Distribution Of Finite-time Lyapunov Exponents Research Articles

Border-collision bifurcation route to strange nonchaotic attractors in the piecewise linear normal form map

A new route to strange nonchaotic attractors (SNAs) is investigated in a quasiperiodically driven nonsmooth map. It is shown that the smooth quasiperiodic torus becomes nonsmooth (continuous and non-differentiable) due to the border-collision bifurcation of the torus. As the coefficients change, the nonsmooth torus gets gradually fractal, forming strange nonchaotic attractors. It is termed the border-collision bifurcation route to SNAs. A novel feature of this route is that a large number of SNAs are formed, and the parameter area of SNAs makes up about 40 % of the given parameter regions. These SNAs are identified by the largest Lyapunov exponents and the phase sensitivity exponents. They are also characterized by the distribution of finite-time Lyapunov exponents. Unlike other types of SNAs, the distribution of finite-time Lyapunov exponents has its maximum at a relatively small negative Lyapunov exponent, which contributes largely to lead to the abundance of SNAs.

Manifestation of strange nonchaotic attractors in extended systems: a study through out-of-time-ordered correlators

We study the spatial spread of out-of-time-ordered correlators (OTOCs) in coupled map lattices (CMLs) of quasiperiodically forced nonlinear maps. We use instantaneous speed (IS) and finite-time Lyapunov exponents (FTLEs) to investigate the role of strange non-chaotic attractors (SNAs) on the spatial spread of the OTOC. We find that these CMLs exhibit a characteristic on and off type of spread of the OTOC for SNA. Further, we provide a broad spectrum of the various dynamical regimes in a two-parameter phase diagram using IS and FTLEs. We substantiate our results by confirming the presence of SNA using established tools and measures, namely the distribution of finite-time Lyapunov exponents, phase sensitivity, spectrum of partial Fourier sums, and $0-1$ test.

Scarring in classical chaotic dynamics with noise.

We report the numerical observation of scarring, which is enhancement of probability density around unstable periodic orbits of a chaotic system, in the eigenfunctions of the classical Perron-Frobenius operator of noisy Anosov ("perturbed cat") maps, as well as in the noisy Bunimovich stadium. A parallel is drawn between classical and quantum scars, based on the unitarity or nonunitarity of the respective propagators. For uniformly hyperbolic systems such as the cat map, we provide a mechanistic explanation for the classical phase-space localization detected, based on the distribution of finite-time Lyapunov exponents, and the interplay of noise with deterministic dynamics. Classical scarring can be measured by studying autocorrelation functions and their power spectra.

INFINITE NUMBER OF PARAMETER REGIONS WITH FRACTAL NONCHAOTIC ATTRACTORS IN A PIECEWISE MAP

We identify a countable infinity of parameter regimes with strange nonchaotic attractors (SNAs). At the edge of each arc parameter area, there is an uncountable infinity of SNAs with torus intermittency. The mechanism for the creation of SNAs in different regime is induced by an [Formula: see text]-frequency quasiperiodic orbit through a quasiperiodic analog of saddle-node bifurcation (Type-[Formula: see text] intermittent route). We describe the transition between tori and SNAs by the largest Lyapunov exponent and phase diagram. These SNAs are characterized by the phase sensitivity exponents, rational approximations, singular-continuous spectra, and distribution of finite-time Lyapunov exponents.

Fractal catastrophes

We analyse the spatial inhomogeneities (‘spatial clustering’) in the distribution of particles accelerated by a force that changes randomly in space and time. To quantify spatial clustering, the phase-space dynamics of the particles must be projected to configuration space. Folds of a smooth phase-space manifold give rise to catastrophes (‘caustics’) in this projection. When the inertial particle dynamics is damped by friction, however, the phase-space manifold converges towards a fractal attractor. It is believed that caustics increase spatial clustering also in this case, but a quantitative theory is missing. We solve this problem by determining how projection affects the distribution of finite-time Lyapunov exponents (FTLEs). Applying our method in one spatial dimension we find that caustics arising from the projection of a dynamical fractal attractor (‘fractal catastrophes’) make a distinct and universal contribution to the distribution of spatial FTLEs. Our results explain a projection formula for the spatial fractal correlation dimension, and how a fluctuation relation for the distribution of FTLEs for white-in-time Gaussian force fields breaks upon projection. We explore the implications of our results for heavy particles in turbulence, and for wave propagation in random media.

Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations

It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A quasiperiodically forced piecewise Logistic system is shown to exhibit many types of routes to the creation of SNAs. We point out that the truncation of border-collision torus-doubling bifurcation can lead to different types of SNAs. We identify and describe the Heagy–Hammel routes, fractalization route and intermittent routes after the two coexisting tori collide at the border and the doubled torus is interrupted in this system. It has been shown that there exist two critical tongue-type regions in the parameter space, where the different mechanisms for the birth of SNAs are investigated. These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents. Different types of SNAs are also characterized by the singular-continuous spectrum, Fourier transform, rational approximations, distribution of finite-time Lyapunov exponents and recurrence analysis.

Dynamic bifurcation and strange nonchaos in a two-frequency parametrically driven nonlinear oscillator

A periodically forced series LCR circuit with Chua’s diode as a nonlinear element exhibits slow passage through Hopf bifurcation. This slow passage leads to a delay in the Hopf bifurcation. The delay in this bifurcation is a unique quantity and it can be predicted using various numerical analysis. We find that when an additional periodic force is added to the system, the delay in bifurcation becomes chaotic which leads to an unpredictability in bifurcation delay. Further, we study the bifurcation of the periodic delay to chaotic delay in the slow passage effect through strange nonchaotic delay. We also report the occurrence of strange nonchaotic dynamics while varying the parameter of the additional force included in the system. We observe that the system exhibits a hitherto unknown dynamical transition to a strange nonchaotic attractor. With the help of Lyapunov exponent, we explain the new transition to strange nonchaotic attractor and its mechanism is studied by making use of rational approximation theory. The birth of SNA has also been confirmed numerically, using Poincaré maps, phase sensitivity exponent, the distribution of finite-time Lyapunov exponents and singular continuous spectrum analysis.

Characteristic distribution of finite-time Lyapunov exponents for chimera states.

Our fascination with chimera states stems partially from the somewhat paradoxical, yet fundamental trait of identical, and identically coupled, oscillators to split into spatially separated, coherently and incoherently oscillating groups. While the list of systems for which various types of chimeras have already been detected continues to grow, there is a corresponding increase in the number of mathematical analyses aimed at elucidating the fundamental reasons for this surprising behaviour. Based on the model systems, there are strong indications that chimera states may generally be ubiquitous in naturally occurring systems containing large numbers of coupled oscillators – certain biological systems and high-Tc superconducting materials, for example. In this work we suggest a new way of detecting and characterising chimera states. Specifically, it is shown that the probability densities of finite-time Lyapunov exponents, corresponding to chimera states, have a definite characteristic shape. Such distributions could be used as signatures of chimera states, particularly in systems for which the phases of all the oscillators cannot be measured directly. For such cases, we suggest that chimera states could perhaps be detected by reconstructing the characteristic distribution via standard embedding techniques, thus making it possible to detect chimera states in systems where they could otherwise exist unnoticed.

Efficiency of Monte Carlo sampling in chaotic systems.

In this paper we investigate how the complexity of chaotic phase spaces affect the efficiency of importance sampling Monte Carlo simulations. We focus on flat-histogram simulations of the distribution of finite-time Lyapunov exponent in a simple chaotic system and obtain analytically that the computational effort: (i) scales polynomially with the finite time, a tremendous improvement over the exponential scaling obtained in uniform sampling simulations; and (ii) the polynomial scaling is suboptimal, a phenomenon known as critical slowing down. We show that critical slowing down appears because of the limited possibilities to issue a local proposal in the Monte Carlo procedure when it is applied to chaotic systems. These results show how generic properties of chaotic systems limit the efficiency of Monte Carlo simulations.

