The dynamics of many important high-dimensional dynamical systems are both chaotic and complex, meaning that strong reducing hypotheses are required to understand the dynamics. The highly influential chaotic hypothesis of Gallavotti and Cohen states that the large-scale dynamics of high-dimensional systems are effectively uniformly hyperbolic, which implies many felicitous statistical properties. We obtain direct and reliable numerical evidence, contrary to the chaotic hypothesis, of the existence of non-hyperbolic large-scale dynamical structures in a mean-field coupled system. To do this we reduce the system to its thermodynamic limit, which we approximate numerically with a Chebyshev basis transfer operator discretisation. This enables us to obtain a high precision estimate of a homoclinic tangency, implying a failure of uniform hyperbolicity. Robust non-hyperbolic behaviour is expected under perturbation. As a result, the chaotic hypothesis should not be {\it a priori} assumed to hold in all systems, and a better understanding of the domain of its validity is required.
We develop an extension of the fast method of angles for hyperbolicity verification in chaotic systems with an arbitrary number of time-delay feedback loops. The adopted method is based on the theory of covariant Lyapunov vectors and provides an efficient algorithm applicable for systems with high-dimensional phase space. Three particular examples of time-delay systems are analyzed and in all cases the expected hyperbolicity is confirmed.
We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.
We consider a chain of oscillators with hyperbolic chaos coupled via diffusion. When the coupling is strong the chain is synchronized and demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent. With the decay of the coupling the second and the third Lyapunov exponents approach zero simultaneously. The second one becomes positive, while the third one remains close to zero. Its finite-time numerical approximation fluctuates changing the sign within a wide range of the coupling parameter. These fluctuations arise due to the unstable dimension variability which is known to b…
We apply to bidimensional chaotic maps the numerical method proposed by Ginelli et al. to approximate the associated Oseledets splitting, i.e. the set of linear subspaces spanned by the so called covariant Lyapunov vectors (CLV) and corresponding to the Lyapunov spectrum. These subspaces are the analog of linearized invariant manifolds for non-periodic points, so the angles between them can be used to quantify the degree of hyperbolicity of generic orbits; however, being such splitting non invariant under smooth transformations of phase space, it is interesting to investigate the properties of…
Departing from a system of two non-autonomous amplitude equations, demonstrating hyperbolic chaotic dynamics, we construct a 1D medium as ensemble of such local elements introducing spatial coupling via diffusion. When the length of the medium is small, all spatial cells oscillate synchronously, reproducing the local hyperbolic dynamics. This regime is characterized by a single positive Lyapunov exponent. The hyperbolicity survives when the system gets larger in length so that the second Lyapunov exponent passes zero, and the oscillations become inhomogeneous in space. However, at a point wher…
This paper uses compressible flow simulation to analyze the hyperbolicity, shadowing directions, and sensitivities of a weakly turbulent three dimensional cylinder flow at Reynolds number 525 and Mach number 0.1. By computing the first 40 Covariant Lyapunov Vectors (CLVs), we find that unstable CLVs are active in the near-wake region, whereas stable CLVs are active in the far-wake region. This phenomenon is related to hyperbolicity since it shows that CLVs point to different directions; it also suggests that for open flows there is a large fraction of CLVs that are stable. However, due to the …
We discuss two mechanical systems with hyperbolic chaotic attractors of Smale - Williams type. Both models are based on Froude pendulums. The first system is composed of two coupled Froude pendulums with alternating periodic braking. The second system is Froude pendulum with time-delayed feedback and periodic braking. We demonstrate by means of numerical simulations that proposed models have chaotic attractors of Smale - Williams type. We specify regions of parameter values at which the dynamics corresponds to Smale - Williams solenoid. We check numerically hyperbolicity of the attractors.
