Diagnosing the structures of attractors

This paper investigates multistability in a 3D autonomous system with different types of chaotic attractors, which are not in the sense of Shil’nikov criteria. First, under some conditions, the system has infinitely many isolated equilibria. Moreover, all equilibria are nonhyperbolic and give the first Lyapunov coefficient. Furthermore, when all equilibria are weak saddle-foci, the system also has infinitely many chaotic attractors. Besides, the Lyapunov exponents spectrum and bifurcation diagram are given. Second, under another condition, all the equilibria constitute a curve and there exist infinitely many singular degenerated heteroclinic orbits. At the same time, the system can show infinitely many chaotic attractors.

The nonlinear dynamics of a generalized higher-order nonlinear Schrodinger (HNLS) equation with a periodic external perturbation is investigated numerically. Via the phase plane analysis, we find that both the homoclinic orbits and heteroclinic orbits can exist for the unperturbed HNLS equation under certain conditions, which respectively corresponds to the bell-shaped and kink-shaped soliton solutions. Moreover, under the effect of the periodic external perturbation, the quasi-periodic bifurcations arise and can evolve into the chaos. The dynamical responses of the perturbed system varying with the perturbation strength and two types of chaotic attractors are discussed to show the existence of the chaotic motions. Via the feedback control methods, such chaotic motions are found to be controlled effectively and finally evolve into the stable quasi-periodic orbits. All the results are helpful to understand the dynamical properties of the nonlinear system.

Complex nonlinear dynamics and applications of a microwave klystron oscillator with delayed feedback are studied in this paper on the basis of a system of delayed differential equations. Results of numerical simulation of self-modulation and chaotic regimes are presented, unveiling a complex bifurcation pattern in a wide range of parameters. Various types of chaotic attractors and transition to chaos scenarios are described. Experimental results for the five-cavity klystron oscillator are presented as well, showing good qualitative agreement with numerical simulations. Special attention is paid to application of the oscillator in direct chaotic communications. Information transmission using chaos shift keying (CSK) modulation and coherent reception is demonstrated via numerical simulation. The driving by an external signal is used to control the chaotic states of the transmitter oscillator for the purpose of generation of CSK basis functions.

Strange attractors with Lyapunov dimension d(L) <3 can be classified by branched manifolds. They can also be classified by the bounding tori that enclose them. Bounding tori organize branched manifolds (classes of strange attractors) in the same way as the branched manifolds organize the periodic orbits in a strange attractor. We describe how bounding tori are constructed and expressed in a useful canonical form. We present the properties of these canonical forms and show that they can be uniquely coded by analogs of periodic orbits of period g-1, where g is the genus. We describe the structure of the global Poincaré surface of section for an attractor enclosed by a genus- g torus and determine the transition matrix for flows between the g-1 components of the Poincaré surface of section. Finally, we show how information about a bounding torus can be extracted from scalar time series.

A single fractal number cannot fully describe the complex structure of a strange attractor embedded in phase space of dynamic systems. Hence it is essential to seek a procedure through which fractal dimension provides more information about the underlying dynamic system, in order to serve such a purpose correlation integrals are calculated at every row or off-diagonal from the distance matrix obtained from a set of points on a strange attractor. The slope of each integral is plotted versus the number of row or off-diagonal considered in the calculation. Hence, an asymptotic convergence to correlation dimension is obtained in this manner. More information is revealed for the dynamic system including its lacunarity.

For three-dimensional dynamical systems with continuous time (flows), a classification of strange homoclinic attractors containing an unique saddle equilibrium state is constructed. The structure and properties of such attractors are determined by the triple of eigenvalues of the equilibrium state. The method of a saddle charts is used for the classification of homoclinic attractors. The essence of this method is in the construction of an extended bifurcation diagram for a wide class of three-dimensional flows (whose linearization matrix is written in the Frobenius form). Regions corresponding to different configurations of eigenvalues are marked in this extended bifurcation diagram. In the space of parameters defining the linear part of the considered class of three-dimensional flows bifurcation surfaces bounding 7 regions are constructed. One region corresponds to the stability of the equilibrium states while other 6 regions correspond to various homoclinic attractors of the following types: Shilnikov attractor, 2 types of spiral figure-eight attractors, Lorenz- like attractor, saddle Shilnikov attractor and attractor of Lyubimov-Zaks-Rovella. The paper also discusses questions related to the pseudohyperbolicity of homoclinic attractors of three-dimensional flows. It is proved that only homoclinic attractors of two types can be pseudohyperbolic: Lorenz-like attractors containing a saddle equilibrium with a two-dimensional stable manifold whose saddle value is positive and saddle Shilnikov attractors containing a saddle equilibrium state with a two-dimensional unstable manifold.

