Partial state feedback control of chaotic neural network and its application

The chaos control in the chaotic neural network is studied using the partial state feedback with a control signal from a few control neurons. The controlled CNN converges to one of the stored patterns with a period which depends on the initial conditions, i.e., the set of control neurons and other control parameters. We show that the controlled CNN can distinguish between two initial patterns even if they have a small difference. This implies that such a controlled CNN can be feasibly applied to information processing such as pattern recognition.     

Dynamics of fractional-order sinusoidally forced simplified Lorenz system and its synchronization

In this paper, dynamical behaviors of the fractional-order sinusoidally forced simplified Lorenz are investigated by employing the time-domain solution algorithm of fractional-order calculus. The system parameters and the fractional derivative orders q are treated as bifurcation parameters. The range of the bifurcation parameters in which the system generates chaos is determined by bifurcation, phase portrait, and Poincaré section, and different bifurcation motions are visualized by virtue of a systematic numerical analysis. We find that the lowest order of this system to yield chaos is 3.903. Based on fractional-order stability theory, synchronization is achieved by using nonlinear feedback control method. Simulation results show the scheme is effective and a chaotic secure communication scheme is present based on this synchronization.     

Dynamical behaviors of a plate activated by an induction motor

Dynamics and chaotification of a system consisting of an induction motor activating a mobile plate (with variable contents) fixed to a spring are studied. The dynamical model of the device is presented and the electromechanical equations are formulated. The oscillations of the plate are analyzed through variations of the following reliable control parameters: phase voltage supply of the motor, frequency of the external source and mass of the plate. The dynamics of the system near the fundamental resonance region presents jump phenomenon. Mapping of the control parameters planes in terms of types of motion reveals period-n motion, quasi-periodicity and chaos. Anti-control of chaos of the induction motor is also obtained using the field-oriented control associated to the time delay feedback control.     

Investigation of chaos synchronization in photonic crystal lasers

Chaotic dynamics and chaos synchronization in photonic crystal (PC) lasers with optical feedback are investigated numerically. The effect of various system parameters such as amplitude reflectivity of the external mirror “r”, external cavity length “L_e”, and injection current “I” on system dynamics is addressed in detail. Simulation results are presented using MATLAB to address system behavior. The parameters r, L_e, and I are varied over the ranges (0.05–0.25), (2.8–3.2 mm), and (1.1I_th–2I_th), respectively. The results indicate that very small parameter mismatches between the transmitter laser and receiver laser affect strongly complete chaos synchronization between them.     

The evolution of the chaotic dynamics of collective modes as a method for the behavioral description of living systems

Human-scaled (in complexity) systems possess a unique feature, viz., the continuous random motion of many components of the state vector x = x(t) of such living systems. Taking this property into consideration causes the rejection of any of the known types of stationary modes (e.g., dx/dt = 0) and requires revision of the concept of chaos. A new approach to the understanding of living systems (as a third paradigm of the natural sciences) and new methods for the study of living systems (as a theory of chaos and self-organization) are proposed. Common grounds of physics and theory of chaos and self-organization are revealed as a generalized uncertainty principle and a limit on the parameters of quasi-attractors.     

THE SWING VIBRATION, TRANSVERSE OSCILLATION OF CRACKED ROTOR AND THE INTERMITTENCE CHAOS

This paper studies the non-linear dynamic response of a cracked rotor by taking the swing vibration of disc into consideration. The results show that if a small crack appears, the frequency of transverse oscillation is synchronous with rotating speed ratio (Ω), and the frequency of swing vibration is N Ω (N=1,2,…). As the crack increases, the response becomes chaotic in some range of Ω. The deeper the crack is, the wider the chaotic range of Ω is. Routes to chaos include intermittence to chaos and quasi-period to chaos. When the crack is fairly deep, some new resonance regions develop. In these regions, the response becomes infinity rapidly. The appearance of intermittence chaos is induced by the frequent frustration of stable oscillation, which is resulted from the continuous increase of swing amplitude. Unbalance parameter U is effective in suppressing chaos. Crack angle β cannot affect the essence of response, but can influence the amplitude of synchronous response.     

Analysis and adaptive synchronization of eight-term 3-D polynomial chaotic systems with three quadratic nonlinearities

This paper proposes a eight-term 3-D polynomial chaotic system with three quadratic nonlinearities and describes its properties. The maximal Lyapunov exponent (MLE) of the proposed 3-D chaotic system is obtained as L ₁ = 6.5294. Next, new results are derived for the global chaos synchronization of the identical eight-term 3-D chaotic systems with unknown system parameters using adaptive control. Lyapunov stability theory has been applied for establishing the adaptive synchronization results. Numerical simulations are shown using MATLAB to describe the main results derived in this paper.     