Navigation satellites are known from numerical studies to reside in a dynamically sensitive environment, which may be of profound importance for their long-term sustainability. We derive the fundamental Hamiltonian of GNSS dynamics and show analytically that near-circular trajectories lie in the neighborhood of a Normally Hyperbolic Invariant Manifold (NHIM), which is the primary source of hyperbolicity. Quasi-circular orbits escape through chaotic transport, regulated by the NHIM's stable and unstable manifolds, following a power-law escape time distribution $P(t) \sim t^{-α}$, with $α\sim 0.…
We consider a self-oscillator whose excitation parameter is varied. Frequency of the variation is much smaller then the natural frequency of the oscillator so that oscillations in the system are periodically excited and decay. Also a time delay as added such that when the oscillations start to grow at a new excitation stage they are influenced via the delay line by the oscillations at the penultimate excitation stage. Due to a nonlinearity the seeding from the past arrives with a doubled phase so that oscillation phase changes from stage to stage according to chaotic Bernoulli-type map. As a r…
The crossing of a transition state in a multidimensional reactive system is mediated by invariant geometric objects in phase space: An invariant hyper-sphere that represents the transition state itself and invariant hyper-cylinders that channel the system towards and away from the transition state. The existence of these structures can only be guaranteed if the invariant hyper-sphere is normally hyperbolic, i.e., the dynamics within the transition state is not too strongly chaotic. We study the dynamics within the transition state for the hydrogen exchange reaction in three degrees of freedom.…
We study an ensemble of identical noisy phase oscillators with a blinking mean-field coupling, where one-cluster and two-cluster synchronous states alternate. In the thermodynamic limit the population is described by a nonlinear Fokker-Planck equation. We show that the dynamics of the order parameters demonstrates hyperbolic chaos. The chaoticity manifests itself in phases of the complex mean field, which obey a strongly chaotic Bernoulli map. Hyperbolicity is confirmed by numerical tests based on the calculations of relevant invariant Lyapunov vectors and Lyapunov exponents. We show how the c…
This report investigates the dynamical stability conjectures of Palis and Smale, and Pugh and Shub from the standpoint of numerical observation and lays the foundation for a stability conjecture. As the dimension of a dissipative dynamical system is increased, it is observed that the number of positive Lyapunov exponents increases monotonically, the Lyapunov exponents tend towards continuous change with respect to parameter variation, the number of observable periodic windows decreases (at least below numerical precision), and a subset of parameter space exists such that topological change is …
We derive a sufficient condition for topological horseshoe and uniform hyperbolicity of a 4-dimensional symplectic map, which is introduced by coupling the two 2-dimensional Hénon maps via linear terms. The coupled Hénon map thus constructed can be viewed as a simple map modeling the horseshoe in higher dimensions. We show that there are two different types of horseshoes, each of which is realized around different anti-integrable limits in the parameter regime.
We test the applicability of the Gallavotti-Cohen fluctuation formula on a nonequilibrium version of the periodic Ehrenfest wind-tree model. This is a one-particle system whose dynamics is rather complex (e.g. it appears to be diffusive at equilibrium), but its Lyapunov exponents are nonpositive. For small applied field, the system exhibits a very long transient, during which the dynamics is roughly chaotic, followed by asymptotic collapse on a periodic orbit. During the transient, the dynamics is diffusive, and the fluctuations of the current are found to be in agreement with the fluctuation …
We show that chimera states, where differentiated subsets of synchronized and desynchronized dynamical elements coexist, can emerge in networks of hyperbolic chaotic oscillators subject to global interactions. As local dynamics we employ Lozi maps which possess hyperbolic chaotic attractors. We consider a globally coupled system of these maps and use two statistical quantities to describe its collective behavior: the average fraction of elements belonging to clusters and the average standard deviation of state variables. Chimera states, clusters, complete synchronization, and incoherence are t…
The effective numerical method is developed performing the test of the hyperbolicity of chaotic dynamics. The method employs ideas of algorithms for covariant Lyapunov vectors but avoids their explicit computation. The outcome is a distribution of a characteristic value which is bounded within the unit interval and whose zero indicate the presence of tangency between expanding and contracting subspaces. To perform the test one needs to solve several copies of equations for infinitesimal perturbations whose amount is equal to the sum of numbers of positive and zero Lyapunov exponents. Since for…
A new information-geometric approach to chaotic dynamics on curved statistical manifolds based on Entropic Dynamics (ED) is proposed. It is shown that the hyperbolicity of a non-maximally symmetric 6N-dimensional statistical manifold M_{s} underlying an ED Gaussian model describing an arbitrary system of 3N degrees of freedom leads to linear information-geometric entropy growth and to exponential divergence of the Jacobi vector field intensity, quantum and classical features of chaos respectively.
An improved approach for estimating the dimension of inertial manifolds in chaotic distributed dynamical systems via analysis of angles between tangent subspaces
A simple and transparent example of a non-autonomous flow system, with hyperbolic strange attractor is suggested. The system is constructed on a basis of two coupled van der Pol oscillators, the characteristic frequencies differ twice, and the parameters controlling generation in both oscillators undergo a slow periodic counter-phase variation in time. In terms of stroboscopic Poincaré section, the respective four-dimensional mapping has a hyperbolic strange attractor of Smale - Williams type. Qualitative reasoning and quantitative data of numerical computations are presented and discussed, e.…
We study a simplified coupled atmosphere-ocean model using the formalism of covariant Lyapunov vectors (CLVs), which link physically-based directions of perturbations to growth/decay rates. The model is obtained via a severe truncation of quasi-geostrophic equations for the two fluids, and includes a simple yet physically meaningful representation of their dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and the parameters are chosen so that a chaotic behaviour is observed. One finds two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen near-zero LEs…
Modified Klein-Gordon-Fock equations were obtained on the basis of one-dimensional chaotic dynamics. The original Lagrangians were found. The concepts of \textit{m}-exponential map and groups with broken symmetry are introduced. A system of bitrial orthogonal functions is considered.
