Use of correlation dimension function in dynamic systems

Be it a physical object or a mathematical model, a nonlinear dynamical system can display complicated aperiodic behavior, or "chaos." In many cases, this chaos is associated with motion on a strange attractor in the system's phase space. And the dimension of the strange attractor indicates the effective number of degrees of freedom in the dynamical system. In this thesis, we investigate numerical issues involved with estimating the dimension of a strange attractor from a finite time series of measurements on the dynamical system. Of the various definitions of dimension, we argue that the correlation dimension is the most efficiently calculable and we remark further that it is the most commonly calculated. We are concerned with the practical problems that arise in attempting to compute the correlation dimension. We deal with geometrical effects (due to the inexact self-similarity of the attractor), dynamical effects (due to the nonindependence of points generated by the dynamical system that defines the attractor), and statistical effects (due to the finite number of points that sample the attractor). We propose a modification of the standard algorithm, which eliminates a specific effect due to autocorrelation, and a new implementation of the correlation algorithm, which is computationally efficient. Finally, we apply the algorithm to chaotic data from the Caltech tokamak and the Texas tokamak (TEXT); we conclude that plasma turbulence is not a low-dimensional phenomenon.

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.

There are numerous phenomena in the space, biological and ecological sciences whose discrete evolution can be effectively modeled by exponentially decaying discrete dynamical systems, which can be represented as follows: Let F : Rn x P → Rn be a smooth map, where Rn is Euclidean n-space representing the state variables x and P is a subset of Rm representing the parameter space comprised of m-vectors denoted by µ . The discrete dynamics of the system is given by the iterates of the map fµ : Rn → Rn , where fµ (x) := F(x,µ), with the exponentially decaying behavior embodied in the property that there exists a positive constant M such that |f µ(x)| ≤ Me -|x| for all (x,µ ) ∈ Rn x P. It is not difficult to show that these exponentially decaying dynamical systems have attracting sets that can have remarkable properties. For example, the attractor may be a strange attractor – a set with non-integer fractal dimension on which the dynamics is chaotic - and variations in the parameter can cause bifurcations that change the fundamental nature of the chaotic regimes. Known results on these strange attractors will be discussed, several new developments will be presented and some challenging open problems will be outlined. AMS Subject Classification: 37C05; 37G10; 37N25; 92D25. Keywords: Bifurcation, chaotic dynamics, discrete dynamical system simulation, fractal dimension, Lyapunov exponents, strange attractors, orbital trajectories, Dynamical Systems, exponentially decaying

The dynamical behaviors of coherent structures in countercurrent axisymmetric shear flows are experimentally studied. The forward velocityU1 and the velocity rationR=(U1-U2)/(U1+U2), whereU2 denotes the suction velocity, are considered as the control parameters. Two kinds of vortex structures, i.e., axisymmetric and helical structures, were discovered with respect to different regimes in theR versusU1 diagram. In the case ofU1 ranging from 3 to 20m/s andR from 1 to 3, the axisymmetric structures play an important role. Based on the dynamical behaviors of axisymmetric structures, a critical forward velocityU1cr=6.8m/s was defined, subsequently, the subcritical velocity regime:U1>U1cr and the supercritical velocity regime:U1 4m/s), correspondingly, the spatial evolution of the temporal asymptotic behavior of a dynamical system can be described as follows: (1) Hopf bifurcation, (2) subharmonic bifurcation, (3) reversed superharmonic bifurcation, (4) superharmonic bifurcation, (5) chaos (“weak turbulence”) in the case ofU1 4m/s). The proposed new terms, superharmonic and reversed superharmonic bifurcations, are characterized of the frequency doubling rather than the period doubling. A kind of unfamiliar vortices referred to as the helical structure was discovered experimentally when the forward velocity around 2m/s and the velocity range from 1.1 to 2.3. There are two base frequencies contained in the flow system and they could coexist as indicated by the Wigner-Ville-Distribution and the temporal asymptotic behavior of the dynamical system corresponding to the helical vortex could be described as 2-torus as indicted by the 3D reconstructed phase trajectory and correlation dimension. The scenario of the spatial evolution of helical structures could be described as follows: the jet column is separated into two parts at a certain spatial location and they entangle each other to form the helical vortex until the occurrence of those separated jet columns to reconnect further downstream with the result that the flow system evolves into turbulence in a catastrophic form. Correspondingly, the dynamical system evolves directly into 2-torus through the supercritical Hopf bifurcation followed by a transition from a quasi-periodic attractor to a strange attractor. In the case ofU1=2m/s, the parametric evolution of the temporal asymptotic behavior of the dynamical system as the velocity ratio increases from 1 to 3 could be described as follows: (1) 2-torus (Hopf bifurcation), (2) limit cycle (reversed Hopf bifurcation), (3) strange attractor (subharmonic bifurcation).

