Extreme value theory for singular measures

In this paper, we perform an analytical and numerical study of the extreme values of specific observables of dynamical systems possessing an invariant singular measure. Such observables are expressed as functions of the distance of the orbit of initial conditions with respect to a given point of the attractor. Using the block maxima approach, we show that the extremes are distributed according to the generalised extreme value distribution, where the parameters can be written as functions of the information dimension of the attractor. The numerical analysis is performed on a few low dimensional maps. For the Cantor ternary set and the Sierpinskij triangle, which can be constructed as iterated function systems, the inferred parameters show a very good agreement with the theoretical values. For strange attractors like those corresponding to the Lozi and He?non maps, a slower convergence to the generalised extreme value distribution is observed. Nevertheless, the results are in good statistical agreement with the theoretical estimates. It is apparent that the analysis of extremes allows for capturing fundamental information of the geometrical structure of the attractor of the underlying dynamical system, the basic reason being that the chosen observables act as magnifying glass in the neighborhood of the point from which the distance is computed.     

Two-cluster bifurcations in systems of globally pulse-coupled oscillators

For a system of globally pulse-coupled phase-oscillators, we derive conditions for stability of the completely synchronous state and all stationary two-cluster states and explain how the different states are naturally connected via bifurcations. The coupling is modeled using the phase-response-curve (PRC), which measures the sensitivity of each oscillator’s phase to perturbations. For large systems with a PRC, which is zero at the spiking threshold, we are able to find the parameter regions where multiple stable two-cluster states coexist and illustrate this by an example. In addition, we explain how a locally unstable one-cluster state may form an attractor together with its homoclinic connections. This leads to the phenomenon of intermittent, asymptotic synchronization with abating beats away from the perfect synchrony.     

Crossing bifurcations and unstable dimension variability

A crisis is a global bifurcation in which a chaotic attractor has a discontinuous change in size or suddenly disappears as a scalar parameter of the system is varied. In this Letter, we describe a global bifurcation in three dimensions which can result in a crisis. This bifurcation does not involve a tangency and cannot occur in maps of dimension smaller than 3. We present evidence of unstable dimension variability as a result of the crisis. We then derive a new scaling law describing the density of the new portion of the attractor formed in the crisis. We illustrate this new type of bifurcation with a specific example of a three-dimensional chaotic attractor undergoing a crisis.     

Universal Behaviour of Extreme Value Statistics for Selected Observables of Dynamical Systems

The main results of the extreme value theory developed for the investigation of the observables of dynamical systems rely, up to now, on the block maxima approach. In this framework, extremes are identified with the block maxima of the time series of the chosen observable, in the limit of infinitely long blocks. It has been proved that, assuming suitable mixing conditions for the underlying dynamical systems, the extremes of a specific class of observables are distributed according to the so called Generalised Extreme Value (GEV) distribution. Direct calculations show that in the case of quasi-periodic dynamics the block maxima are not distributed according to the GEV distribution. In this paper we show that considering the exceedances over a given threshold instead of the block-maxima approach it is possible to obtain a Generalised Pareto Distribution also for extremes computed in systems which do not satisfy mixing conditions. Requiring that the invariant measure locally scales with a well defined exponent—the local dimension—, we show that the limiting distribution for the exceedances of the observables previously studied with the block maxima approach is a Generalised Pareto distribution where the parameters depend only on the local dimensions and the values of the threshold but not on the number of observations considered. We also provide connections with the results obtained with the block maxima approach. In order to provide further support to our findings, we present the results of numerical experiments carried out considering the well-known Chirikov standard map.     

On the Stability of a Quantum Dynamics of a Bose-Einstein Condensate Trapped in a One-Dimensional Toroidal Geometry

We rigorously analyze the stability of the “quasi-classical” dynamics of a Bose-Einstein condensate with repulsive and attractive interactions, trapped in an effective 1D toroidal geometry. The “classical” dynamics, which corresponds to the Gross-Pitaevskii mean field theory, is stable in the case of repulsive interaction, and unstable (under some conditions) in the case of attractive interaction. The corresponding quantum dynamics for observables is described by using a closed system of linear partial differential equations. In both cases of stable and unstable quasi-classical dynamics the quantum effects represent a singular perturbation to the quasi-classical solutions, and are described by the terms in these equations which consist of a small quasi-classical parameter which multiplies high-order “spatial” derivatives. We demonstrate that as a result of the quantum singularity for observables a convergence of quantum solutions to the corresponding classical solutions exists only for limited times, and estimate the characteristic time-scales of the convergence.     

