Recurrence Time Statistics Research Articles

Multiple moving agents on complex networks: From intermittent synchronization to complete synchronization

We investigate multiple moving agents on complex networks. Each of them takes a random walk strategy and carries a chaotic oscillator. Remarkably, we find that with increasing the number of agents, a significant transition occurs from intermittent synchronization to complete synchronization. In particular, we observe that the distribution of laminar length presents a clearly power-law behavior in the intermittent synchronization stage. While reaching a complete synchronization state, correlation dimension and recurrence time statistics of synchronous orbits are in excellent agreement with that of their carrying chaotic system under consideration. Our work reveals that the number of moving agents has a profound effect on shaping synchronization behaviors.

Estimating entropy rate from censored symbolic time series: A test for time-irreversibility

In this work, we introduce a method for estimating the entropy rate and the entropy production rate from a finite symbolic time series. From the point of view of statistics, estimating entropy from a finite series can be interpreted as a problem of estimating parameters of a distribution with a censored or truncated sample. We use this point of view to give estimations of the entropy rate and the entropy production rate, assuming that they are parameters of a (limit) distribution. The last statement is actually a consequence of the fact that the distribution of estimations obtained from recurrence-time statistics satisfies the central limit theorem. We test our method using a time series coming from Markov chain models, discrete-time chaotic maps, and a real DNA sequence from the human genome.

Bifurcations, time-series analysis of observables, and network properties in a tripartite quantum system

In a tripartite system comprising a Λ-atom interacting with two radiation fields in the presence of field nonlinearities and an intensity-dependent field-atom coupling, striking features have been shown to occur in the dynamics of the mean photon number 〈Ni(t)〉 (i=1,2) corresponding to either field. In this Letter, we carry out a detailed time-series analysis and establish an interesting correlation between the short-time and long-time dynamics of 〈Ni(t)〉. Lyapunov exponents, return maps, recurrence plots, recurrence-time statistics, as well as the clustering coefficient and the transitivity of networks constructed from the time series, are studied as functions of the intensity parameter κ. These are shown to carry signatures of a special value κ=κ‾. Our work also exhibits how techniques from nonlinear dynamics help analyze the behavior of observables in multipartite quantum systems.

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.

Recurrence time correlations in random walks with preferential relocation to visited places.

Random walks with memory usually involve rules where a preference for either revisiting or avoiding those sites visited in the past are introduced somehow. Such effects have a direct consequence on the transport properties as well as on the statistics of first-passage and subsequent recurrence times through a site. A preference for revisiting sites is thus expected to result in a positive correlation between consecutive recurrence times. Here we derive a continuous-time generalization of the random walk model with preferential relocation to visited sites proposed in Phys. Rev. Lett. 112, 240601 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.240601 to explore this effect, together with the main transport properties induced by the long-range memory. Despite the long-range memory effects governing the process, our analytical treatment allows us to (i) observe the existence of an asymptotic logarithmic (ultraslow) growth for the mean square displacement, in accordance to the results found for the original discrete-time model, and (ii) confirm the existence of positive correlations between first-passage and subsequent recurrence times. This analysis is completed with a comprehensive numerical study which reveals, among other results, that these correlations between first-passage and recurrence times also exhibit clear signatures of this ultraslow relaxation dynamics.

Collapse of hierarchical phase space and mixing rates in Hamiltonian systems

We employ statistical properties of Poincaré recurrences to investigate dynamical behaviors of coupled Hamiltonian maps with a mixed phase space, where sticking to regular (or resonant) islands degrades chaotic properties of the system. In particular we investigate two and three coupled Chirikov–Taylor standard mappings (SM) and choose nonlinearity parameters corresponding to a mixed phase space. Our first set of findings are: (i) the collapse of hierarchical phase space depends of the coupling intensity, (ii) the penetration of resonance islands generates a trapping regime responsible for a long intermediate algebraic decay which is suppressed as coupling increases; and (iii) the projection of recurrence-time statistics (RTS) onto two-dimensional planes of the high-dimensional phase space of coupled maps is useful to understand different dynamical features, when coupling strengths are varied. We also estimate the asymptotic polynomial decay exponent of RTS γ for both cases, and elaborate an indirect way to estimate the decay exponent of correlations χ by using large deviations theory applied to the probability distributions of finite-time largest Lyapunov exponents (FTLLEs). In the asymptotic regime both methods yield γ∼1.20 (2 SM) and γ∼1.10 (3 SM): the corresponding power-law exponents for time correlation decay χ is given by the known relationship χ=γ−1. As higher-dimensional Hamiltonian systems (with mixed phase space) share the crucial property that resonance islands are no more forbidden domains in phase space, the results obtained here for two and three coupled maps should be extended to even higher-dimensional systems.

