Taming Chaotic Circuits

Control algorithms that exploit chaotic behavior can vastly improve the performance of many practical and useful systems. The program Perfect Moment is built around a collection of such techniques. It autonomously explores a dynamical system's behavior, using rules embodying theorems and definitions from nonlinear dynamics to zero in on interesting and useful parameter ranges and state-space regions. It then constructs a reference trajectory based on that information and causes the system to follow it. This program and its results are illustrated with several examples, among them the phase-locked loop, where sections of chaotic attractors are used to increase the capture range of the circuit.

Parameter Estimation in Chaotic Systems

This report examines how to estimate the parameters of a chaotic system given noisy observations of the state behavior of the system. Investigating parameter estimation for chaotic systems is interesting because of possible applications for high-precision measurement and for use in other signal processing, communication, and control applications involving chaotic systems. In this report, we examine theoretical issues regarding parameter estimation in chaotic systems and develop an efficient algorithm to perform parameter estimation. We discover two properties that are helpful for performing parameter estimation on non-structurally stable systems. First, it turns out that most data in a time series of state observations contribute very little information about the underlying parameters of a system, while a few sections of data may be extraordinarily sensitive to parameter changes. Second, for one-parameter families of systems, we demonstrate that there is often a preferred direction in parameter space governing how easily trajectories of one system can "shadow'" trajectories of nearby systems. This asymmetry of shadowing behavior in parameter space is proved for certain families of maps of the interval. Numerical evidence indicates that similar results may be true for a wide variety of other systems. Using the two properties cited above, we devise an algorithm for performing parameter estimation. Standard parameter estimation techniques such as the extended Kalman filter perform poorly on chaotic systems because of divergence problems. The proposed algorithm achieves accuracies several orders of magnitude better than the Kalman filter and has good convergence properties for large data sets.

[Ressenya del llibre Power/resistance: local politics and the chaotic state d’Andrew Kirby]

Ressenya del llibre Power/resistance: local politics and the chaotic state. Es tracta d’una reflexió serena i meditada sobre l’essència i la presència de -i resistència a- l’estat en les societats occidentals contemporànies des d’una perspectiva espacial

Dynamic scaling of bred vectors in spatially extended chaotic systems

We unfold a profound relationship between the dynamics of finite-size perturbations in spatially extended chaotic systems and the universality class of Kardar-Parisi-Zhang (KPZ). We show how this relationship can be exploited to obtain a complete theoretical description of the bred vectors dynamics. The existence of characteristic length/time scales, the spatial extent of spatial correlations and how to time it, and the role of the breeding amplitude are all analyzed in the light of our theory. Implications to weather forecasting based on ensembles of initial conditions are also discussed.

Evidence for the chaotic origin of Northern Annular Mode variability

Exponential spectra are found to characterize variability of the Northern Annular Mode (NAM) for periods less than 36 days. This corresponds to the observed rounding of the autocorrelation function at lags of a few days. The characteristic persistence timescales during winter and summer is found to be ∼5 days for these high frequencies. Beyond periods of 36 days the characteristic decorrelation timescale is ∼20 days during winter and ∼6 days in summer. We conclude that the NAM cannot be described by autoregressive models for high frequencies; the spectra are more consistent with low-order chaos. We also propose that the NAM exhibits regime behaviour, however the nature of this has yet to be identified.

A closer look at chaotic advection in the stratosphere: part I: geometric structure

The relevance of chaotic advection to stratospheric mixing and transport is addressed in the context of (i) a numerical model of forced shallow-water flow on the sphere, and (ii) a middle-atmosphere general circulation model. It is argued that chaotic advection applies to both these models if there is suitable large-scale spatial structure in the velocity field and if the velocity field is temporally quasi-regular. This spatial structure is manifested in the form of “cat’s eyes” in the surf zone, such as are commonly seen in numerical simulations of Rossby wave critical layers; by analogy with the heteroclinic structure of a temporally aperiodic chaotic system the cat’s eyes may be thought of as an “organizing structure” for mixing and transport in the surf zone. When this organizing structure exists, Eulerian and Lagrangian autocorrelations of the velocity derivatives indicate that velocity derivatives decorrelate more rapidly along particle trajectories than at fixed spatial locations (i.e., the velocity field is temporally quasi-regular). This phenomenon is referred to as Lagrangian random strain.

