49 pages, 20 figures; previous lecture notes extended by a new third chapter on anomalous diffusion; to be published in Reviews of Nonlinear Dynamics and Complexity, Vol.3, as a book chapter
发表信息：
book chapter (60 pages) in: Reviews of Nonlinear Dynamics and Complexity Volume 3, Heinz Georg Schuster (Editor), ISBN: 978-3-527-40945-7, Wiley-VCH 2010, http://www.wiley.com/WileyCDA/WileyTitle/productCd-3527409459.html
This is an easy-to-read introduction to foundations of deterministic chaos, deterministic diffusion and anomalous diffusion. The first part introduces to deterministic chaos in one-dimensional maps in form of Ljapunov exponents and dynamical entropies. The second part outlines the concept of deterministic diffusion. Then the escape rate formalism for deterministic diffusion, which expresses the diffusion coefficient in terms of the above two chaos quantities, is worked out for a simple map. Part three explains basics of anomalous diffusion by demonstrating the stochastic approach of continuous time random walk theory for an intermittent map. As an example of experimental applications, the anomalous dynamics of biological cell migration is discussed.
In this letter we consider the phase diffusion of a harmonically driven undamped pendulum and show that it is anomalous in the strong sense. The role played by the fractal properties of the phase space is highlighted, providing an illustration of the link between deterministic chaos and anomalous transport. Finally, we build a stochastic model which reproduces most properties of the original Hamiltonian system by alternating ballistic flights and random diffusion.
This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory description, and properties of the anomalous dynamics associated with it. In the first half of the chapter we study simple one-dimensional deterministic maps, in the second half basic stochastic models and eventually an experiment. We start by reminding the reader of fundamental chaos quantities and their relation to each other, exemplified by the paradigmatic Bernoulli shift. Using the intermittent Pomeau-Manneville map the problem of weak chaos and infinite ergodic theory is outlined, defining a very rec…
We show that the fluctuations of the periodic orbits of deterministically chaotic systems can be captured by supersymmetry, in the sense that they are repackaged in the contribution of the absolute value of the determinant of the noise fields, defined by the equations of motion.
We discuss the necessity and demonstrate the validity of introduction the notion of deterministic chaos in quantum field theory. Brief review of the existing approaches to this problem is given. We compare proposed chaos criterion for quantum fields with existing ones. Its consequences in particle physics are also discussed.
This paper shows that with mechanistic primary budget rules and with some simple assumptions on interest rates the well-known debt dynamics equation transforms into the infamous logistic map. The logistic map has very peculiar and rich nonlinear behaviour and it can exhibit deterministic chaos with certain parameter regimes. Deterministic chaos means the existence of the butterfly effect which in turn is qualitatively very important, as it shows that even deterministic budget rules produce unpredictable behaviour of the debt-to-GDP ratio, as chaotic systems are extremely sensitive to initial c…
We show that defect-mediated turbulence can exist in media where the underlying local dynamics is deterministically chaotic. While many of the characteristics of defect-mediated turbulence, such as the exponential decay of correlations and a squared Poissonian distribution for the number of defects, are identical to those seen in oscillatory media, the fluctuations in the number of defects differ significantly. The power spectra suggest the existence of underlying correlations that lead to a different and non-universal scaling structure in chaotic media.
Non-deterministic chaos is a form of low-dimensional dynamics which is characterized by the existence of a countable set of {\em sensitive decision points} (SDP's). Away from these points, the dynamics is well-behaved. Near these points, however, perturbations (e.g., thermal noise) may cause the outgoing trajectory to be chosen randomly. An example of a non-deterministic chaotic system is given, and a statistical method of analyzing the resultant dynamics is developed.
We study statistical properties of the irregular bursting arising in a class of neuronal models close to the transition from spiking to bursting. Prior to the transition to bursting, the systems in this class develop chaotic attractors, which generate irregular spiking. The chaotic spiking gives rise to irregular bursting. The duration of bursts near the transition can be very long. We describe the statistics of the number of spikes and the interspike interval distributions within one burst as functions of the distance from criticality.
