities of the mapping averaged along the chaotic sea. The critical exponents are obtained via extensive numerical simulations. Given the mappings considered are parametrized by an exponent gamma in one of the dynamical variables, the critical exponents that characterize the scaling functions are obtained for many different values of gamma . Therefore classes of universality are defined.

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.