Using rotation number to detect sticky orbits in Hamiltonian systems.

In Hamiltonian systems, depending on the control parameter, orbits can stay for very long times around islands, the so-called stickiness effect caused by a temporary trapping mechanism. Different methods have been used to identify sticky orbits, such as recurrence analysis, recurrence time statistics, and finite-time Lyapunov exponent. However, these methods require a large number of map iterations and to know the island positions in the phase space. Here, we show how to use the small divergence of bursts in the rotation number calculation as a tool to identify stickiness without knowing the island positions. This new procedure is applied to the standard map, a map that has been used to describe the dynamic behavior of several nonlinear systems. Moreover, our procedure uses a small number of map iterations and is proper to identify the presence of stickiness phenomenon for different values of the control parameter.

On the stochastic parametrization of short‐scale processes

Stochastic parametrization of short‐scale processes is revisited in an idealized setting in which the short scales are manifested as small‐amplitude deterministic forcings acting on the evolution laws of the resolved variables. Conditions under which the invariant probability density of the original deterministic system tends to that induced by the action of appropriately parametrized Gaussian white noise are identified. On the other hand, substantial differences are revealed between the deterministic description and its stochastic substitute when time‐dependent properties are considered such as short‐time behaviour of the variance, recurrence and first passage time statistics. Illustrations and explicit evaluations are provided on low‐order model systems.

Exploring conservative islands using correlated and uncorrelated noise.

In this work, noise is used to analyze the penetration of regular islands in conservative dynamical systems. For this purpose we use the standard map choosing nonlinearity parameters for which a mixed phase space is present. The random variable which simulates noise assumes three distributions, namely equally distributed, normal or Gaussian, and power law (obtained from the same standard map but for other parameters). To investigate the penetration process and explore distinct dynamical behaviors which may occur, we use recurrence time statistics (RTS), Lyapunov exponents and the occupation rate of the phase space. Our main findings are as follows: (i) the standard deviations of the distributions are the most relevant quantity to induce the penetration; (ii) the penetration of islands induce power-law decays in the RTS as a consequence of enhanced trapping; (iii) for the power-law correlated noise an algebraic decay of the RTS is observed, even though sticky motion is absent; and (iv) although strong noise intensities induce an ergodic-like behavior with exponential decays of RTS, the largest Lyapunov exponent is reminiscent of the regular islands.

Computing two dimensional Poincaré maps for hyperchaotic dynamics

Poincaré map (PM) is one of the felicitous discrete approximation of the continuous dynamics. To compute PM, the discrete relation(s) between the successive point of interactions of the trajectories on the suitable Poincaré section (PS) are found out. These discrete relations act as an amanuensis of the nature of the continuous dynamics. In this article, we propose a computational scheme to find a hyperchaotic PM (HPM) from an equivalent three dimensional (3D) subsystem of a 4D (or higher) hyperchaotic model. For the experimental purpose, a standard four dimensional (4D) hyperchaotic Lorenz-Stenflo system (HLSS) and a five dimensional (5D) hyperchaotic laser model (HLM) is considered. Equivalent 3D subsystem is obtained by comparing the movements of the trajectories of the original hyperchaotic systems with all of their 3D subsystems. The quantitative measurement of this comparison is made promising by recurrence quantification analysis (RQA). Various two dimensional (2D) Poincaré mas are computed for several suitable Poincaré sections for both the systems. But, only some of them are hyperchaotic in nature. The hyperchaotic behavior is verified by positive values of both one dimensional (1D) Lyapunov Exponent (LE-I) and 2D Lyapunov Exponent (LE-II). At the end, similarity of the dynamics between the hyperchaotic systems and their 2D hyperchaotic Poincaré maps (HPM) has been established through mean recurrence time (MRT) statistics for both of 4D HLSS and 5D HLM and the best approximated discrete dynamics for both the hyperchaotic systems are found out.