Chaos theory can be used for quantitative dynamics of uncertainty and finding order in its perturbances. Chaos theory is largely a colloquial notion and is referred to as an analysis of non-linear dynamical systems. The studies devoted to linear dynamical systems and theories of complexity involve studies of turbulence, or, more precisely, the transfer from stability to turbulence. The dynamical system by its nature does not follow long-term forecasts. There are two reasons for the unpredictability. Dynamical systems involve both feedback and critical levels. To paraphrase, one can say that a dynamical system forms a system of non-linear feedback. The most important properties of the system include: sensitivity to the change of the initial conditions, occurrence of critical points and a fractal dimension. Non-linear dynamical systems tend to have more than one solution. The case often is that the number of solutions is huge or even infinite. A visual representation of the data forms a finite space called the phase space of the system. The number of dimensions in the space is determined by the number of variable presents in the system. If there are two or three variables, it is possible to visually examine the data. In case of a larger number of dimensions, the data are examined with the use of mathematical data. Another term used in non-linear dynamical systems is the attractor. This is an area of a equilibrium of non-linear system within a time series. A system which aims to achieve an equilibrium in the form of a single value has a point attractor. Besides, there are phase attractors, which form periodic cycles or orbits in space forming a limit cycle. An attractor, in which none of the points in the space overlaps and whose orbits do not cross but both originate in the same area of phase space, is called strange attractor. Such attractors in contrast to point attractors are non-periodic and predominantly have a fractal dimension [10,11].

Most dynamical systems in real-life applications are usually nonlinear and nonstationary, such as biomechanical systems, neuroscience, healthcare, economics, and engineering. Nowadays, the availability of data makes it possible to investigate and discover the hidden dynamics of target systems. However, numerous concerns arise in modeling and analysis of such systems, including complex system structure, data-inherent uncertainty in observations, poor interpretability, etc. To this end, we will investigate dynamical behaviors by proposing data-driven machine learning and dynamical system techniques. Specifically, we present various approaches for extracting characteristics or patterns that have good interpretability in various complex dynamical system applications.In the first study, we consider a pattern recognition problem in biomechanical system using multivariate human kinematic data and propose data-driven analytical methods to model human movement. We first perform recurrence quantification analysis on the phase space of the kinematic system to visualize and evaluate the recurrence behavior of the folded phase space trajectory. However, this method could miss out the relationship between variables. We therefore introduce a new modeling approach based on the self-expressive assumption to discover the interactions between variables and characterize the patterns of human motion. Our model measures correlations between variables by underlying linear subspaces, which can be further extended to nonlinear subspaces. Based on these correlations, our model improves the classification accuracy by 20% - 30% over other standard state-of-art correlation measurements and clustering methods. In the second study, we consider a specific neuroscience problem, seizure detection, to recognize abnormal brain states by learning from highly nonlinear electroencephalography. Based on dynamical systems theory, Koopman operator theory, and dynamic mode decomposition, we project the complex brain dynamics into an infinite-dimensional linear space, where the intrinsic linear dynamic shares the same dynamical behavior as the original nonlinear dynamical system. We propose a new system measurement to unify the projections in different Koopman spaces. Our measurement indicates the stability of the dynamical system, which represents the system state trajectory with respect to the corresponding attractor. We demonstrate the effectiveness of our approach with a seizure onset detection task. In this case study, our model gives high accuracy with a low false-positive rate. In the last study, we consider dynamical systems with multiple modalities. We design a Koopman-based deep learning network to identify and simplify the intrinsic nonlinear dynamics from different sources/observations of a multi-modal dynamical system. Since these different modalities describe the same system, they jointly share the same intrinsic dynamics. We develop an autoencoder framework with custom loss functions to find proper project functions/operators and to obtain the intrinsic Koopman coordinates. Our framework can reconstruct the original system states based on the joint dynamics. Once the intrinsic linear dynamics is learned, future system states can be predicted by the evolution in the Koopman space. This approach allows us to uncover and linearize the hidden nonlinear dynamics, which enables linear dynamical tools, for example, system control, spectrum analysis, to analyze complex dynamical systems.--Author's abstract