Dynamical equations are formulated and a numerical study is provided for self-oscillatory model systems based on the triple linkage hinge mechanism of Thurston -- Weeks -- Hunt -- MacKay. We consider systems with holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-osc…
We investigate dynamical tunneling in many dimensional systems using a quasi-periodically modulated kicked rotor, and find that the tunneling rate from the torus to the chaotic region is drastically enhanced when the chaotic states become delocalized as a result of the Anderson transition. This result strongly suggests that amphibious states, which were discovered for a one-dimensional kicked rotor with transporting islands [L. Hufnagel et al., Phys. Rev. Lett. 89, 154101 (2002)], quite commonly appear in many dimensional systems.
We analyse the infinite-dimensional limit of the maximally chaotic dynamical systems that are defined on N-dimensional tori. These hyperbolic systems found successful application in computer algorithms that generate high-quality pseudorandom numbers for advanced Monte Carlo simulations. The chaotic properties of these systems are increasing with $N$ because the corresponding Kolmogorov-Sinai entropy grows linearly with $N$. We calculated the spectrum and the entropy of the system that appears in the infinite dimensional limit. We demonstrated that the limiting system has exponentially expandin…
We consider the ultradiscretization of a solvable one-dimensional chaotic map which arises from the duplication formula of the elliptic functions. It is shown that ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously. A geometric interpretation of the dynamics of the tent map is given in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding to the $m$-th multiplication formula of the elliptic functions is also discussed.
We present numerical verification of hyperbolic nature for chaotic attractor in a system of two coupled non-autonomous van der Pol oscillators (Kuznetsov, Phys. Rev. Lett., 95, 144101, 2005). At certain parameter values, in the four-dimensional phase space of the Poincare map a toroidal domain (a direct product of a circle and a three-dimensional ball) is determined, which is mapped into itself and contains the attractor we analyze. In accordance with the computations, in this absorbing domain the conditions of hyperbolicity are valid, which are formulated in terms of contracting and expanding…
The authors present two results on infinite-dimensional linear dynamical systems with chaoticity. One is about the chaoticity of the backward shift map in the space of infinite sequences on a general Fréchet space. The other is about the chaoticity of a translation map in the space of real continuous functions. The chaos is shown in the senses of both Li-Yorke and Wiggins. Treating dimensions as freedoms, the two results imply that in the case of an infinite number of freedoms, a system may exhibit complexity even when the action is linear. Finally, the authors discuss physical applications of…
A system of $N$ rotators is investigated with a constraint given by a condition of vanishing sum of the cosines of the rotation angles. Equations of the dynamics are formulated and results of numerical simulation for the cases $N$=3, 4, and 5 are presented relating to the geodesic flows on a two-dimensional, three-dimensional, and four-dimensional manifold, respectively, in a compact region (due to the periodicity of the configuration space in angular variables). It is shown that a system of three rotators demonstrates chaos, characterized by one positive Lyapunov exponent, and for systems of …
Sensitivity analysis in chaotic dynamical systems is a challenging task from a computational point of view. In this work, we present a numerical investigation of a novel approach, known as the space-split sensitivity or S3 algorithm. The S3 algorithm is an ergodic-averaging method to differentiate statistics in ergodic, chaotic systems, rigorously based on the theory of hyperbolic dynamics. We illustrate S3 on one-dimensional chaotic maps, revealing its computational advantage over naive finite difference computations of the same statistical response. In addition, we provide an intuitive expla…
Chaotic dynamics of the dripping faucet was investigated both experimentally and theoretically. We measured continuous change in drop position and velocity using a high-speed camera. Continuous trajectories of a low-dimensional chaotic attractor were reconstructed from these data, which was not previously obtained but predicted in our fluid dynamic simulation. From the simulation, we further obtained an approximate potential function with only two variables, the drop mass and its position of the center of mass. The potential landscape helps one to understand intuitively how the dripping dynami…
Using examples we test formulae previously conjectured to give the fractal information dimension of chaotic repellors and their stable and unstable manifolds in ``typical'' dynamical systems in terms of the Lyapunov exponents and the characteristic escape time from the repellor. Our main example, a three-dimensional chaotic scattering billiard, yields a new structure for its invariant manifolds. This system also provides an example of a system which is not typical and illustrates how perturbation to the system restores typicality and the applicability of the dimension formulae.