We investigate the high dimensional Hamiltonian chaotic dynamics in $N$ coupled area-preserving maps. We show the existence of an enhanced trapping regime caused by trajectories performing a random walk {\em inside} the area corresponding to regular islands of the uncoupled maps. As a consequence, we observe long intermediate regimes of power-law decay of the recurrence time statistics (with exponent $γ=0.5$) and of ballistic motion. The asymptotic decay of correlations and anomalous diffusion depend on the stickiness of the $N$-dimensional invariant tori. Detailed numerical simulations show w…
Statistical properties of Wigner delay times and the effect of evanescent modes on the deterministic scattering of an electron matter wave from a classically chaotic 2-d electron waveguide are studied for the case of 2, 6, and 16 propagating modes. Deterministic reaction matrix theory for this system is generalized to include the effect of evanescent modes on the scattering process. Inclusion of evanescent modes can change the positions of resonance poles in the complex energy plane and can delay electron scattering, particularly at energies where new channels open. The statistical properties …
Non-deterministic chaos is a new dynamical paradigm where a non-deterministic system is influenced by random perturbations to produce the appearance of complexity. The non-determinism is envisioned to occur only at a single point in phase space, where many trajectories intersect. In the presence of external random perturbations (noise), whenever the phase space trajectory approaches the singularity, it will jump in an unpredictable way to a different solution. This behavior, while similar in appearance to deterministic chaos, has rather different implications for prediction and control.
In the inertial range of fully developed turbulence, we model the vertex network dynamics by an iterated unimodular map having the universal behavior. Inertial range anomalous scaling for the pair correlation functions of the velocity and the local energy dissipation is established as a consequence of the chaotic behavior of the unimodular map when the Feigenbaum attractor looses stability. The anomalous exponents determined by the Feigenbaum constant $η$ to the Kolmogorov's spectra are larger than those observed in experiments.
Recently, we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal Lyapunov exponent. In this paper, we investigate the capability of the test to cope with moderate amounts of noisy data. Comparisons are made between an improved version of our test and both the ``tangent space'' and ``direct method'' for computing the maximal Lyapunov exponent. The evidence of numerical experiments, ranging from the logistic map to an eight-dimensional Lorenz…
Although deterministic chaos has been predicted to occur in the triply resonant optical parametric oscillator (TROPO) fifteen years ago, experimental evidence of chaotic behavior in this system has been lacking so far, in marked contrast with most nonlinear systems, where chaos has been actively tracked and found. This situation is probably linked to the high sensitivity of the TROPO to perturbations, which adversely affects stationary operation at high power. We report the experimental observation in this system of a burst of irregular behavior of duration 80 microseconds. Although the system…
We experimentally study the nonlinear dynamics of a femtosecond ytterbium doped mode-locked fiber laser. With the laser operating in the pulsed regime a route to chaos is presented, starting from stable mode-locking, period two, period four, chaos and period three regimes. Return maps and bifurcation diagrams were extracted from time series for each regime. The analysis of the time series with the laser operating in the quasi mode-locked regime presents deterministic chaos described by an unidimensional Rossler map. A positive Lyapunov exponent $λ= 0.14$ confirms the deterministic chaos of the…
We investigate the long-term diffusion transport and chaos properties of single and coupled standard maps. We consider model parameters that are known to induce anomalous diffusion in the maps' phase spaces, as opposed to normal diffusion which is associated with Gaussian distribution properties of the kinematic variables. This type of transport originates in the presence of the so-called accelerator modes, i.e.~non-chaotic initial conditions which exhibit ballistic transport, which also affect the dynamics in their vicinity. We first systematically study the dynamics of single standard maps, …
Dynamics of coupled chaotic oscillators on a network are studied using coupled maps. Within a broad range of parameter values representing the coupling strength or the degree of elements, the system repeats formation and split of coherent clusters. The distribution of the cluster size follows a power law with the exponent $α$, which changes with the parameter values. The number of positive Lyapunov exponents and their spectra are scaled anomalously with the power of the system size with the exponent $β$, which also changes with the parameters. The scaling relation $α\sim 2(β+1)$ is uncovered, …