Theoretical modelling of the local ionospheric medium (LIM) is made difficult by the occurrence of irregular ionospheric behaviours at many space and time scales, making prior hypotheses uncertain. Investigating the LIM from scratch with the tools of dynamical system theory may be an option, using the vertical total electron content (vTEC) as an appropriate tracer of the system variability. An embedding procedure is applied to vTEC time series to obtain the finite dimension (m∈N) of the phase space of an LIM-equivalent dynamical system, as well as its correlation dimension (D2) and Kolmogorov entropy rate (K2). In this paper, the dynamical features (m,D2,K2) are studied for the vTEC on the top of three GNSS stations depending on the time scale (τ) at which the vTEC is observed. First, the vTEC undergoes empirical mode decomposition; then (m,D2,K2) are calculated as functions of τ. This captures the multi-scale structure of the Earth’s ionospheric dynamics, demonstrating a net distinction between the behaviour at τ≤24h and τ≥24h. In particular, sub-diurnal-scale modes are assimilated to much more chaotic systems than over-diurnal-scale modes.

Parallel computation of the correlation dimension from a time series

In this paper we investigate numerically the possibility of conversion of a chaotic attractor into a nonchaotic but strange attractor in both a discrete system (an one dimensional map) and in a continuous dynamical system — Bonhoeffer—van der Pol oscillator. In these systems we show suppression of chaotic property, namely, the sensitive dependence on initial states, by adding appropriate i) chaotic signal and ii) Gaussian white noise. The controlled orbit is found to be strange but nonchaotic with largest Lyapunov exponent negative and noninteger correlation dimension. Return map and power spectrum are also used to characterize the strange nonchaotic attractor.

Application of rotational spectrum for correlation dimension estimation

The spatial patterns and processes of cities can be described with various entropy functions. However, spatial entropy always depends on the scale of measurement, and it is difficult to find a characteristic value for it. In contrast, fractal parameters can be employed to characterize scale-free phenomena and reflect the local features of random multi-scaling structure. This paper is devoted to exploring the similarities and differences between spatial entropy and fractal dimension in urban description. Drawing an analogy between cities and growing fractals, we illustrate the definitions of fractal dimension based on different entropy concepts. Three representative fractal dimensions in the multifractal dimension set, capacity dimension, information dimension, and correlation dimension, are utilized to make empirical analyses of the urban form of two Chinese cities, Beijing and Hangzhou. The results show that the entropy values vary with the measurement scale, but the fractal dimension value is stable is method and study area are fixed; if the linear size of boxes is small enough (e.g., <1/25), the linear correlation between entropy and fractal dimension is significant (based on the confidence level of 99%). Further empirical analysis indicates that fractal dimension is close to the characteristic values of spatial entropy. This suggests that the physical meaning of fractal dimension can be interpreted by the ideas from entropy and scaling and the conclusion is revealing for future spatial analysis of cities.