Smooth and filamental structures in chaotically advected chemical fields

This paper studies the spatial structure of decaying chemical fields generated by a chaotic-advection flow and maintained by a spatially smooth chemical source. Previous work showed that in a regime where diffusion can be neglected (large Péclet number), the structures are filamental or smooth depending on the relative strength of the chemical dynamics and the stirring induced by the flow. The scaling exponent, gamma(q), of the qth -order structure function depends, at leading order, linearly on the ratio of the rate of decay of the chemical processes, alpha , and the average rate of divergence of neighboring fluid parcel trajectories (Lyapunov exponent), h. Under a homogeneous stretching approximation, gamma(q)/q=max[alpha/h,1] which implies that a well-defined filamental-smooth transition occurs at alpha=h. This approximation has been improved by using the distribution of finite-time Lyapunov exponents to characterize the inhomogeneous stretching of the flow. However, previous work focused more on the behavior of the exponents as q varies and less on the effects of alpha and hence the implications for the filamental-smooth transition. Here we set out the precise relation between the stretching rate statistics and the scaling exponents and emphasize that the latter are determined by the distribution of the finite-size (rather than finite-time) Lyapunov exponents. We clarify the relation between the two distributions. We show that the corrected exponents, [symbol: see text] depend nonlinearly on alpha with [formula: see text]. The magnitude of the correction to the homogeneous stretching approximation, [formula: see text], grows as alpha increases, reaching a maximum when the leading-order transition is reached (alpha=h). The implication of these results is that there is no well-defined bulk filamental-smooth transition. Instead it is the case that the chemical field is unambiguously smooth for alpha>h(max), where h(max) denotes the maximum finite-time Lyapunov exponent and unambiguously filamental for alpha<h, with an intermediate character for alpha between these two values. Theoretical predictions are confirmed by numerical results obtained for a linearly decaying chemistry coupled to a renewing type of flow together with careful calculations of the Crámer function.

Transition to weak generalized synchrony in chaotically driven flows

We study regimes of strong and weak generalized synchronization in chaotically forced nonlinear flows. The transition between these dynamical states can occur via a number of different routes, and here we examine the onset of weak generalized synchrony through intermittency and blowout bifurcations. The quantitative characterization of this dynamical transition is facilitated by measures that have been developed for the study of strange nonchaotic motion. Weak and strong generalized synchronous motion show contrasting sensitivity to parametric variation and have distinct distributions of finite-time Lyapunov exponents.

Detecting dynamical boundaries from kinematic data in biomechanics

Ridges in the state space distribution of finite-time Lyapunov exponents can be used to locate dynamical boundaries. We describe a method for obtaining dynamical boundaries using only trajectories reconstructed from time series, expanding on the current approach which requires a vector field in the phase space. We analyze problems in musculoskeletal biomechanics, considered as exemplars of a class of experimental systems that contain separatrix features. Particular focus is given to postural control and balance, considering both models and experimental data. Our success in determining the boundary between recovery and failure in human balance activities suggests this approach will provide new robust stability measures, as well as measures of fall risk, that currently are not available and may have benefits for the analysis and prevention of low back pain and falls leading to injury, both of which affect a significant portion of the population.

EXPERIMENTAL REALIZATION OF STRANGE NONCHAOTIC ATTRACTORS IN A NONLINEAR SERIES LCR CIRCUIT WITH NONSINUSOIDAL FORCE

We have identified several prominent routes, namely, fractalization, fractalization followed by intermittency, intermittency and Heagy–Hammel routes, for the birth of strange nonchaotic attractors (SNAs) in a quasiperiodically forced electronic system with nonsinusoidal (square wave) force as one of the quasiperiodic forces [Senthilkumar et al., 2008]. In addition, a new bubbling route has also been identified in this circuit. Although some of these prominent routes have been reported experimentally [Thamilmaran et al., 2006] in a quasiperiodically forced electronic circuit with both the forcings as sinusoidal forces, experimental identification of all these routes is reported here in a quasiperiodically forced electronic circuit with one of the forcings as a nonsinusoidal (square wave) force. The birth of SNAs by these routes are characterized from both the experimental and numerical data by the maximal Lyapunov exponents and their variance, Poincaré maps, Fourier amplitude spectra, spectral distribution functions and the distribution of finite-time Lyapunov exponents.

Separatrices and basins of stability from time series data: an application to biodynamics

An approach is presented for identifying separatrices in phase space generated from noisy time series data sets representative of measured experimental data. These separatrices are identified as ridges in the phase space distribution of finite-time Lyapunov exponents, i.e., Lagrangian coherent structures (LCS). As opposed to previous approaches, the LCS is identified using only trajectories since no analytical or data-defined vector field is available. The method is applied to a biological simulation in which the separatrix reveals a basin of stability. These results suggest that the method will be a fruitful approach to time series analysis, particularly in cases where a limited number of trajectories are available as might be encountered in experiments.