The relative stability and ergodicity of deterministic time-reversible thermostats, both singly and in coupled pairs, are assessed through their Lyapunov spectra. Five types of thermostat are coupled to one another through a single Hooke's-Law harmonic spring. The resulting dynamics shows that three specific thermostat types, Hoover-Holian, Ju-Bulgac, and Martyna-Klein-Tuckerman, have very similar Lyapunov spectra in their equilibrium four-dimensional phase spaces and when coupled in equilibrium or nonequilibrium pairs. All three of these oscillator-based thermostats are shown to be ergodic, w…
We examine characteristic properties of deterministic and stochastic diffusion in low-dimensional chaotic dynamical systems. As an example, we consider a periodic array of scatterers defined by a simple chaotic map on the line. Adding different types of time-dependent noise to this model we compute the diffusion coefficient from simulations. We find that there is a crossover from deterministic to stochastic diffusion under variation of the perturbation strength related to different asymptotic laws for the diffusion coefficient. Typical signatures of this scenario are suppression and enhancemen…
The peculiar phase-ordering properties of a lattice of coupled chaotic maps studied recently (A. Lema\^ıtre & H. Chaté, {\em Phys. Rev. Lett.} {\bf 82}, 1140 (1999)) are revisited with the help of detailed investigations of interface motion. It is shown that ``normal'', curvature-driven-like domain growth is recovered at larger scales than considered before, and that the persistence exponent seems to be universal. Using generalized persistence spectra, the properties of interface motion in this deterministic, chaotic, lattice system are found to ``interpolate'' between those of the two canonic…
We demonstrate aging behavior in a simple non-linear system. Our model is a chaotic map which generates deterministically sub-diffusion. Asymptotic behaviors of the diffusion process are described using aging continuous time random walks, introduced previously to model diffusion in glasses.
Fifty years after the invention of the laser diode and fourty years after the report of the butterfly effect - i.e. the unpredictability of deterministic chaos, it is said that a laser diode behaves like a damped nonlinear oscillator. Hence no chaos can be generated unless with additional forcing or parameter modulation. Here we report the first counter-example of a free-running laser diode generating chaos. The underlying physics is a nonlinear coupling between two elliptically polarized modes in a vertical-cavity surface-emitting laser. We identify chaos in experimental time-series and show …
A two-dimensional system of non-locally coupled complex Ginzburg-Landau oscillators is investigated numerically for the first time. As already known for the one-dimensional case, the system exhibits anomalous spatio-temporal chaos characterized by power-law spatial correlations. In this chaotic regime, the amplitude difference between neighboring elements shows temporal noisy on-off intermittency. The system is also spatially intermittent in this regime, which is revealed by multi-scaling analysis; the amplitude field is multi-affine and the difference field is multi-fractal. Correspondingly, …
Chaotic internal degrees of freedom of a molecule can act as noise and affect the diffusion of the molecule on a substrate. A separation of time scales between the fast internal dynamics and the slow motion of the centre of mass on the substrate makes it possible to directly link chaos to diffusion. We discuss the conditions under which this is possible, and show that in simple atomistic models with pair-wise harmonic potentials, strong chaos can arise through the geometry. Using molecular-dynamics simulations, we demonstrate that a realistic model of benzene is indeed chaotic, and that the in…
Consider a chaotic dynamical system generating Brownian motion-like diffusion. Consider a second, non-chaotic system in which all particles localize. Let a particle experience a random combination of both systems by sampling between them in time. What type of diffusion is exhibited by this {\em random dynamical system}? We show that the resulting dynamics can generate anomalous diffusion, where in contrast to Brownian normal diffusion the mean square displacement of an ensemble of particles increases nonlinearly in time. Randomly mixing simple deterministic walks on the line we find anomalous …
We introduce a cycle-expansion (fully deterministic) technique to compute the asymptotic behavior of arbitrary order transport moments. The theory is applied to different kinds of one-dimensional intermittent maps, and Lorentz gas with infinite horizon, confirming the typical appearance of phase transitions in the transport spectrum.