Analytical solutions to the chaotic and ergodic motion of a certain class of one-dimensional dissipative and discrete dynamical systems are derived. This allows us to obtain exact expressions for physical properties like the time correlation function. We illustrate our solutions by means of a few examples for which conventional numerical trajectory calculations fail to predict the correct behaviour.
The fractal Weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller. We provide the first test for the fractal Weyl law for a three-dimensional open scattering system. For the four-sphere billiard, we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition. We test the fractal Weyl law for various symmetry subspaces and sphere-to-sphere separations.
Dominance of Milnor attractors in high-dimensional dynamical systems is reviewed, with the use of globally coupled maps. From numerical simulations, the threshold number of degrees of freedom for such prevalence of Milnor attractors is suggested to be $5 \sim 10$, which is also estimated from an argument of combinatorial explosion of basin boundaries. Chaotic itinerancy is revisited from the viewpoint of Milnor attractors. Relevance to neural networks is discussed.
We use a recently found parametrization of the solutions of the inverse Frobenius-Perron problem within the class of complete unimodal maps to develop a Monte-Carlo approach for the construction of one-dimensional chaotic dynamical laws with given statistical properties, i.e. invariant density and autocorrelation function. A variety of different examples are presented to demonstrate the power of our method.
The small-scale kinematic dynamo in a two-dimensional chaotic flow is studied. The analytic approach is developed in framework of the Kraichnan-Kazantsev model. It is shown that the growth of magnetic field $\bm{B}$ fluctuations stops at large times in accordance with so-called anti-dynamo theorems. The value of $\bm{B}^2$ increased therewith in square of the magnetic Prandtl number times. The spatial structure of the correlation tensor of the magnetic field is found.
Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatio-temporal systems including fluid turbulence is supported by non-chaotic, exactly recurring time-periodic solutions of the governing equations. These unstable periodic orbits capture key features of the turbulent dynamics and sufficiently large sets of orbits promise a framework to predict the statistics of the chaotic flow. Computing periodic orbits for high-dimensional spatio-temporally chaotic systems remains challenging as known methods either show poor convergence properties…
The probability current statistics of two-dimensional open chaotic ballistic billiards is studied both analytically and numerically. Assuming that the real and imaginary parts of the scattering wave function are both random Gaussian fields, we find a universal distribution function for the probability current. In by-passing we recover previous analytic forms for wave function statistics. The expressions bridge the entire region from GOE to GUE type statistics. Our analytic expressions are verified numerically by explicit quantum-mechanical calculations of transport through a Bunimovich billiar…
The spatiotemporal dynamics of an excitable medium with multiple spiral defects is shown to vary smoothly with system size from short-lived transients for small systems to extensive chaos for large systems. A comparison of the Lyapunov dimension density with the average spiral defect density suggests an average dimension per spiral defect varying between three and seven. We discuss some implications of these results for experimental studies of excitable media.
We have measured resonance spectra in a superconducting microwave cavity with the shape of a three-dimensional generalized Bunimovich stadium billiard and analyzed their spectral fluctuation properties. The experimental length spectrum exhibits contributions from periodic orbits of non-generic modes and from unstable periodic orbit of the underlying classical system. It is well reproduced by our theoretical calculations based on the trace formula derived by Balian and Duplantier for chaotic electromagnetic cavities.
We experimentally observe the nonlinear dynamics of an optoelectronic time-delayed feedback loop designed for chaotic communication using commercial fiber optic links, and we simulate the system using delay differential equations. We show that synchronization of a numerical model to experimental measurements provides a new way to assimilate data and forecast the future of this time-delayed high-dimensional system. For this system, which has a feedback time delay of 22 ns, we show that one can predict the time series for up to several delay periods, when the dynamics is about 15 dimensional.
We investigate the origin of various convective patterns using bifurcation diagrams that are constructed using direct numerical simulations. We perform two-dimensional pseudospectral simulations for a Prandtl number 6.8 fluid that is confined in a box with aspect ratio $Γ= 2\sqrt{2}$. Steady convective rolls are born from the conduction state through a pitchfork bifurcation at $r=1$, where $r$ is the reduced Rayleigh number. These fixed points bifurcate successively to time-periodic and quasiperiodic rolls through Hopf and Neimark-Sacker bifurcations at $r \simeq 80$ and $r \simeq 500 $ respec…