A closer look at chaotic advection in the stratosphere: part II: statistical diagnostics

Statistical diagnostics of mixing and transport are computed for a numerical model of forced shallow-water flow on the sphere and a middle-atmosphere general circulation model. In particular, particle dispersion statistics, transport fluxes, Liapunov exponents (probability density functions and ensemble averages), and tracer concentration statistics are considered. It is shown that the behavior of the diagnostics is in accord with that of kinematic chaotic advection models so long as stochasticity is sufficiently weak. Comparisons with random-strain theory are made.

Chaotic mixing and transport in Rossby-wave critical layers

A simple, dynamically consistent model of mixing and transport in Rossby-wave critical layers is obtained from the well-known Stewartson–Warn–Warn (SWW) solution of Rossby-wave critical-layer theory. The SWW solution is thought to be a useful conceptual model of Rossby-wave breaking in the stratosphere. Chaotic advection in the model is a consequence of the interaction between a stationary and a transient Rossby wave. Mixing and transport are characterized separately with a number of quantitative diagnostics (e.g. mean-square dispersion, lobe dynamics, and spectral moments), and with particular emphasis on the dynamics of the tracer field itself. The parameter dependences of the diagnostics are examined: transport tends to increase monotonically with increasing perturbation amplitude whereas mixing does not. The robustness of the results is investigated by stochastically perturbing the transient-wave phase speed. The two-wave chaotic advection model is contrasted with a stochastic single-wave model. It is shown that the effects of chaotic advection cannot be captured by stochasticity alone.

Chaotic destruction of Anderson localization in a nonlinear lattice

We consider a scattering problem for a nonlinear disordered lattice layer governed by the discrete nonlinear Schrodinger equation. The linear state with exponentially small transparency, due to the Anderson localization, is followed for an increasing nonlinearity, until it is destroyed via a bifurcation. The critical nonlinearity is shown to decay with the lattice length as a power law. We demonstrate that in the chaotic regimes beyond the bifurcation the field is delocalized and this leads to a drastic increase of transparency. Copyright (C) EPLA, 2008

The chaotic frequency hopping signals for MIMO radar

In multiple-input multiple-output (MIMO) radar systems, the transmitters emit orthogonal waveforms to increase the spatial resolution. New frequency hopping (FH) codes based on chaotic sequences are proposed. The chaotic sequences have the characteristics of good encryption, anti-jamming properties and anti-intercept capabilities. The main idea of chaotic FH is based on queuing theory. According to the sensitivity to initial condition, these sequences can achieve good Hamming auto-correlation while also preserving good average correlation. Simulation results show that the proposed FH signals can achieve lower autocorrelation side lobe level and peak cross-correlation level with the increasing of iterations. Compared to the LFM signals, this sequence has higher range-doppler resolution.

A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor

In this paper we consider the strongly damped wave equation with time-dependent terms u(tt) - Delta u - gamma(t)Delta u(t) + beta(epsilon)(t)u(t) = f(u), in a bounded domain Omega subset of R(n), under some restrictions on beta(epsilon)(t), gamma(t) and growth restrictions on the nonlinear term f. The function beta(epsilon)(t) depends on a parameter epsilon, beta(epsilon)(t) -> 0. We will prove, under suitable assumptions, local and global well-posedness (using the uniform sectorial operators theory), the existence and regularity of pullback attractors {A(epsilon)(t) : t is an element of R}, uniform bounds for these pullback attractors, characterization of these pullback attractors and their upper and lower semicontinuity at epsilon = 0. (C) 2010 Elsevier Ltd. All rights reserved.

