Chaos Behavior Research Articles

Parameter identification and synchronization of spatiotemporal chaos in globally coupled network

Synchronization and parameter identification of globally coupled network constructed by discrete spatiotemporal chaos systems with unknown parameters are studied. The control law designed by Milosavljevic is extended, and applied to the identification of unknown parameters in spatiotemporal chaos systems at the nodes of the network and the study of global synchronization of the network. The one-dimensional convective equation with spatiotemporal chaos behaviour in physics is further taken for example to imitate analysis.

Comprehensive Research on the Origin of the Solar System Structure by Quantum-like Model

A quantum-like model of gravitational system is introduced to explore the formation of the solar system structure. In this model, the chaos behavior of a large number of original nebular particles in a gravitational field can be described in terms of the wave function satisfying formal Schrodinger equation, in which the Planck constant is replaced by a constant on cosmic scale. Numerical calculation shows that the radial distribution density of the particles has the character of wave curves with decreasing amplitudes and elongating wavelengths. By means of this model, many questions of the solar system, such as the planetary distance, mass, energy, angular momentum, the distribution of satellites, the structure of the planetary rings, and the asteroid belt and the Kuiper belt etc., can be explained in reason. In addition, the abnormal rotations of Venus and Mercury can be naturally explained by means of the quantum-like model.

UNIFORMITY AND SPATIAL CHAOS OF SPATIAL PHYSICS KINEMATIC SYSTEM

This article summarizes the uniformity law of spatial physics kinematic systems, and studies the chaos and bifurcation behavior of the uniform system in space. In particular, it also fully explains the relation among the uniform system, the coupled map lattice model which has attracted considerable interest currently, and one-dimensional nonlinear dynamical systems.

Nonlinear flexural waves and chaos behavior in finite-deflection Timoshenko beam

Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov’s method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.

Multi-pulse orbits and chaotic dynamics in a nonlinear forced dynamics of suspended cables

The global bifurcations in mode of a nonlinear forced dynamics of suspended cables are investigated with the case of the 1:1 internal resonance. After determining the equations of motion in a suitable form, the energy phase method proposed by Haller and Wiggins is employed to show the existence of the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the two cases of Hamiltonian and dissipative perturbation. Furthermore, some complex chaos behaviors are revealed for this class of systems.

Generalized synchronization of spatiotemporal chaos in a weighted complex network

The nodes of the network are composed of the spatiotemporal chaos systems. The relations between the nodes are built through a weighted connection and the nonlinear terms of the chaos systems themselves are taken as coupling functions. The structure of the coupling functions between the connected nodes and the range of the control gain are obtained based on Lyapunov stability theory. It is proven that generalized chaos synchronization of the weight complex network can be realized even if the coupling strength between the nodes is adopted as any weight value. Subsequently, the catalytic reaction diffusion system which has spatiotemporal chaos behavior is taken as example, and simulation results show the effectiveness of the synchronization principle.

Projective synchronization of spatiotemporal chaos in a weighted complex network

Projective synchronization of a weighted complex network is studied in which nodes are spatiotemporal chaos systems and all nodes are coupled not with the nonlinear terms of the system but through a weighted connection. The range of the linear coefficient matrix of separated configuration, when the synchronization is implemented, is determined according to Lyapunov stability theory. It is found that projective synchronization can be realized for unidirectional star-connection even if the coupling strength between the nodes is a given arbitrary weight value. The Gray–Scott models having spatiotemporal chaos behaviours are taken as nodes in the weighted complex network, and simulation results of spatiotemporal synchronization show the effectiveness of the method.

Four-scroll hyperchaos and four-scroll chaos evolved from a novel 4D nonlinear smooth autonomous system

This Letter presents a novel four-dimensional complex nonlinear autonomous system. This system has very rich dynamics and can exhibit hyperchaos, chaos, quasi-periodic and periodic behavior. Applying two symmetrical initial conditions, this system can generate two coexist double-scroll hyperchaotic attractors. More interestingly, this new system can generate not only a four-scroll hyperchaotic but also a four-scroll chaotic attractor. This nonlinear complex system is analyzed by Lyapunov exponent spectrum, bifurcation diagram and Poincaré mapping.

