Coherence and stochastic resonance in threshold crossing detectors with delayed feedback

We consider the dynamical behavior of threshold systems driven by external periodic and stochastic signals and internal delayed feedback. Specifically, the effect of positive delayed feedback on the sensitivity of a threshold crossing detector (TCD) to periodic forcing embedded in noise is investigated. The system has an intrinsic ability to oscillate in the presence of positive feedback. We first show conditions under which such reverberatory behavior is enhanced by noise, which is a form of coherence resonance (CR) for this system. Further, for input signals that are subthreshold in the absence of feedback, the open-loop stochastic resonance (SR) characteristic can be sharply enhanced by positive delayed feedback. This enhancement is shown to depend on the stimulus period, and is maximal when this period is matched to an integer multiple of the delay. Reverberatory oscillations, which are particularly prominent after the offset of periodic forcing, are shown to be eliminated by a summing network of such TCDs with local delayed feedback. Theoretical analysis of the crossing rate dynamics qualitatively accounts for the existence of CR and the resonant behavior of the SR effect as a function of delay and forcing frequency.     

Stochastic Perturbations to Dynamical Systems: A Response Theory Approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.     

Analytic approximations of statistical quantities and response of noisy oscillators

Nonlinear oscillators have been utilized in many contexts because they encompass a large class of phenomena. For a reduced phase oscillator model with weak noise forcing that is necessarily multiplicative, we provide analytic formulas for the stationary statistical quantities of the random period. This is an important quantity which we term ‘response’ (i.e., the spike times, instantaneous frequency in neuroscience, the cycle time in chemical reactions, etc.) that is often analytically intractable in noisy oscillator systems. The analytic formulas are accurate in the weak noise limit so that one does not have to numerically solve a time-varying Fokker-Planck equation. The steady-state and dynamic responses are also analyzed with deterministic forcing. A second order analytic formula is derived for the steady-state response, whereas the dynamic response with time-varying forcing is more complicated. We focus on the specific case where the forcing is sinusoidal and accurately capture the frequency response with an analytic approximation that is obtained with a rescaling of the equation. By utilizing various techniques in the weak noise regime, this work leads to a better understanding of how the random period of nonlinear oscillators are affected by multiplicative noise and external forcing. Comparisons of the asymptotic formulas with a full oscillator system confirm the qualitative accurateness of the theory.     

In-out intermittency in partial differential equation and ordinary differential equation models

We find concrete evidence for a recently discovered form of intermittency, referred to as in-out intermittency, in both partial differential equation (PDE) and ordinary differential equation (ODE) models of mean field dynamos. This type of intermittency [introduced in P. Ashwin, E. Covas, and R. Tavakol, Nonlinearity 9, 563 (1999)] occurs in systems with invariant submanifolds and, as opposed to on-off intermittency which can also occur in skew product systems, it requires an absence of skew product structure. By this we mean that the dynamics on the attractor intermittent to the invariant manifold cannot be expressed simply as the dynamics on the invariant subspace forcing the transverse dynamics; the transverse dynamics will alter that tangential to the invariant subspace when one is far enough away from the invariant manifold. Since general systems with invariant submanifolds are not likely to have skew product structure, this type of behavior may be of physical relevance in a variety of dynamical settings. The models employed here to demonstrate in-out intermittency are axisymmetric mean-field dynamo models which are often used to study the observed large-scale magnetic variability in the Sun and solar-type stars. The occurrence of this type of intermittency in such models may be of interest in understanding some aspects of such variabilities. (c) 2001 American Institute of Physics.     

The invariant density of a chaotic dynamical system with small noise

We consider a chaotic dynamical system perturbed by noise and calculate an approximate invariant density when the noise level is small. Because of the special structure of the dynamical system, the effective support of the invariant density is much smaller than the noiseless attractor. This behavior is captured by the asymptotic form of the invariant density, which is given explicitly.     

Chaotic motion of the dynamical system under both additive and multiplicative noise excitations

With both additive and multiplicative noise excitations, the effect on the chaotic behaviour of the dynamical system is investigated in this paper. The random Melnikov theorem with the mean-square criterion that applies to a type of dynamical systems is analysed in order to obtain the conditions for the possible occurrence of chaos. As an example, for the Duffing system, we deduce its concrete expression for the threshold of multiplicative noise amplitude for the rising of chaos, and by combining figures, we discuss the influences of the amplitude, intensity and frequency of both bounded noises on the dynamical behaviour of the Duffing system separately. Finally, numerical simulations are illustrated to verify the theoretical analysis according to the largest Lyapunov exponent and Poincaré map.     