We derive a locally projective noise reduction scheme for nonlinear time series using concepts from deterministic dynamical systems, or chaos theory. We will demonstrate its effectiveness with an example with known deterministic dynamics and discuss methods for the verification of the results in the case of an unknown deterministic system.
The scope of this teaching package is to make a brief introduction to some notions and properties of chaotic systems. We first make a brief introduction to chaos in general and then we show some important properties of chaotic systems using the logistic map and its bifurcation diagram. We also show the universality found in "the route to chaos". The user is only required to have notions of algebra, so it is quite accessible. The formal basis of chaos theory are not covered in this introduction, but are pointed out for the reader interested in them. Therefore, this package is also useful for pe…
Several years after the pioneering work by Fermi Pasta and Ulam, fundamental questions about the link between dynamical and statistical properties remain still open in modern statistical mechanics. Particularly controversial is the role of deterministic chaos for the validity and consistency of statistical approaches. This contribution reexamines such a debated issue taking inspiration from the problem of diffusion and heat conduction in deterministic systems. Is microscopic chaos a necessary ingredient to observe such macroscopic phenomena?
We study the consequences of deterministic chaos for diffusion-controlled reaction. As an example, we analyze a diffusive-reactive deterministic multibaker and a parameter-dependent variation of it. We construct the diffusive and the reactive modes of the models as eigenstates of the Frobenius-Perron operator. The associated eigenvalues provide the dispersion relations of diffusion and reaction and, hence, they determine the reaction rate. For the simplest model we show explicitly that the reaction rate behaves as phenomenologically expected for one-dimensional diffusion-controlled reaction. U…
We consider the motion of ballistic electrons within a superlattice miniband under the influence of an alternating electric field. We show that the interaction of electrons with the self-consistent electromagnetic field generated by the electron current may lead to the transition from regular to chaotic dynamics. We estimate the conditions for the experimental observation of this deterministic chaos and discuss the similarities of the superlattice system with the other condensed matter and quantum optical systems.
We analyze, from a theoretical viewpoint, the bidirectional interdisciplinary relation between mathematics and psychology, focused on the mathematical theory of deterministic dynamical systems, and in particular, on the theory of chaos. On one hand, there is the direct classic relation: the application of mathematics to psychology. On the other hand, we propose the converse relation which consists in the formulation of new abstract mathematical problems appearing from processes and structures under research of psychology. The bidirectional multidisciplinary relation from-to pure mathematics, l…
The phenomenon of Stochastic Resonance (SR) is observed in a completely deterministic setting - with thermal noise being replaced by one-dimensional chaos. The piecewise linear map investigated in the paper shows a transition from symmetry-broken to symmetric chaos on increasing a system parameter. In the latter state, the chaotic trajectory switches between the two formerly disjoint attractors, driven by the map's inherent dynamics. This chaotic switching rate is found to `resonate' with the frequency of an externally applied periodic perturbation (multiplicative or additive). By periodically…
Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto the whole real line which preserves distances except at a countable set of points. This property, which leads to vanishing Lyapunov exponents, is designed to mimic diffusion in non-chaotic polygonal billiards that give rise to normal and anomalous diffusion in a fully deterministic setting. As these billiards are typically too complicated to be analyzed from f…
We adopt the '0-1' test for chaos using Brownian motion chains to identify the dynamics of the Ricker's population model. In the '0-1' test '0' is related to regular motion while '1' is associated with chaotic motion. The identified regular and chaotic types of solutions have been confirmed by means of recurrence plots.
