Chaotic Behavior Of System Research Articles

Hyperbolic octagons and Teichmüller space in genus 2

In this paper we study different models of the Teichmüller space of compact hyperbolic surfaces with special emphasis on their construction by geodesic octagons. Such surfaces have become increasingly interesting due to their applications to the physical behavior of chaotic quantum systems. They are, on the one hand, complex enough to show the relevant features, and on the other hand, they possess a simple mathematical structure allowing practical implementations. We give a new description of Teichmüller space and the mapping class group in terms of geometric data of the octagons. This provides modellings based on the vertices and also on the generators of the associated isometry group. In addition, we explicitly connect our approach with other models currently used in the literature (Helling matrices, Fenchel–Nielsen parameters). The result of the paper may be considered as a computational tool kit to work with a specific model and to relate it with the others.

Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

This paper deals with forced vibrations of two-DOF systems with more than one equilibrium positions. Such systems may be obtained by digitization of elastic post-buckling systems. A vibration mode, which is periodic at small force amplitudes and becomes chaotic as the force amplitudes are slowly increased, is selected. It is possible to formulate and solve the problem of stability of a periodic or chaotic vibration mode in a space with greater dimension using the classical Lyapunov stability definition and some calculating procedures. Instability of phase trajectories is used as a criterion of the chaotic behavior in dynamical systems. Trajectories with very close initial values are compared. Use of the Lyapunov stability definition shows mutual stability/instability of the trajectories. Calculations permit to observe an appearance and enlargement of the chaotic behavior regions. Specific results are obtained for the nonautonomous Duffing equation and pendulum system.

CONTROLLING CHAOS IN HIGHER-ORDER DYNAMICAL SYSTEMS

In this paper, we investigate a continuous extension of the Ott, Grebogi and Yorke control method [Ott et al., 1990]. This extension allows the control of chaotic behavior in higher-order dynamical systems. To begin with, studying the so-called Poincaré Sections (PS) enables determining fixed points and the effect of small changes in the control parameter on the system trajectory. Introduction of a stabilizer parameter in the control law keeps the dynamical properties and maintains them on a cyclic trajectory. The effectiveness of the proposed method is tested by numerical examples on the two typical Chen and Chua chaotic systems.

Lyapunov spectrum of the many-dimensional dilute random Lorentz gas

For a better understanding of the chaotic behavior of systems of many moving particles, it is useful to look at other systems with many degrees of freedom. An interesting example is the high-dimensional Lorentz gas, which, just like a system of moving hard spheres, may be interpreted as a dynamical system consisting of a point particle in a high-dimensional phase space, moving among fixed scatterers. In this paper, we calculate the full spectrum of Lyapunov exponents for the dilute random Lorentz gas in an arbitrary number of dimensions. We find that the spectrum becomes flatter with increasing dimensionality. Furthermore, for fixed collision frequency the separation between the largest Lyapunov exponent and the second largest one increases logarithmically with dimensionality, whereas the separations between Lyapunov exponents of given indices not involving the largest one go to fixed limits.

CHAOTIC BEHAVIOR OF INTERVAL MAPS AND TOTAL VARIATIONS OF ITERATES

Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f:I→I is chaotic, then its iterates fnwill display heightened oscillatory behavior or profiles as n→∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of fnon subintervals. There are four distinctive cases of the growth of total variations of fnas n→∞:(i) the total variations of fnon I remain bounded;(ii) they grow unbounded, but not exponentially with respect to n;(iii) they grow with an exponential rate with respect to n;(iv) they grow unbounded on every subinterval of I.We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of fnare more extreme, the more complex is the behavior of the system.

Chaos control in a nonlinear pendulum using a semi-continuous method

Chaotic behavior of dynamical systems offers a rich variety of orbits, which can be controlled by small perturbations in either a specific parameter of the system or a dynamical variable. Chaos control usually involves two steps. In the first, unstable periodic orbits (UPOs) that are embedded in the chaotic set are identified. After that, a control technique is employed in order to stabilize a desirable orbit. This contribution employs the close-return method to identify UPOs and a semi-continuous control method, which is built up on the OGY method, to stabilize some desirable UPO. As an application to a mechanical system, a nonlinear pendulum is considered and, based on parameters obtained from an experimental setup, analyses are carried out. At first, it is considered signals generated by numerical integration of the mathematical model. After that, the analysis is done from scalar time series and therefore, it is important to evaluate the effect of state space reconstruction. Delay coordinates method is employed with this aim. Finally, an analysis related to the effect of noise in controlling chaos is of concern. Results show situations where these techniques may be used to control chaos in mechanical systems.