(Non)Invariance of Dynamical Quantities for Orbit Equivalent Flows

We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are shown to either remain invariant, transform according to a multiplicative factor or transform through a convoluted dependence that may take the form of an integral over the initial local values. We discuss the significance of these results for the apparent non-invariance of chaos in general relativity and explore applications to the synchronization of equilibrium states and the elimination of expansions.     

Preserving synchronization under matrix product modifications

In this article we present a methodology under which stability and synchronization of a dynamical master/slave system configuration are preserved under modification through matrix multiplication. The objective is to show that under a defined multiplicative group, hyperbolic critical points are preserved along the stable and unstable manifolds. The properties of this multiplicative group were determined through the use of simultaneous Jordan decomposition. It is also shown that a consequence of this approach is the preservation of the signature of the Jacobian matrix associated with the dynamical system. To illustrate the results we present several examples of different modified systems.     

3D Stochastic Primitive Equations of the Large-Scale Ocean: Global Well-Posedness and Attractors

In this paper, we consider the global well-posedness and long-time dynamics for the three-dimensional viscous primitive equations describing the large-scale oceanic motion under a random forcing, which is an additive white in time noise. We firstly prove the existence and uniqueness of global strong solutions to the initial boundary value problem for the stochastic primitive equations. Subsequently, by studying the asymptotic behavior of strong solutions, we obtain the existence of random attractors for the corresponding random dynamical system.     

四维相空间中非中心势动力学系统的Poisson 结构和Casimir 函数

将非中心势动力学系统的运动微分方程写成Ermakov形式，得到Ermakov不变量. 运用Hamilton理论，把Ermakov不变量当作Hamiltonian 函数，在四维相空间中建立了非中心势动力学系统的Poisson 结构。结果表明：此Poisson 结构是一退化的结构，而系统具有四个不变量，即Hamiltonian 函数，Ermakov不变量及两个Casimir函数。     

Bifurcations of a lattice gas flow under external forcing

We study the behavior of a Frisch-Hasslacher-Pomeau lattice gas automaton under the effect of a spatially periodic forcing. It is shown that the lattice gas dynamics reproduces the steady-state features of the bifurcation pattern predicted by a properly truncated model of the Navier-Stokes equations. In addition, we show that the dynamical evolution of the instabilities driving the bifurcation can be modeled by supplementing the truncated Navier-Stokes equation with a random force chosen on the basis of the automaton noise.     

非线性热弹耦合Sine-Gordon型系统的整体吸引子

研究受Peierls-Nabarro力作用的非线性热弹耦 合Sine-Gordon型系统的动力行为.利用算子半群理论证明了 在一定的初边界条件下系统存在连续解, 利用算子半群分解技巧构造了渐近紧的不变吸收集, 进而证明了系统存在整体吸引子.     

Direct transition from a stable equilibrium to quasiperiodicity in non-smooth systems

The purpose of this Letter is to show how a border-collision bifurcation in a piecewise-smooth dynamical system can produce a direct transition from a stable equilibrium point to a two-dimensional invariant torus. Considering a system of nonautonomous differential equations describing the behavior of a power electronic DC/DC converter, we first determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in phase space. When this happens, one can observe a number of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 cycle. Another is the transformation of the stable equilibrium into an unstable period-1 cycle with complex conjugate multipliers, and the associated formation of a two-dimensional (ergodic or resonant) torus.     