Anomalous dynamics and the choice of Poincaré recurrence set.

We investigate the dependence of Poincaré recurrence-time statistics on the choice of recurrence set by sampling the dynamics of two- and four-dimensional Hamiltonian maps. We derive a method that allows us to visualize the direct relation between the shape of a recurrence set and the values of its return probability distribution in arbitrary phase-space dimensions. Such a procedure, which is shown to be quite effective in the detection of tiny regions of regular motion, allows us to explain why similar recurrence sets have very different distributions and how to modify them in order to enhance their return probabilities. Applied to data, this enables us to understand the coexistence of extremely long, transient powerlike decays whose anomalous exponent depends on the chosen recurrence set.

Fault detection via recurrence time statistics and one-class classification

Predictive maintenance has emerged as a fundamental practice to preserve production assets in many industrial environments. Of a wide set of approaches, vibration analysis is one of the most used for high-speed rotating machinery, especially when fault detection is to be automatic. Traditionally, this task has been studied as a classification problem using data extracted from the frequency domain. This approach, however, has two main limitations: (a) manufacture and mounting procedures can vary the vibration spectra of a machine, even when these share the same design; and (b) incipient fault signatures may be concealed in the frequency domain by noise and vibration from other parts of the system. For these reasons, the application of a classifier obtained for one machine to another machine is pointless, making early fault detection difficult. In this paper, a bearing fault detection problem is tackled using one-class classifiers and features extracted from vibration capture in the time domain using recurrence time statistics. We also describe a study of the behavior of the proposed method in real conditions. Our method shows high detection accuracy accompanied by a reduced number of false positives and negatives.

Quantitative recurrence statistics and convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems

For non-uniformly hyperbolic dynamical systems we consider the time series of maxima along typical orbits. Using ideas based upon quantitative recurrence time statistics we prove convergence of the maxima (under suitable normalization) to an extreme value distribution, and obtain estimates on the rate of convergence. We show that our results are applicable to a range of examples, and include new results for Lorenz maps, certain partially hyperbolic systems, and non-uniformly expanding systems with sub-exponential decay of correlations. For applications where analytic results are not readily available we show how to estimate the rate of convergence to an extreme value distribution based upon numerical information of the quantitative recurrence statistics. We envisage that such information will lead to more efficient statistical parameter estimation schemes based upon the block-maxima method.

Speed of convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems

Suppose (f, 𝒳, ν) is a dynamical system and ϕ : 𝒳 → ℝ is an observation with a unique maximum at a (generic) point in 𝒳. We consider the time series of successive maxima Mn(x) := max {ϕ(x),…,ϕ ◦ fn-1(x)}. Recent works have focused on the distributional convergence of such maxima (under suitable normalization) to an extreme value distribution. In this paper, for certain dynamical systems, we establish convergence rates to the limiting distribution. In contrast to the case of i.i.d. random variables, the convergence rates depend on the rate of mixing and the recurrence time statistics. For a range of applications, including uniformly expanding maps, quadratic maps, and intermittent maps, we establish corresponding convergence rates. We also establish convergence rates for certain hyperbolic systems such as Anosov systems, and discuss convergence rates for non-uniformly hyperbolic systems, such as Hénon maps.

Recurrence-time statistics in non-Hamiltonian volume-preserving maps and flows.

We analyze the recurrence-time statistics (RTS) in three-dimensional non-Hamiltonian volume-preserving systems (VPS): an extended standard map and a fluid model. The extended map is a standard map weakly coupled to an extra dimension which contains a deterministic regular, mixed (regular and chaotic), or chaotic motion. The extra dimension strongly enhances the trapping times inducing plateaus and distinct algebraic and exponential decays in the RTS plots. The combined analysis of the RTS with the classification of ordered and chaotic regimes and scaling properties allows us to describe the intricate way trajectories penetrate the previously impenetrable regular islands from the uncoupled case. Essentially the plateaus found in the RTS are related to trajectories that stay for long times inside trapping tubes, not allowing recurrences, and then penetrate diffusively the islands (from the uncoupled case) by a diffusive motion along such tubes in the extra dimension. All asymptotic exponential decays for the RTS are related to an ordered regime (quasiregular motion), and a mixing dynamics is conjectured for the model. These results are compared to the RTS of the standard map with dissipation or noise, showing the peculiarities obtained by using three-dimensional VPS. We also analyze the RTS for a fluid model and show remarkable similarities to the RTS in the extended standard map problem.