Crisis-induced unstable dimension variability in a dynamical system

Unstable dimension variability is an extreme form of non-hyperbolic behavior in chaotic systems whose attractors have periodic orbits with a different number of unstable directions. We propose a new mechanism for the onset of unstable dimension variability based on an interior crisis, or a collision between a chaotic attractor and an unstable periodic orbit. We give a physical example by considering a high-dimensional dissipative physical system driven by impulsive periodic forcing.     

Template analysis of a Faraday disk dynamo

In a recent paper Moroz [1] returned to a nonlinear three-dimensional model of dynamo action for a self-exciting Faraday disk dynamo introduced by Hide et al. [2]. Since only two examples of chaotic behaviour were shown in [2], Moroz [1] performed a more extensive analysis of the dynamo model, producing a selection of bifurcation transition diagrams, including those encompassing the two examples of chaotic behaviour in [2]. Unstable periodic orbits were extracted and presented in [1], but no attempt was made to identify the underlying chaotic attractor. We rectify that here. Illustrating the procedure with one of the cases considered in [1], we use some of the unstable periodic orbits to identify a possible template for the chaotic attractor, using ideas from topology [3]. In particular, we investigate how the template is affected by changes in bifurcation parameter.     

Formulation of the third-order Grüneisen parameter at extreme compression

We present a direct method using the basic principles of calculus to derive the expression for the third-order Grüneisen parameter in terms of the pressure derivatives of bulk modulus at extreme compression. The derivation presented here does not depend on the assumptions regarding the values of free-volume parameter and its variation with pressure. The identities used in the present analysis are valid at extreme compression for all physically acceptable equations of state.     

Homoclinic tangency and chaotic attractor disappearance in a dripping faucet experiment

A sequence of attractors, reconstructed from interdrops time series data of a leaky faucet experiment, is analyzed as a function of the mean dripping rate. We established the presence of a saddle point and its manifolds in the attractors and we explained the dynamical changes in the system using the evolution of the manifolds of the saddle point, as suggested by the orbits traced in first return maps. The sequence starts at a fixed point and evolves to an invariant torus of increasing diameter (a Hopf bifurcation) that pushes the unstable manifold towards the stable one. The torus breaks up and the system shows a chaotic attractor bounded by the unstable manifold of the saddle. With the attractor expansion the unstable manifold becomes tangential to the stable one, giving rise to the sudden disappearance of the chaotic attractor, which is an experimental observation of a so called chaotic blue sky catastrophe.     

Properties of maps related to flows around a saddle point

General properties of maps associated with systems in which trajectories of the flow get close to a hyperbolic fixed point with a two-dimensional stable and a one-dimensional unstable manifold are examined in the chaotic region.Exponents characterizing power law singular behaviour of the Jacobian, of the shape of and of the stationary probability distribution on the chaotic attractor are expressed in terms of the ratios of the eigenvalues of the linearized flow at the hyperbolic point. Emphasis is laid on the study of the limiting case of strong dissipation leading to a simple one-dimensional attractor but to a dynamics with interesting features.     

Sharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence

In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to log  ν ⁻¹ for all values of the governing parameter ε, except for ε =1. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, the complexity of the dynamics of the shell model increases as the viscosity ν tends to zero, and we describe a precise scenario of successive bifurcations for different parameters regimes. In the “three-dimensional” regime of parameters this scenario changes when the parameter ε becomes sufficiently close to 0 or to 1. We also show that in the “two-dimensional” regime of parameters, for a certain non-zero forcing term, the long-term dynamics of the model becomes trivial for every value of the viscosity. AMS Subject Classifications: 76F20, 76D05, 35Q30     

Attractors with controllable basin sizes from cooperation of contracting and expanding dynamics in pulse-coupled oscillators

Unstable attractors are a novel type of attractor with local unstable dynamics, but with positive measures of basins.Here, we introduce local contracting dynamics by slightly modifying the function which mediates the interactions among the oscillators. Thus, the property of unstable attractors can be controlled through the cooperation of expanding and contracting dynamics. We demonstrate that one certain type of unstable attractor is successfully controlled through this simple modification. Specifically, the staying time for unstable attractors can be prolonged, and we can even turn the unstable attractors into stable attractors with predictable basin sizes. As an application, we demonstrate how to realize the switching dynamics that is only sensitive to the finite size perturbations.     

Collision-number statistics for transport processes

Physical observables are often represented as walkers performing random displacements. When the number of collisions before leaving the explored domain is small, the diffusion approximation leads to incongruous results. In this Letter, we explicitly derive an explicit formula for the moments of the number of particle collisions in an arbitrary volume, for a broad class of transport processes. This approach is shown to generalize the celebrated Kac formula for the moments of residence times. Some applications are illustrated for bounded, unbounded and absorbing domains.     