A neural network model that exhibits stochastic population bursting is studied by simulation. First return maps of inter-burst intervals exhibit recurrent unstable periodic orbit (UPO)-like trajectories similar to those found in experiments on hippocampal slices. Applications of various control methods and surrogate analysis for UPO-detection also yield results similar to those of experiments. Our results question the interpretation of the experimental data as evidence for deterministic chaos and suggest caution in the use of UPO-based methods for detecting determinism in time-series data.
A fundamental issue in nonlinear dynamics and statistical physics is how to distinguish chaotic from stochastic fluctuations in short experimental recordings. This dilemma underlies many complex systems models from stochastic gene expression or stock exchange to quantum chaos. Traditionally, deterministic chaos is characterized by "sensitive dependence on initial conditions" as indicated by a positive Lyapunov exponent. However, ambiguity arises when applying this criterion to real-world data that are corrupted by measurement noise or perturbed nonautonomously by exogenous deterministic or sto…
It is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is possible to control (reduce) the chaotic diffusion in the phase space of a Hamiltonian system with 1.5 degrees of freedom which models the diffusion of charged test particles in a turbulent electric field across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications, this result suggests that some strategy to control turbulent transport in magnetized plasmas, in partic…
Chaotic deterministic dynamics of a particle can give rise to diffusive Brownian motion. In this paper, we compute analytically the diffusion coefficient for a particular two-dimensional stochastic layer induced by the kicked Harper map. The variations of the transport coefficient as a parameter is varied are analyzed in terms of the underlying classical trajectories with particular emphasis in the appearance and bifurcations of periodic orbits. When accelerator modes are present, anomalous diffusion of the Lévy type can occur. The exponent characterizing the anomalous diffusion is computed nu…
Recent work in dynamical systems theory has shown that many properties that are associated with irreversible processes in fluids can be understood in terms of the dynamical properties of reversible, Hamiltonian systems. That is, stochastic-like behavior is possible for these systems. Here we review the basic theory for this stochastic-like behavior and show how it may be used to obtain an understanding of irreversible processes in gases and fluids. Recent, closely related, work on the use of kinetic theory to calculate dynamical quantities such as Lyapunov exponents is also discussed.
We propose a novel approach to generate chaotic business cycles in a deterministic setting. Rather than producing chaos endogenously, we consider aggregate economic models with limit cycles and equilibriums, subject them to chaotic exogenous shocks and obtain chaotic cyclical motions. Thus, we emphasize that chaotic cycles, which are inevitable in economics, are not only interior properties of economic models, but also can be considered as a result of interaction of several economical systems. This provides a comprehension of chaos (unpredictability, lack of forecasting) and control of chaos a…
Symbolic dynamics has proven to be an invaluable tool in analyzing the mechanisms that lead to unpredictability and random behavior in nonlinear dynamical systems. Surprisingly, a discrete partition of continuous state space can produce a coarse-grained description of the behavior that accurately describes the invariant properties of an underlying chaotic attractor. In particular, measures of the rate of information production--the topological and metric entropy rates--can be estimated from the outputs of Markov or generating partitions. Here we develop Bayesian inference for k-th order Markov…
It is shown that a paradigm of classical statistical mechanics --- the thermalization of a Brownian particle --- has a low-dimensional, deterministic analogue: when a heavy, slow system is coupled to fast deterministic chaos, the resultant forces drive the slow degrees of freedom toward a state of statistical equilibrium with the fast degrees. This illustrates how concepts useful in statistical mechanics may apply in situations where low-dimensional chaos exists.