Simple chaotic oscillator’s chaos behavior and its feedback control circuit design

The aim of this paper is to study a chaotic oscillators stability and chaos behavior and to determine the conditions for the stability and chaotic behavior of the chaotic oscillator by theoretical analysis. Furthermore,this study also aims to control the chaotic oscillator by an exact feedback linearization method. Finally,both numerical simulations and circuit experiments verify the validity of the theoretical analysis.

Quantum chaos and critical behavior on a chip

The Dicke model describes N qubits (or two-level atoms) homogenously coupled to a bosonic mode. Here we examine an open-system realization of the Dicke model, which contains critical and chaotic behaviour. In particular, we extend this model to include an additional open transport qubit (TQ) (coupled to the bosonic mode) for passive and active measurements. We illustrate how the scaling (in the number of qubits N) of the superradiant phase transition can be observed in both current and current-noise measurements through the transport qubit. Using a master equation, we also investigate how the phase transition is affected by the back-action from the transport qubit and losses in the cavity. In addition, we show that the non-integrable quantum chaotic character of the Dicke model is retained in an open-system environment. We propose how all of these effects could been seen in a circuit QED system formed from an array of superconducting qubits, or an atom chip, coupled to a quantized resonant cavity (e.g., a microwave transmission line).

Deterministic Chaos and Fractal Complexity in the Dynamics of Cardiovascular Behavior: Perspectives on a New Frontier

Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts.

HIGH-DIMENSIONAL CHAOS IN DISSIPATIVE AND DRIVEN DYNAMICAL SYSTEMS

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

Period-two discharge characteristics in argon atmospheric dielectric-barrier discharges

Period multiplication and chaos behaviors in atmospheric glow discharges are important to understand atmospheric glow plasma stability and to optimize its applications. In this paper, we study the period-two discharge characteristics in argon atmospheric dielectric-barrier discharges using a one-dimension fluid model. Under certain conditions, period-two discharges can occur at different excitation frequencies and exhibit different current and voltage behaviors. When the discharge current becomes highly symmetrical the period-two discharge can reach to a steady state, which sustains over a broad frequency range. At the sufficiently high excitation frequency, the period-two discharge in atmospheric argon shows noticeably different behaviors from the discharges generated at kilohertz frequency. The spatial characteristics of period-two discharge as well as the influence of driving frequency on period-two discharge behaviors are also investigated.

On Sliding-Mode Control of Chaotic Systems

Chaos control means to design a controller that is able to mitigate or eliminate the chaos behaviour of nonlinear systems that experience such phenomenon. In this paper, a nonlinear Sliding-Mode Controller is presented and applied to nonlinear chaotic systems. Two nonlinear chaotic systems are chosen to be our case study in this paper, the first one, is the well known Chua's circuit and the second is Lorenz system. The study shows the effectiveness of the designed nonlinear Sliding-Mode Controller.

A novel delayed chaotic neural model and its circuitry implementation

In this paper, we investigate a novel delayed chaotic neural model, in which a non-monotonously increasing transfer function is employed as activation function. Local stability and existence of Hopf bifurcation are analyzed in details. Chaos behavior of the neuron model is observed in computer simulations. An electronic implementation of the neuron is also considered. The dynamical behavior of the designed circuits is closely similar to the results simulated by numerical experiments.

Real-time deterministic chaos control by means of selected evolutionary techniques

This contribution introduces continuation of an investigation on deterministic spatiotemporal chaos real-time control by means of selected evolutionary techniques. Real-time-like behavior is specially defined and simulated with the spatiotemporal chaos model based on mutually nonlinearly joined n equations, so-called coupled map lattices (CML). Four evolutionary algorithms are used for chaos control here: differential evolution, self-organizing migrating algorithm, genetic algorithm and simulated annealing in a total of 12 versions. For modeling of real-time spatiotemporal chaos behavior, the so-called CML were used based on logistic equation to generate chaos. The main aim of this investigation was to show that evolutionary algorithms, under certain conditions, are capable of real-time control of deterministic chaos, when the cost function is properly defined as well as parameters of selected evolutionary algorithm. Investigation consists of four different case studies with increasing simulation complexity. For all algorithms, each simulation was repeated 100 times to show and check robustness of used methods. All data were processed and used in order to get summarizing results and graphs.