Bubbling transition to spatial mode excitation in an extended dynamical system

We investigated the transition to spatio-temporal chaos in spatially extended nonlinear dynamical systems possessing an invariant subspace with a low-dimensional attractor. When the latter is chaotic and the subspace is transversely stable we have a spatially homogeneous state only. The onset of spatio-temporal chaos, i.e. the excitation of spatially inhomogeneous modes, occur through the loss of transversal stability of some unstable periodic orbit embedded in the chaotic attractor lying in the invariant subspace. This is a bubbling transition, since there is a switching between spatially homogeneous and nonhomogeneous states with statistical properties of on-off intermittency. Hence the onset of spatio-temporal chaos depends critically both on the existence of a chaotic attractor in the invariant subspace and its being transversely stable or unstable.     

Extreme value laws in dynamical systems under physical observables

Extreme value theory for chaotic deterministic dynamical systems is a rapidly expanding area of research. Given a system and a real function (observable) defined on its phase space, extreme value theory studies the limit probabilistic laws obeyed by large values attained by the observable along orbits of the system. Based on this theory, the so-called block maximum method is often used in applications for statistical prediction of large value occurrences. In this method, one performs statistical inference for the parameters of the Generalised Extreme Value (GEV) distribution, using maxima over blocks of regularly sampled observable values along an orbit of the system. The observables studied so far in the theory are expressed as functions of the distance with respect to a point, which is assumed to be a density point of the system’s invariant measure. However, at least with respect to the ambient (usually Euclidean) metric, this is not the structure of the observables typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. In this paper we consider extreme value limit laws for observables which are not expressed as functions of the distance (in the ambient metric) from a density point of the dynamical system. In such cases, the limit laws are no longer determined by the functional form of the observable and the dimension of the invariant measure: they also depend on the specific geometry of the underlying attractor and of the observable’s level sets. We present a collection of analytical and numerical results, starting with a toral hyperbolic automorphism as a simple template to illustrate the main ideas. We then formulate our main results for a uniformly hyperbolic system, the solenoid map. We also discuss non-uniformly hyperbolic examples of maps (Hénon and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models). Our purpose is to outline the main ideas and to highlight several serious problems found in the numerical estimation of the limit laws.     

Stability analysis and observational measurement in 5-D Brans–Dicke cosmology

This Letter is a study of the effects of higher dimensional gravity and Brans–Dicke (BD) scalar field on cosmic acceleration in 5-D BD cosmological model. We assume a flat cosmological model in which the matter content of the universe is either cold dark matter or radiation. In a framework to study attractor solutions in the phase space we simultaneously constrain the model parameters with the observational data for distance modulus. The phase space analysis illustrates that the universe begins from an unstable state in the past and eventually reaches an asymptotically stable state (attractor). We examine the model by performing Hubble parameter test in addition to statefinder diagnosis. We also reconstruct the equation of state parameter, the scale factor in 3-D space and along extra dimension. The results show that due to the presence of extra dimension and Brans–Dicke scalar field in the model, the universe undergoes a period of acceleration.     

Multifractal dimensions in QCD cascades

Particle production in small rapidity or angular intervals have fractal structures similar to a Cantor dust. In this paper we present analytical result for multifractal dimensions valid for high energies. For high moments the dimension is given by \(\sqrt {6\alpha _s /\pi } \) . The scaling properties seen in the partonic state are not so well reflected in the hadronic multiplicity moments or factorial moments. We show how to define new observables on the final hadronic state, which do scale well. This means that the multifractal dimensions can be well determined and compared with results from QCD.     

Scroll wave dynamics in a model of the heterogeneous heart

Scroll waves are found in physical, chemical and biological systems and underlie many significant processes including life-threatening cardiac arrhythmias. The theory of scroll waves predicts scroll wave dynamics should be substantially affected by heterogeneity of cardiac tissue together with other factors including shape and anisotropy. In this study, we used our recently developed analytical model of the human ventricle to identify effects of shape, anisotropy, and regional heterogeneity of myocardium on scroll wave dynamics. We found that the main effects of apical-base heterogeneity were an increased scroll wave drift velocity and a shift towards the region of maximum action potential duration. We also found that transmural heterogeneity does not substantially affect scroll wave dynamics and only in extreme cases changes the attractor position.     

Unstable attractors induce perpetual synchronization and desynchronization

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which unstable attractors arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.     

How to define physical properties of unstable particles

In the framework of effective quantum field theory we address the definition of physical quantities characterizing unstable particles. With the aid of a one-loop calculation, we study this issue in terms of the charge and the magnetic moment of a spin-1/2 resonance. By appealing to the invariance of physical observables under field redefinitions we demonstrate that physical properties of unstable particles should be extracted from the residues at complex (double) poles of the corresponding S -matrix.