Extensive chaos in the Lorenz-96 model

We explore the high-dimensional chaotic dynamics of the Lorenz-96 model by computing the variation of the fractal dimension with system parameters. The Lorenz-96 model is a continuous in time and discrete in space model first proposed by Lorenz to study fundamental issues regarding the forecasting of spatially extended chaotic systems such as the atmosphere. First, we explore the spatiotemporal chaos limit by increasing the system size while holding the magnitude of the external forcing constant. Second, we explore the strong driving limit by increasing the external forcing while holding the system size fixed. As the system size is increased for small values of the forcing we find dynamical states that alternate between periodic and chaotic dynamics. The windows of chaos are extensive, on average, with relative deviations from extensivity on the order of 20%. For intermediate values of the forcing we find chaotic dynamics for all system sizes past a critical value. The fractal dimension exhibits a maximum deviation from extensivity on the order of 5% for small changes in system size and the deviation from extensivity decreases nonmonotonically with increasing system size. The length scale describing the deviations from extensivity is consistent with the natural chaotic length scale in support of the suggestion that deviations from extensivity are due to the addition of chaotic degrees of freedom as the system size is increased. We find that each wavelength of the deviation from extensive chaos contains on the order of two chaotic degrees of freedom. As the forcing is increased, at constant system size, the dimension density grows monotonically and saturates at a value less than unity. We use this to quantify the decreasing size of chaotic degrees of freedom with increased forcing which we compare with spatial features of the patterns.     

Dynamic control and information processing in chemical reaction systems by tuning self-organization behavior

Specific external control of chemical reaction systems and both dynamic control and signal processing as central functions in biochemical reaction systems are important issues of modern nonlinear science. For example nonlinear input-output behavior and its regulation are crucial for the maintainance of the life process that requires extensive communication between cells and their environment. An important question is how the dynamical behavior of biochemical systems is controlled and how they process information transmitted by incoming signals. But also from a general point of view external forcing of complex chemical reaction processes is important in many application areas ranging from chemical engineering to biomedicine. In order to study such control issues numerically, here, we choose a well characterized chemical system, the CO oxidation on Pt(110), which is interesting per se as an externally forced chemical oscillator model. We show numerically that tuning of temporal self-organization by input signals in this simple nonlinear chemical reaction exhibiting oscillatory behavior can in principle be exploited for both specific external control of dynamical system behavior and processing of complex information.     

On the Recurrence and Robust Properties of Lorenz’63 Model

Lie-Poisson structure of the Lorenz’63 system gives a physical insight on its dynamical and statistical behavior considering the evolution of the associated Casimir functions. We study the invariant density and other recurrence features of a Markov expanding Lorenz-like map of the interval arising in the analysis of the predictability of the extreme values reached by particular physical observables evolving in time under the Lorenz’63 dynamics with the classical set of parameters. Moreover, we prove the statistical stability of such an invariant measure. This will allow us to further characterize the SRB measure of the system.     

THE UNIVERSAL TRANSITION OF PLATEAU STRUCTURE OF LYAPUNOV EXPONENTS

The universal transition of Lyapunov exponents between conservative limit and dissipa-tire limit of nonlinear dynamical system is studied. It is discovered numerically and proved analytically that for homogeneous dissipative two-dimensional maps, along the equal dissi-pation line in parameter space, the Lyapunov exponents of attractor orbits possess a plateau structure and strict symmetry about its plateau value, The ratios between the plateau width and the stable window width of period 1-4 orbits for Henon map are calculated. The result shows that the plateau structure of Lyapunov exponents remains invariant for the attractor orbits belonging to a period doubling bifurcation sequence. This fact reveals a new universal transition behavior between order and chaos when the dissipation of the dynamical system is weakened to zero.     

Heteroclinic synchronization: ultrasubharmonic locking

According to the traditional view of synchronization, a weak periodic input is able to lock a nonlinear oscillator at a frequency close to that of the input (1:1 zone). If the forcing increases, it is possible to achieve synchronization at subharmonic bands also. Using a competitive dynamical system we show the inverse phenomenon: with a weak signal the 1:1 zone is narrow, but the synchronization of ultrasubharmonics is dominant. In the system's phase space, there exists a heteroclinic contour in the autonomous regime, which is the image of sequential dynamics. Under the action of a weak periodic forcing, in the vicinity of the contour a stable limit cycle with long period appears. This results in the locking of very low-frequency oscillations with the finite frequency of the forcing. We hypothesize that this phenomenon can be the origin for the synchronization of slow and fast brain rhythms.     

Defining quantum dynamical entropy

We propose an elementary definition of the dynamical entropy for a discrete-time quantum dynamical system. We apply our construction to classical dynamical systems and to the shift on a quantum spin chain. In the first case, we recover the Kolmogorov-Sinai invariant and, for the second, we find the mean entropy of the invariant state plus the logarithm of the dimension of the single-spin space.