Stabilization of the chaotic behavior of dynamical systems

Stabilization of the chaotic behavior of dynamical systems

Control of chaos in laser-induced vapor capillaries

Vapor capillaries associated with deep penetration laser welding (keyholes) are inherently unstable and it is well known that the bounded evolution of unstable modes of nonlinear dynamic systems can lead to complex fluctuations of state and/or output variables. In the case of keyholes, the magnitude of self-induced fluctuations is sometimes large enough to cause occasional closure (“collapse”) and trapping of voids in the solidifying weld metal. Since voids are serious and frequent defects in partial penetration laser welds, methods to improve the stability of the keyhole, and thereby reduce or eliminate their incidence are of considerable practical interest. One technique that is known to simplify the complicated behavior of chaotic systems involves the application of periodic perturbations to accessible system inputs or boundaries. Consequently, this work investigated the effect of power modulation on porosity formation in laser welds. Frozen glycerin was used as the medium for welding since its transparency allowed visualization of the keyhole, facilitating control parameter development. Statistical and spectral analysis of penetration depth signals for the uncontrolled system was performed to characterize the nonlinear keyhole dynamics. In the control studies, it was found that the optimum frequency of power modulation at 400 Hz was very effective for reducing porosity. A 15° leading beam incidence angle was found to provide further improvements. Three-dimensional phase portraits of the keyhole depth for uncontrolled and controlled signals showed that the control did not convert depth fluctuations to a completely predictable limit cycle but that the remaining fluctuations were more predictable than those of the uncontrolled system.

Application of a New Theory of Fuzzy Chaos for the Simulation and Control of NonLinear Dynamical Systems

We describe in this article 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 sets theory. Using fuzzy sets, we can build a new theory of Fuzzy Chaos, where we can use fuzzy set to describe the behaviors of a system. We illustrate our approach with two cases: Chua's circuit and Duffing's oscillator.

Fuzzy clustering approach for accurate embedding dimension identification in chaotic time series

The dynamics of many physically complex systems cannot be represented by a set of differential equations. Measured sensor signals in the form of time-series can be used to investigate complexities and chaotic behavior in such systems. In order to cha

Normal forms and nonlocal chaotic behavior in Sprott systems

The Sprott systems are used as benchmarks for investigating the applicability of the normal form transformation in estimating nonlocal properties of attractors such as positive and zero Liapunov exponents. Possibility of a relation between complex conjugate eigenvalue pairs and zero Liapunov exponents; conditions under which the normal form expansion can represent the attractor; an averaging relation for the largest Liapunov exponent based on this representation are studied. Nonlinear transformations that can change the order of a resonance are considered. In spite of their convergence problems, it is seen that the normal form approach can give reasonable estimates of nonlocal properties of attractors near Hopf bifurcations.

Chaotic behaviour of charges in crossed fields

The study of numerical model for behaviour of charges in crossed electric and magnetic fields at the change of control parameters (disturbing field, cyclotron frequency and initial speeds of charge) was carried out. On the basis of Lyapunov exponents calculation the areas of existence of regular and chaotic behaviour of system are determined. Features of the electrons trajectories in the case of radial nonhomogeneous H field are discussed also. It is shown the movement of electrons in the area with strong H field leads to appearance of regular dynamics instead of chaotic one. The calculation of the Lyapunov exponents, power spectra, Lyapunov maps of regimes has been done.

A coupled oscillator model of disordered interlimb coordination in patients with Parkinson's disease.

Coordination between the left and right limbs during cyclic movements, which can be characterized by the amplitude of each limb's oscillatory movement and relative phase, is impaired in patients with Parkinson's disease (PD). A pedaling exercise on an ergometer in a recent clinical study revealed several types of coordination disorder in PD patients. These include an irregular and burst-like amplitude modulation with intermittent changes in its relative phase, a typical sign of chaotic behavior in nonlinear dynamical systems. This clinical observation leads us to hypothesize that emergence of the rhythmic motor behaviors might be concerned with nonlinearity of an underlying dynamical system. In order to gain insight into this hypothesis, we consider a simple hard-wired central pattern generator model consisting of two identical oscillators connected by reciprocal inhibition. In the model, each oscillator acts as a neural half-center controlling movement of a single limb, either left or right, and receives a control input modeling a flow of descending signals from higher motor centers. When these two control inputs are tonic-constant and identical, the model has left-right symmetry and basically exhibits ordered coordination with an alternating periodic oscillation. We show that, depending on the intensities of these two control inputs and on the difference between them that introduces asymmetry into the model, the model can reproduce several behaviors observed in the clinical study. Bifurcation analysis of the model clarifies two possible mechanisms for the generation of disordered coordination in the model: one is the spontaneous symmetry-breaking bifurcation in the model with the left-right symmetry. The other is related to the degree of asymmetry reflecting the difference between the two control inputs. Finally, clinical implications by the model's dynamics are briefly discussed.

Targeting in dissipative chaotic systems: A survey.