Global attractor for a model of extensible beam with nonlinear damping and source terms

This paper is concerned with the existence of a global attractor for the nonlinear beam equation, with nonlinear damping and source terms, u(tt) + Delta(2)u -M (integral(Omega)vertical bar del u vertical bar(2)dx) Delta u + f(u) + g(u(t)) = h in Omega x R(+), where Omega is a bounded domain of R(N), M is a nonnegative real function and h is an element of L(2)(Omega). The nonlinearities f(u) and g(u(t)) are essentially vertical bar u vertical bar(rho) u - vertical bar u vertical bar(sigma) u and vertical bar u(t)vertical bar(r) u(t) respectively, with rho, sigma, r > 0 and sigma < rho. This kind of problem models vibrations of extensible beams and plates. (C) 2010 Elsevier Ltd. All rights reserved.

Regularity of solutions on the global attractor for a semilinear damped wave equation

We consider attractors A(eta), eta epsilon [0, 1], corresponding to a singularly perturbed damped wave equation u(tt) + 2 eta A(1/2)u(t) + au(t) + Au = f (u) in H-0(1)(Omega) x L-2 (Omega), where Omega is a bounded smooth domain in R-3. For dissipative nonlinearity f epsilon C-2(R, R) satisfying vertical bar f ``(s)vertical bar <= c(1 + vertical bar s vertical bar) with some c > 0, we prove that the family of attractors {A(eta), eta >= 0} is upper semicontinuous at eta = 0 in H1+s (Omega) x H-s (Omega) for any s epsilon (0, 1). For dissipative f epsilon C-3 (R, R) satisfying lim(vertical bar s vertical bar) (->) (infinity) f ``(s)/s = 0 we prove that the attractor A(0) for the damped wave equation u(tt) + au(t) + Au = f (u) (case eta = 0) is bounded in H-4(Omega) x H-3(Omega) and thus is compact in the Holder spaces C2+mu ((Omega) over bar) x C1+mu((Omega) over bar) for every mu epsilon (0, 1/2). As a consequence of the uniform bounds we obtain that the family of attractors {A(eta), eta epsilon [0, 1]} is upper and lower semicontinuous in C2+mu ((Omega) over bar) x C1+mu ((Omega) over bar) for every mu epsilon (0, 1/2). (c) 2007 Elsevier Inc. All rights reserved.

Chaotic phase synchronization and desynchronization in an oscillator network for object selection

Object selection refers to the mechanism of extracting objects of interest while ignoring other objects and background in a given visual scene. It is a fundamental issue for many computer vision and image analysis techniques and it is still a challenging task to artificial Visual systems. Chaotic phase synchronization takes place in cases involving almost identical dynamical systems and it means that the phase difference between the systems is kept bounded over the time, while their amplitudes remain chaotic and may be uncorrelated. Instead of complete synchronization, phase synchronization is believed to be a mechanism for neural integration in brain. In this paper, an object selection model is proposed. Oscillators in the network representing the salient object in a given scene are phase synchronized, while no phase synchronization occurs for background objects. In this way, the salient object can be extracted. In this model, a shift mechanism is also introduced to change attention from one object to another. Computer simulations show that the model produces some results similar to those observed in natural vision systems.

Chaotic synchronization in general network topology for scene segmentation

Chaotic synchronization has been discovered to be an important property of neural activities, which in turn has encouraged many researchers to develop chaotic neural networks for scene and data analysis. In this paper, we study the synchronization role of coupled chaotic oscillators in networks of general topology. Specifically, a rigorous proof is presented to show that a large number of oscillators with arbitrary geometrical connections can be synchronized by providing a sufficiently strong coupling strength. Moreover, the results presented in this paper not only are valid to a wide class of chaotic oscillators, but also cover the parameter mismatch case. Finally, we show how the obtained result can be applied to construct an oscillatory network for scene segmentation.