Turbocharger rotor dynamics with foundation excitation

To investigate the effect of foundation excitation on the dynamical behavior of a turbocharger, a dynamic model of a turbocharger rotor-bearing system is established which includes the engine’s foundation excitation and nonlinear lubricant force. The rotor vibration response of eccentricity is simulated by numerical calculation. The bifurcation and chaos behaviors of nonlinear rotor dynamics with various rotational speeds are studied. The results obtained by numerical simulation show that the differences of dynamic behavior between the turbocharger rotor systems with/without foundation excitation are obviously. With the foundation excitation, the dynamic behavior of rotor becomes more complicated, and develops into chaos state at a very low rotational speed.

Quasi-cycles and sensitive dependence on seed values in edge of chaos behaviour in a class of self-evolving maps

In a recent paper [Melby P, Kaidel J, Weber N, Hubler A. Adaptation to the edge of chaos in the self-adjusting logistic map. Phys Rev Lett 2000;84:5991–3], Melby et al. attempted to understand edge of chaos behaviour through a very simple model. Based on our exhaustive numerical experiments, here we show that the model, with the definition of the edge of chaos given in the paper, cannot unequivocally support the idea of adaptation to the edge of chaos, let alone allow a conjecture of its generic presence in systems having the same characteristic features.

Characterization of state variables in post-stall dynamics of high-performance aircraft

The behaviour of the state variables in post stall flight region of high-performance aircraft is presented in this paper. To demonstrate the state variables of aircraft in post-stall dynamics, the non-linear state equations of aircraft are solved. The numerical investigation has shown that a sufficiently negative lift slope can lead to limit cycle oscillation motion. The behaviour of limit cycle is demonstrated with the action attributed to relay action caused by decrement of lift coefficient jump at stall. The time-history of state variables and the effects of decrement of lift coefficient on limit-cycle amplitudes are achieved. A numerical model constructed for a high-performance aircraft F-18 is solved to illustrate the results. The effects of significant factors such as mass, thrust on limit cycle behaviour in post stall region are considered in this analysis. The couple of effects including the mass and velocity on limit-cycle amplitude of attack angle and pitch angle are obtained in this region. The results indicate that chaos behaviour of attack angle exists in the case of a 20 per cent decrease in mass and a 5—10 per cent increase in velocity. Multi-point attractor is predicted with the change in the values of attack angle. A good agreement between the numerical results and the published work is obtained for limit cycle oscillation existence at different values of lift coefficient decrement.

The higher order Schwarzian derivative: Its applications for chaotic behavior and new invariant sufficient condition of chaos

The Schwarzian derivative of a function f ( x ) which is defined in the interval ( a , b ) having higher order derivatives is given by S f ( x ) = ( f ″ ( x ) f ′ ( x ) ) ′ − 1 2 ( f ″ ( x ) f ′ ( x ) ) 2 . A sufficient condition for a function to behave chaotically is that its Schwarzian derivative is negative. In this paper, we try to find a sufficient condition for a non-linear dynamical system to behave chaotically. The solution function of this system is a higher degree polynomial. We define the n th Schwarzian derivative to examine its general properties. Our analysis shows that the sufficient condition for chaotic behavior of higher order polynomial is provided if its highest order three terms satisfy an inequality which is invariant under the degree of the polynomial and the condition is represented by the Hankel determinant of order 2. Also the n th order polynomial can be considered to be the partial sum of real variable analytic function. Let this analytic function be the solution of a non-linear differential equation, then the sufficient condition for the chaotical behavior of this function is that the Hankel determinant of order 2 should be negative, where the elements of this determinant are the coefficients of the terms of n , n − 1 , n − 2 in the Taylor expansion.