The large number of unstable equilibrium modes embedded in the strange attractor of dissipative chaotic systems usually presents a sufficiently rich repertoire for the choice of the desirable motion as a target. Once the system is close enough to the chosen target local stabilization techniques can be employed to capture the system within the desired motion. The ergodic behavior of chaotic systems on their strange attractors guarantees that the system will eventually visit a close neighborhood of the target. However, for arbitrary initial conditions within the basin of attraction of the strange attractor the waiting time for such a visit may be intolerably long. In order to reduce the long waiting time it usually becomes indispensable to employ an appropriate method of targeting, which refers to the task of steering the system toward the close neighborhood of the target. This paper provides a survey of targeting methods proposed in the literature for dissipative chaotic systems. (c) 2002 American Institute of Physics.

Evidence of a nanomechanical resonator being driven into chaotic response via the Ruelle–Takens route

Nanomechanical resonators, of importance for signal filtering and transduction, are investigated within the limit of extreme nonlinear excitation. The Ruelle–Takens route establishes the transition of a system into chaos with n frequencies present. By providing a set of three sources, this transition from linear to nonlinear and finally to chaotic response can be traced, and it is found in our experiment. Knowledge of potentially chaotic behavior of nanomechanical systems is crucial for their application. Our resonator system has an overall length of 4 μm and a cross section of about 100 nm×300 nm with natural frequencies of ∼50 MHz.

CHUA'S PERIODIC TABLE

An interdisciplinary analysis is presented in this paper on the relationship between the bifurcation behavior of chaotic systems, specifically Chua's circuit, and the quantized energy level of atoms. We show that it is possible to associate special capacitance values in Chua's circuits with the electronic energy levels Enof atoms calculated from Bohr's Law. In particular, these particular capacitance values correspond to the bifurcation points αnof Chua's circuits. We found a functional relation associating a specific Chua's circuit to the Hydrogen atom and obtained a map of "Atom" to " Chua's Circuit". This map can be extended to other elements in the Periodic Table, thereby demonstrating an almost universal relation between the two different physical systems. With this map we can calculate the energy levels of different atoms from bifurcation diagrams (or vice-versa) and express the analytical relations in terms of the two variables αnand En.

Noise-Induced Chaotic Behavior in Nonlinear Vibration Systems. Bifurcation of Random Invariant Measures.

Noise-induced chaotic bifurcations of random attractors arising in a Duffing oscillator and a parametrically exited pendulum with noisy harmonic excitations are investigated. We first follow the standard pullback procedure to obtain random invariant measures and then intuitively construct Poincare maps of them. The results on the stability and fractal analysis show that the noise-induced chaotic behavior which is not strange is generated from strange but non-chaotic invariant measures.

Effect of parameter calculation in direct estimation of the Lyapunov exponent in short time series

The literature about non‐linear dynamics offers a few recommendations, which sometimes are divergent, about the criteria to be used in order to select the optimal calculus parameters in the estimation of Lyapunov exponents by direct methods. These few recommendations are circumscribed to the analysis of chaotic systems. We have found no recommendation for the estimation of λ starting from the time series of classic systems. The reason for this is the interest in distinguishing variability due to a chaotic behavior of determinist dynamic systems of variability caused by white noise or linear stochastic processes, and less in the identification of non‐linear terms from the analysis of time series. In this study we have centered in the dependence of the Lyapunov exponent, obtained by means of direct estimation, of the initial distance and the time evolution. We have used generated series of chaotic systems and generated series of classic systems with varying complexity. To generate the series we have used the logistic map.

Kurt Vonnegut's The Sirens of Titan: Human Will in a Newtonian Narrative Gone Chaotic

The paper “Kurt Vonnegut’s The Sirens of Titan: Human Will in a Newtonian Narrative Gone Chaotic” explores the notion of human identity in its relation with the issue of free will as portrayed by Vonnegut’s 1959 novel The Sirens of Titan. Written at the very threshold of the postmodernist period, Vonnegut’s novel invites a shift of perspective in the interpretation of ‘reality’ —namely, a transition from the modern Newtonian to a postmodernist chaotic scientific paradigm. The novel suggests an implicit counter-message parallel to and subtly undermining its apparent affirmation of determinism. For this purpose, the paper focuses on the way in which some passages in the novel reflect some of the major theories about the chaotic behavior of molecular systems, basically temporal irreversibility and the combination of chance and necessity in molecular creative processes. Finally, the paper analyzes the metaphysical macrocosmic implications of these theories as depicted in Vonnegut’s novel —firstly, as an answer to the individuals’ felt inability to find their identity through decision-making; and secondly, as a way out of constraining, totalizing explanations of the world and the collective paranoia which the search for those explanations may bring about.

Making chaotic behavior in a damped linear harmonic oscillator

The present Letter proposes a simple control method which makes chaotic behavior in a damped linear harmonic oscillator. This method is a modified scheme proposed in paper by Wang and Chen (IEEE CAS-I 47 (2000) 410) which presents an anti-control method for making chaotic behavior in discrete-time linear systems. We provide a systematic procedure to design parameters and sampling period of a feedback controller. Furthermore, we show that our method works well on numerical simulations.