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.

This tutorial presents an in-depth introduction to chaos in dynamical systems, and presents several practical techniques for recognizing and classifying chaotic behavior. These techniques include the poincaré map, Lyapunov exponents, capacity, information dimension, correlation dimension, Lyapunov dimension, and the reconstruction of attractors from a single time series.

This paper presents the novel approach to evaluate the effects of different motor activation tasks of the human electroencephalogram (EEG). The applications of chaos and fractal properties that are the most important tools in nonlinear analysis are been presented for four tasks of EEG during the real and imaginary motor movement. Three subjects, aged 23–30 years, participated in the experiment. Correlation dimension ( D 2 ), Lyapunov spectrum (λ i ), and Lyapunov dimension ( D L ) are been estimated to characterize the movement related EEG signals. Experimental results show that these nonlinear measures are good discriminators of EEG signals. There are significant differences in all conditions of subjective task. The fractal dimension appeared to be higher in movement conditions compared to the baseline condition. It is concluded that chaos and fractal analysis could be powerful methods in investigating brain activities during motor movements.

Abstract. Archives of composite weather radar images represent an invaluable resource to study the predictability of precipitation. In this paper, we compare two distinct approaches to construct empirical low-dimensional attractors from radar precipitation fields. In the first approach, the phase space variables of the attractor are defined using the domain-scale statistics of precipitation fields, such as the mean precipitation, fraction of rain, and spatial and temporal correlations. The second type of attractor considers the spatial distribution of precipitation and is built by principal component analysis (PCA). For both attractors, we investigate the density of trajectories in phase space, growth of errors from analogue states, and fractal properties. To represent different scales and climatic and orographic conditions, the analyses are done using multi-year radar archives over the continental United States (≈4000×4000 km2, 21 years) and the Swiss Alpine region (≈500×500 km2, 6 years).

Dynamical analysis and more detailed numerical analysis for understanding the compound structure of Chen's attractor are proposed in this paper. Routes to chaos and bifurcation of the Chen's system under a constant controller are demonstrated with numerical experiments. The systematic analysis method can also be applied to investigate chaotic attractor's compound structure shared by some typical attractors such as the Lorenz attractor, modified Lorenz attractor, Lu attractor, and PWL Chen's attractor, etc.

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.

This work is a continuation of our efforts to quantify the irregular scalar stress signals from the Ananthakrishna model for the Portevin-Le Chatelier instability observed under constant strain rate deformation conditions. Stress related to the spatial average of the dislocation activity is a dynamical variable that also determines the time evolution of dislocation densities. We carry out detailed investigations on the nature of spatiotemporal patterns of the model realized in the form of different types of dislocation bands seen in the entire instability domain and establish their connection to the nature of stress serrations. We then characterize the spatiotemporal dynamics of the model equations by computing the Lyapunov dimension as a function of the drive parameter. The latter scales with the system size only for low strain rates, where isolated dislocation bands are seen, and at high strain rates, where fully propagating bands are seen. At intermediate applied strain rates corresponding to the partially propagating bands, the Lyapunov dimension exhibits two distinct slopes, one for small system sizes and another for large. This feature is rationalized by demonstrating that the spatiotemporal patterns for small system sizes are altered from the partially propagating band types to isolated burst type. This in turn allows us to reconfirm that low-dimensional chaos is projected from the stress signals as long as there is a one-to-one correspondence between the bursts of dislocation bands and the stress drops. We then show that the stress signals in the regime of partially to fully propagative bands have features of extensive chaos by calculating the correlation dimension density. We also show that the correlation dimension density also depends on the system size. A number of issues related to the system size dependence of the Lyapunov dimension density and the correlation dimension density are discussed.

In this paper phase space reconstruction technique is used for distinguishing the combustion states in the boiler furnace. The phase space maps and the correlation dimensions under the stable combustion & the unstable combusition are charted and calculated. The simulation and calculation results of the actual flame signals show that the phase space maps and the correlation dimensions under both combustion states are different at all time. Therefore it is presented that the correlation dimension ought to be regarded as the characteristic parameter to distinguish the combustion states of the flame in the boiler furnace. And the approach presented in this paper may be applied to develop the novel optical flame detector.

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.