Chaos: A tutorial for engineers

A new method for studying the problem of emergence of chaos in dynamical systems, the right-hand sides of which are continuously differentiable functions, is proposed. The application of obtained results to the investigation of chaotic dynamical systems without homoclinic or heteroclinic orbits and also without use of the concept of equilibrium points (for example, it can be memristive systems) is shown. In addition, the offered method can be used as a tool for search of hidden chaotic attractors.

We have seen briefly in the previous chapter how tori and so-called "strange attractors" in phase space may provide the underlying mathematical representations of the processes involved in the creation of turbulence. Stationary and temporally periodic solutions, which we have covered in depth in Chapters 2 to 14, provide the simplest examples of attractors--the phase space limits of model trajectories--and their properties are relatively easy to describe analytically. However, with the need to consider more complicated temporal flows when modeling the transition to turbulence, it becomes essential to define quantitative measures for characterizing and categorizing the various types of attractors. In this chapter we introduce the Lyapunov exponents (Oseledec, 1968), the Lyapunov dimension (Kaplan and Yorke, 1979), and the correlation dimension (Grassberger and Procaccia, 1983a,b). These quantities give information about the attractor structure, solution transitions, and predictability characteristics of nonlinear models. We investigate how these quantitative measures of attractor dynamics might be helpful in distinguishing different types of chaotic attractors, and thereby possibly distinguishing some of the different routes to chaos. Moreover, because the seventh modeling principle suggests that a model must have sufficient complexity to represent a phenomenon in adequate detail, these quantitative measures may help us evaluate the adequacy of a particular model. Specifically, we illustrate the utility of the Lyapunov and correlation dimensions in characterizing attractors in several low-order spectral models, including the Lorenz (1963) model, the seven-coefficient generalized Lorenz system of Chang and Shirer (1984), and a related eleven-coefficient model.

Multiwavelets have recently been used for analysis of transient oscillations. Here we use them for transient oscillation detection by incorporating them into two existing Morlet-based self-normalizing detectors, one that operates on a single time series and another on two correlated time series. The Morlet-based detectors are effective in detecting weak, transient oscillations, especially in strong clutter where FFT based methods fail. Simulated receiver operating curves reveal that the new multiwavelet detectors show marked improvement when applied to complex, transient oscillations in white Gaussian noise and 60 dB clutter. At 0.1% false alarm rate for a (signal on) signal to noise ratio of 0.25, detection rates are: 95.5% and 99.6% for the single and dual time series detectors, respectively. These compare to 75% and 95.5% for the Morlet wavelet detectors.

We apply a dimensional analysis to a neural network model with chaotic dynamics, or a mathematical model of chaotic neural networks. First, Lyapunov spectra are examined to confirm existence of chaotic network behavior and to calculate the Lyapunov dimension. Then, the correlation dimension is estimated in the following three cases: (1) reconstruction from single variable time series, (2) observation in state subspaces and (3) observation of the macroscopic average activity. The results show good correlation dimension obtained by observation in state subspaces and the characteristics of the correlation dimension estimated by macroscopic observation.

Prolactin (PRL) circadian profiles were analyzed using methods of nonlinear dynamics, directly from the experimental data, by combining in a single time-series (432 measurements), six individual 24-hour PRL profiles (72 measurements per profile, sampling interval = 20 min), obtained from young healthy human volunteers (4 males, 2 females), under basal conditions. Significant autocorrelation exists between any given point of the time series and a limited number of its successors. Fourier analysis showed a dominant frequency of 1 cycle/24 h, without sub-24-hour harmonics. Poincaré section indicated the presence of a fractal attractor and a sketch of the attractor revealed a highly convoluted geometric structure with a conical contour. The box-counting dimension (D0), information dimension (D1) and correlation dimension (D2) of the attractor had low, fractal values, did not differ significantly from each other, and exhibited saturation at an embedding dimension of 2. The evidence taken together suggests that, under basal conditions, the daily changes in the peripheral blood levels of PRL are governed by nonlinear deterministic dynamics, with a dominant rhythm of 1 cycle/day mixed with a higher-frequency, low-amplitude signal.

We study how spatiotemporal chaos in dynamical systems can be controlled by stochastically returning them to their initial conditions. Focusing on discrete nonlinear maps, we analyze how key measures of chaos-the Lyapunov exponent and butterfly velocity, which quantify sensitivity to initial perturbations and the ballistic spread of information, respectively-are reduced by stochastic resetting. We identify a critical resetting rate that induces a dynamical phase transition, characterized by the simultaneous vanishing of the Lyapunov exponent and butterfly velocity, effectively arresting the spread of information. These theoretical predictions are validated and illustrated with numerical simulations of the celebrated logistic map and its lattice extension. Beyond discrete maps, our findings are applicable to virtually any chaotic extended classical many-body system.

This chapter introduces the basic concepts of dynamical systems theory and several basic mathematical methods for controlling chaos. The main goal of this chapter is to provide an introduction to and a summary of the theory of dynamical systems, with particular emphasis on fractal theory, chaos theory, and chaos control. We first define what is meant by a dynamical system, then we define an attractor, and then the concept of the fractal dimension of a geometrical object. We also define the Lyapunov exponents as a measure of the chaotic behavior of a dynamical system. On the other hand, the fractal dimension can be used to classify geometrical objects because it measures the complexity of an object. The chapter also describes mathematical methods for controlling chaos in dynamic systems. These methods can be used to control a real dynamic system; however, due to efficiency and accuracy requirements we were forced to use fuzzy logic to model the uncertainty, which is present when numerical simulations are performed. We also describe in this chapter a new theory of chaos using fuzzy logic techniques. Chaotic behavior in nonlinear dynamical systems is very difficult to detect and control. Part of the problem is that mathematical results for chaos are difficult to use in many cases, and even if one could use them there is an underlying uncertainty in the accuracy of the numerical simulations of the dynamical systems. For this reason, we can model the uncertainty of detecting the range of values where chaos occurs, using fuzzy set theory. Using fuzzy sets, we can build a theory of fuzzy chaos, where we can use fuzzy sets to describe the behaviors of a system. We illustrate our approach with two cases: Chua’s circuit and Duffing’s oscillator.