Extensive numerical investigations on the ergodic properties of two coupled Pomeau–Manneville maps

We present extensive numerical investigations on the ergodic properties of two identical Pomeau–Manneville maps interacting on the unit square through a diffusive linear coupling. The system exhibits anomalous statistics, as expected, but with strong deviations from the single intermittent map: Such differences are characterized by numerical experiments with densities which do not have singularities in the marginal fixed point, escape and Poincaré recurrence time statistics that share a power-law decay exponent modified by a clear dimensional scaling, while the rate of phase-space filling and the convergence of ensembles of Lyapunov exponents show a stretched instead of pure exponential behavior. In spite of the lack of rigorous results about this system, the dependence on both the intermittency and the coupling parameters appears to be smooth, paving the way for further analytical development. We remark that dynamical exponents appear to be independent of the (nonzero) coupling strength.

Poincaré recurrence statistics as an indicator of chaos synchronization.

The dynamics of the autonomous and non-autonomous Rössler system is studied using the Poincaré recurrence time statistics. It is shown that the probability distribution density of Poincaré recurrences represents a set of equidistant peaks with the distance that is equal to the oscillation period and the envelope obeys an exponential distribution. The dimension of the spatially uniform Rössler attractor is estimated using Poincaré recurrence times. The mean Poincaré recurrence time in the non-autonomous Rössler system is locked by the external frequency, and this enables us to detect the effect of phase-frequency synchronization.

Recurrence time distribution and temporal clustering properties of a cellular automaton modelling landslide events

Abstract. Reasonable prediction of landslide occurrences in a given area requires the choice of an appropriate probability distribution of recurrence time intervals. Although landslides are widespread and frequent in many parts of the world, complete databases of landslide occurrences over large periods are missing and often such natural disasters are treated as processes uncorrelated in time and, therefore, Poisson distributed. In this paper, we examine the recurrence time statistics of landslide events simulated by a cellular automaton model that reproduces well the actual frequency-size statistics of landslide catalogues. The complex time series are analysed by varying both the threshold above which the time between events is recorded and the values of the key model parameters. The synthetic recurrence time probability distribution is shown to be strongly dependent on the rate at which instability is approached, providing a smooth crossover from a power-law regime to a Weibull regime. Moreover, a Fano factor analysis shows a clear indication of different degrees of correlation in landslide time series. Such a finding supports, at least in part, a recent analysis performed for the first time of an historical landslide time series over a time window of fifty years.

A recurrence‐based technique for detecting genuine extremes in instrumental temperature records

In this paper, we analyze several instrumental records of temperatures at different locations by using new techniques originally developed for the analysis of extreme values of dynamical systems. We show that they have the same recurrence time statistics as a chaotic dynamical system perturbed with dynamical noise and by instrument errors. The technique provides a criterion to discriminate whether the recurrence of a certain temperature belongs to the normal variability or can be considered as a genuine extreme event with respect to a specific timescale fixed as parameter. The method gives a self‐consistent estimation of the convergence of the statistics of recurrences toward the theoretical extreme value laws.

Multifractality, stickiness, and recurrence-time statistics

We identify the fine structure of resonance islands and the stickiness in chaos through recurrence time statistics (RTS), which is based on the concept of Poincaré recurrences. The projection of recurrence time statistics onto the phase space does give relevant information on the hierarchical and microstructures of the chaotic beach around the islands of a near-integrable system, the annular billiard. These microstructures interfere in the effective transport of a particle in the phase space, which can be observed through RTS. This technique proves also to be a powerful tool to describe the homoclinic tangle of the manifolds within the chaotic sea.