Origin of chaos in soft interactions and signatures of nonergodicity

The emergence of chaotic motion is discussed for hard-point like and soft collisions between two particles in a one-dimensional box. It is known that ergodicity may be obtained in hard-point like collisions for specific mass ratios gamma=m(2)/m(1) of the two particles and that Lyapunov exponents are zero. However, if a Yukawa interaction between the particles is introduced, we show analytically that positive Lyapunov exponents are generated due to double collisions close to the walls. While the largest finite-time Lyapunov exponent changes smoothly with gamma , the number of occurrences of the most probable one, extracted from the distribution of finite-time Lyapunov exponents over initial conditions, reveals details about the phase-space dynamics. In particular, the influence of the integrable and pseudointegrable dynamics without Yukawa interaction for specific mass ratios can be clearly identified and demonstrates the sensitivity of the finite-time Lyapunov exponents as a phase-space probe. Being not restricted to two-dimensional problems such as Poincaré sections, the number of occurrences of the most probable Lyapunov exponents suggests itself as a suitable tool to characterize phase-space dynamics in higher dimensions. This is shown for the problem of two interacting particles in a circular billiard.

Spatial Variations of Stirring in the Surface Ocean: A Case Study of the Tasman Sea

Abstract Stirring in the Tasman Sea is examined using surface geostrophic currents derived from satellite altimeter measurements. Calculations of the distribution of finite-time Lyapunov exponents (FTLEs) indicate that the stirring in this region is not uniform and stretching rates over 15 days vary from less than 0.02 day−1 to over 0.3 day−1. These variations occur at both small (∼10 km) and large (∼1000 km) scales and in both cases are linked to dynamical features of the flow. The small-scale variations are related to the characteristics of coherent vortex structures, and there are low FTLEs inside vortices and filaments of high FTLEs in strain-dominated regions surrounding these vortices. Regional variations in the stirring are closely related to variations in mesoscale activity and eddy kinetic energy (EKE). High values of mean FTLE occur in regions of high EKE (highest mean values of around 0.2 day−1 occur in the East Australia Current separation region) whereas small values occur in regions with low EKE (mean values around 0.03 day−1 in the east Tasman Sea). There is a compact relationship between the mean FTLEs and EKE, raising the possibility of using the easily calculated EKE to estimate the stirring. This possibility is even more intriguing because the FTLE distributions can be approximated, for the time scales considered here, by Weibull distributions with shape parameter equal to 1.6, which can be defined from the mean value alone.

Two-dimensional turbulence of dilute polymer solutions.

We investigate theoretically and numerically the effect of polymer additives on two-dimensional turbulence by means of a viscoelastic model. We provide compelling evidence that, at vanishingly small concentrations, such that the polymers are passively transported, the probability distribution of polymer elongation has a power law tail: Its slope is related to the statistics of finite-time Lyapunov exponents of the flow, in quantitative agreement with theoretical predictions. We show that at finite concentrations and sufficiently large elasticity the polymers react on the flow with manifold consequences: Velocity fluctuations are drastically depleted, as observed in soap film experiments; the velocity statistics becomes strongly intermittent; the distribution of finite-time Lyapunov exponents shifts to lower values, signaling the reduction of Lagrangian chaos.

Characterization of the local instability in the Hénon–Heiles Hamiltonian

Several prototypical distributions of finite-time Lyapunov exponents have been computed for the two-dimensional Hénon–Heiles Hamiltonian system. Different shapes are obtained for each dynamical state. Even when an evolution is observed in the morphology of the distributions for the smallest integration intervals, they can still serve for characterizing the dynamical state of the system.

Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map: Mechanisms and their characterizations

A simple quasiperiodically forced one-dimensional cubic map is shown to exhibit very many types of routes to chaos via strange nonchaotic attractors (SNAs) in a two-parameter (A-f) space. The routes include transitions to chaos via SNAs from both a one-frequency torus and a period-doubled torus. In the former case, we identify the fractalization and type-I intermittency routes. In the latter case, we point out that at least four distinct routes for the truncation of the torus-doubling bifurcation and the creation of SNAs occur in this model. In particular, the formation of SNAs through Heagy-Hammel, fractalization, and type-III intermittent mechanisms is described. In addition, it has been found that in this system there are some regions in the parameter space where a dynamics involving a sudden expansion of the attractor, which tames the growth of period-doubling bifurcation, takes place, creating the SNA. The SNAs created through different mechanisms are characterized by the behavior of the Lyapunov exponents and their variance, by the estimation of the phase sensitivity exponent, and through the distribution of finite-time Lyapunov exponents.

The effects of finite lifetime of passive scalars and vorticity on their power spectra

The power law exponent for the wavenumber power spectrum of a passive scalar field with finite lifetime in Lagrangian chaotic flows and the enstrophy spectrum of two-dimensional Navier–Stokes turbulence with drag are investigated. The power law exponents are related to the distribution of finite-time Lyapunov exponents and the finite lifetime (or the drag coefficient).