Analysis and characterization of complex spatio-temporal patterns in nonlinear reaction-diffusion systems

On the evidence for low-dimensional chaos in an epileptic electroencephalogram

This chapter introduces the basic concepts of dynamical systems theory, and several basic mathematical methods for controlling chaos. The main goal of this chapter is to provide an introduction to and a summary to the theory of dynamical systems with particular emphasis on fractal theory, chaos theory, and chaos control. We first define what is meant by a dynamical system, then we define an attractor, and the concept of the fractal dimension of a geometrical object. Also, we define the Lyapunov exponents as a measure of the chaotic behavior of a dynamical system. On the other hand, the fractal dimension can be used to classify geometrical objects because it measures the complexity of an object. We finish the chapter by reviewing mathematical methods for controlling chaos in dynamic systems. These methods can be used to control a real dynamic system, however, due to efficiency and accuracy requirement we will be forced to consider more advanced methods in the following chapters.KeywordsFractal DimensionPeriodic OrbitLyapunov ExponentChaotic SystemChaotic BehaviorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nonlinear MHD waves forced into a bounded cavity exhibit structural instability of their equilibria and periodic orbits as the forcing varies. For some values of the forcing the computed Poincaré maps, Lyapunov exponents and phase-space correlation reveal that the waves become chaotic; the Haussdorff dimension of the waves’ attracting set lies between its correlation dimension (≈ 2.2) and its information dimension (≈ 2.25).

It is well known that the structural phase transitions in ferroelectric materials are connected with strong nonlinear properties. So we should expect all features of nonlinear dynamical systems such as period-doubling-cascades and chaotic behaviour in a dynamical system that contains ferroelectric materials. The experimentally investigated nonlinear dynamical system is a series resonance circuit, consisting of a linear inductance and a nonlinear capacitance. Different ferroelectric materials have been used as a nonlinear capacitance. There are two directions for our investigations: (i) The response-function of the nonlinear dynamical system and analysis methods of the nonlinear dynamics for dissipative chaotic systems (e. g. phase portrait, Fourier spectra, correlation dimension and Lyapunov exponents) are used for the study of structural phase-transitions and for the characterization of the large signal behaviour of ferroelectric materials. (ii) Controlling the chaotic behaviour of the system by applying a small perturbation of an external control parameter to it. This way the chaotic motion of the system can be switched over to a regular one by stabilizing one of the many regular instable states of the system.

Dynamics of pressure oscillations in flow boiling and condensation in the minichannel

Chaos and spectral analyses of heart rate variability during head-up tilting in essential hypertension

Over large areas, land cover classification has conventionally been undertaken using satellite time series. Typically temporal metric percentiles derived from single pixel location time series have been used to take advantage of spectral differences among land cover classes over time and to minimize the impact of missing observations. Deep convolutional neural networks (CNNs) have demonstrated potential for land cover classification of single date images. However, over large areas and using time series their application is complicated because they are sensitive to missing observations and they may misclassify small and spatially fragmented surface features due to their spatial patch-based implementation. This study demonstrates, for the first time, a one-dimensional (1D) CNN single pixel time series land classification approach that uses temporal percentile metrics and that does not have these issues. This is demonstrated for all the Conterminous United States (CONUS) considering two different 1D CNN structures with 5 and 8 layers, respectively. CONUS 30 m land cover classifications were derived using all the available Landsat-5 and -7 imagery over a seven-month growing season in 2011 with 3.3 million 30 m land cover class labelled samples extracted from the contemporaneous CONUS National Land Cover Database (NLCD) 16 class land cover product. The 1D CNNs and, a conventional random forest model, were trained using 10%, 50% and 90% samples, and the classification accuracies were evaluated with an independent 10% proportion. Temporal metrics were classified using 5, 7 and 9 percentiles for each of five Landsat reflective wavelength bands and their eight band ratios. The CONUS and detailed 150 × 150 km classification results demonstrate that the approach is effective at scale and locally. The 1D CNN classification land cover class boundaries were preserved for small axis dimension features, such as roads and rivers, with no stripes or anomalous spatial patterns. The 8-layer 1D CNN provided the highest overall classification accuracies and both the 5-layer and 8-layer 1D CNN architectures provided higher accuracies than the random forest by 1.9% - 2.8% which as all the accuracies were > 83% is a meaningful increase. The CONUS overall classification accuracies increased marginally with the number of percentiles (86.21%, 86.40%, and 86.43% for 5, 7 and 9 percentiles, respectively) using the 8-layer 1D-CNN. Class specific producer and user accuracies were quantified, with lower accuracies for the developed land, crop and pasture/hay classes, but no systematic pattern among classes with respect to the number of temporal percentiles used. Application of the trained model to a different year of CONUS Landsat ARD showed moderately decreased accuracy (80.79% for 7 percentiles) that we illustrate is likely due to different intra-annual surface variations between years. These encouraging results are discussed with recommended research for deep learning using temporal metric percentiles.