Extreme events such as rogue wave in optics and fluids are often associated with the merging dynamics of coherent structures. We present experimental and numerical results on the physics of extreme events appearance in a spatially extended semiconductor microcavity laser with intracavity saturable absorber. This system can display deterministic irregular dynamics only thanks to spatial coupling through diffraction of light. We have identified parameter regions where extreme events are encountered and established the origin of this dynamics in the emergence of deterministic spatiotemporal chaos…
Recurrence analysis is a well settled method allowing to discern chaos from order, and determinism from noise. We apply this tool to study time series representing geodesic and inspiraling motion of a test particle in a deformed Kerr spacetime, when deterministic chaos and different levels of stochastic noise are present. In particular, we suggest a recurrence-based criterion to reveal whether the time series comes from a deterministic source and find a noise-level threshold of its applicability.
We exhibit a remarkable equivalence between the dynamics of an intermittent nonlinear map and the electronic transport properties (obtained via the scattering matrix) of a crystal defined on a double Cayley tree. This strict analogy reveals in detail the nature of the mobility edge normally studied near (not at) the metal-insulator transition in electronic systems. We provide an analytical expression for the conductance as function of system size that at the transition obeys a q-exponential form. This manifests as power-law decay or few and far between large spike oscillations according to dif…
Thermal Control of Hysteresis and Deterministic Chaos in a Memristive MEMS Resonator
ICTP South American Institute for Fundamental Research, Instituto de Física Teórica（ICTP南美基础物理研究所，理论物理研究所）；Institut de Mathématiques et de Sciences Physiques (IMSP), Université d’Abomey Calavi (UAC)（数学与物理科学研究所（IMSP），阿博米-卡利维大学（UAC））；IMT School for Advanced Studies Lucca（卢卡高等研究院）；École Normale Supérieure de Natitingou, Université Nationale des Sciences, Technologies, Ingénierie et Mathématiques d’Abomey（纳蒂廷古师范学院，阿博美国立科学技术工程与数学大学）
Using algorithms of Higuchi and of Grassberger and Procaccia, we study numerically how fractal dimensions cross over from finite-dimensional Brownian noise at short time scales to finite values of deterministic chaos at longer time scales for data generated from a Langevin equation that has a strange attractor in the limit of zero noise. Our results suggest that the crossover occurs at such short time scales that there is little chance of finite-dimensional Brownian noise being incorrectly identified as deterministic chaos.
In recent years there has been a considerable increase in the publishing of textbooks and monographs covering what was formerly known as random or irregular deterministic motion, now named by the more fashionable term of deterministic chaos. There is still substantial interest in a matter that is included in many graduate and even undergraduate courses on classical mechanics. Based on the Hamiltonian formalism, the main objective of this article is to provide, from the physicist's point of view, an overall and intuitive review of this broad subject (with some emphasis on the KAM theorem and th…
This brief report presents a master-slave scheme to demonstrate how control chaos in the Braiman and Goldhirsch method for the one- dimensional map system. The scheme also naturally explain why the anomalous response arises in a periodically perturbed, unimodal map system.
A semiclassical analysis based on concepts developed in quantum chaos reveals that anomalous magneto-oscillations in quasi two-dimensional systems with spin-orbit interaction reflect the non-adiabatic spin precession of a classical spin vector along the cyclotron orbits.
Institute of Risk Analysis, Prediction and Management (Risks-X), Academy for Advanced Interdisciplinary Sciences, Southern University of Science and Technology（南方科技大学前沿交叉科学研究院风险分析与预测管理研究所（Risks-X））
It is shown, that the both angular CMB Doppler spectrum: $C_l$ (Planck space telescope - cosmic microwave background [1]) and the 3D galaxy-galaxy power spectrum: $P(k)$ (Sloan Digital Sky Survey SDSS-II [2]), exhibit a considerable range with an exponential decay: $370 < l < 2500$ and $0.05 < k < 0.27~~(h/Mpc)$, respectively. The rates of the exponential decay are $l_c \simeq 300$ for $C_l \sim \exp-(l/l_c)$ and $k_c \simeq 0.09~~(h/Mpc)$ for $P(k) \sim \exp-(k/k_c)$. A waviness is observed along a straight line representing the exponential decay in the log-linear scale graphs of these spectr…