Abstract. Neural networks are able to approximate chaotic dynamical systems when provided with training data that cover all relevant regions of the system's phase space. However, many practical applications diverge from this idealized scenario. Here, we investigate the ability of feed-forward neural networks to (1) learn the behavior of dynamical systems from incomplete training data and (2) learn the influence of an external forcing on the dynamics. Climate science is a real-world example where these questions may be relevant: it is concerned with a non-stationary chaotic system subject to external forcing and whose behavior is known only through comparatively short data series. Our analysis is performed on the Lorenz63 and Lorenz95 models. We show that for the Lorenz63 system, neural networks trained on data covering only part of the system's phase space struggle to make skillful short-term forecasts in the regions excluded from the training. Additionally, when making long series of consecutive forecasts, the networks struggle to reproduce trajectories exploring regions beyond those seen in the training data, except for cases where only small parts are left out during training. We find this is due to the neural network learning a localized mapping for each region of phase space in the training data rather than a global mapping. This manifests itself in that parts of the networks learn only particular parts of the phase space. In contrast, for the Lorenz95 system the networks succeed in generalizing to new parts of the phase space not seen in the training data. We also find that the networks are able to learn the influence of an external forcing, but only when given relatively large ranges of the forcing in the training. These results point to potential limitations of feed-forward neural networks in generalizing a system's behavior given limited initial information. Much attention must therefore be given to designing appropriate train-test splits for real-world applications.

In this work, we apply a nonlinear chaos analysis to the two phase flow in polymer electrolyte fuel cell cathode air-delivery microchannels. The fuel cell voltage signal is analyzed using techniques designed to estimate invariants typical of deterministic systems with high sensitivity to initial conditions, such as chaotic two phase flow. Voltage data are taken under varying fuel cell operating conditions, and noise in the data is reduced using a nonlinear noise reduction algorithm. The chaotic strange attractor of the system is reconstructed in phase space using time-delay embedding. Correlation sums over the strange attractor are calculated to estimate the fractal correlation dimension of the system. Estimations of the Kolmogorov entropy provide an additional measure of the complexity of the strange attractor. The values of the chaotic invariants are compared across varying degrees of cathode flooding to discern how they change with two phase flow regimes and fuel cell operating conditions. Future work will involve leveraging the chaotic understanding of two phase flow with chaos control methods to increase the power stability.

Summary The method of calculation of fractal correlation dimension D values is applied for examination of statistics of large and extra-large ore deposits. The used approach differs from the commonly used one. Firstly, we permit the different D values for different spatial scales that seems to be useful in distinguishing of regularities in the location of ore deposits at different scales. Secondly, we introduce a new notion, a mixed correlation dimension and use it for examination of pairs of objects of different typ (e.g., deposits of Au and Ag). The standard formula for calculating the correlation dimension values can be trivially generalized for such cases. The “mixed D” values can be both lower and higher than the dimension of the hosting medium. The cases when this dimension value is higher than that of the hosting medium are interpreted as a “mutual repulsion” of the deposits of corresponding two types. Small correlation dimensions indicate that the deposits of the corresponding types tend to be located closely.The calculations are conducted for the spherical Earth. Different patterns of the studied behavior are illustrated by the model examples.

This paper compares measures derived from linear power spectral analysis and nonlinear chaotic analysis of helicopter vibratory data obtained during different flight tests. In particular, the strange attractor behavior of the trajectory of reconstructed state vectors in phase space was examined and quantified using the correlation dimension, which provides a fractal dimension measure in regards to the number of active degrees of freedom of helicopter vibration. We have examined the helicopter vibratory data collected from a well tuned four-blade servo-flap rotor system conducting hover, low-speed, and cruising-speed forward flights. In this paper, we examined servo-flap mid-span bending and rotor blade flat-wise bending. The results have shown that during helicopter hover, vibratory data of servo-flap bending and rotor flat-wise bending are more random than chaotic. On the other hand, the complexity (fractal dimension) of vibratory data increase when the helicopter switched from the minimum forward speed flight to cruising-speed forward flight.