Quantum chaos in nuclear physics

A definition of classical and quantum chaos on the basis of the Liouville–Arnold theorem is proposed. According to this definition, a chaotic quantum system that has N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) that are determined by the symmetry of the Hamiltonian for the system being considered. Quantitative measures of quantum chaos are established. In the classical limit, they go over to the Lyapunov exponent or the classical stability parameter. The use of quantum-chaos parameters in nuclear physics is demonstrated.     

Limit cycles and chaos in realistic models of the Belousov-Zhabotinskii reaction system

Effects of spatial variation in the Belousov-Zhabotinskii reaction is studied numerically by adopting the Field-Noyes kinetics (Oregonator) and the Zhabotinskii-Zaikin-Korzukhin-Kreitser kinetics. This is carried out for a spatially-discrete model composed ofN equivalent cells interacting through gradient coupling. When the system is near the boundary at which a uniform steady state bifurcates into a limit cycle, it is found with the aid of a perturbation expansion that the above models withN=3 exhibit various types of oscillations depending on the interaction strength between cells. Chaotic characteristics are also observed for a certain region of parameters. It is shown that the ZZKK model withN=2 exhibits a different kind of chaos when the size of the limit cycle becomes sensitive to external parameters, e.g., the concentrations of bromate ion or bromomalonic acid. Although each cell is equivalent, symmetry about cell numbers usually breaks down in a periodic state. It is found, however, that symmetry is recovered for the former kind of chaos, while the latter kind of chaos, there exists an asymmetric chaos as well as symmetric chaos. This has been examined by the time evolution of a certain concentration variable and by its Lorenz plot. In the asymmetric chaos, the Lorenz plot constitutes approximately a one-dimensional map. Furthermore, possible connections of the present limit cycles and chaos with the experiments of Zhabotinskii and Vavilin-Zhabotinskii-Zaikin are suggested.     

Frequency and peak discontinuities in self-pulsations of a CO2 laser with feedback

We show that self-pulsations observed in a CO₂ laser with feedback display two types of recurrent period discontinuities when control parameters are changed. Periodic self-pulsations emerge organized in wide adjacent phases in which oscillations differ by a constant number of peaks in their period. The number of peaks increases through characteristic pulse deformations of the signal that we describe in detail. The passage across the boundaries delimiting adjacent phases is abrupt and not mediated by windows of chaos. In addition, we provide an explicit criterion for locating the discontinuity boundaries between adjacent phases.     

A one dimensional model für quantum Chaos

Spectral rigidityΔ ₃ (L) is calculated for the energy spectra of¹³¹Te,¹³⁵Ba and¹³⁷Ce, obtained in the IBFM calculation. Results reveal an intermediate situation between regularity and chaos, but remarkably closer to the chaotic limit. Influence of the model parameters on the fluctuation pattern is discussed.     

Period-doubling bifurcations,chaos, phase-locking and devil's staircase in a Bonhoeffer-van der Pol oscillator

The effect of a periodic input current A₁ cos t in the Bonhoeffer-van der Pol oscillator along with a bias A₀ is investigated numerically. As the parameter A₁ is varied in the absence of bias by holding the other parameters at constant values, typical period-doubling bifurcation sequence is found to occur leading to chaotic motion in agreement with the Feigenbaum scenario. When the bias is switched on at the transition to chaos, frequency-locking is observed in the system. The frequency-locked intervals exhibit complete devil's staircase similar to the one observed in Belousov-Zhabotinsky reaction.     

Estimating ultimate bound and finding topological horseshoe for a new chaotic system

This paper presents a new three-dimensional autonomous chaotic system with only one positive term. Basic dynamical properties of the new attractor are demonstrated in terms of phase portraits, equilibria, Lyapunov exponents, Poincare mapping, bifurcation diagram. Furthermore, we derive a three-dimensional spheriform ultimate bound and positively invariant set for all the positive values of its parameters a, b, c. At last, the horseshoe chaos in this system is investigated based on the topological theory.     

Modeling nonlinear dissipative chemical dynamics by a forced modified Van der Pol-Duffing oscillator with asymmetric potential: Chaotic behaviors predictions

This paper addresses the issues of nonlinear chemical dynamics modeled by a modified Van der Pol-Duffing oscillator with asymmetric potential. The Melnikov method is utilized to analytically determine the domains boundaries where Melnikov’s chaos appears in chemical oscillations. Routes to chaos are investigated through bifurcations structures, Lyapunov exponent, phase portraits and Poincaré section. The effects of parameters in general and in particular the effect of the constraint parameter β which shows the difference between a nonlinear chemical dynamics order two differential equation and ordinary Van der Pol-Duffing equation are analyzed. Results of analytical investigations are validated and complemented by numerical simulations.     

Quantum signatures of chaos or quantum chaos?

A critical analysis of the present-day concept of chaos in quantum systems as nothing but a “quantum signature” of chaos in classical mechanics is given. In contrast to the existing semi-intuitive guesses, a definition of classical and quantum chaos is proposed on the basis of the Liouville–Arnold theorem: a quantum chaotic system featuring N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) specified by the symmetry of the Hamiltonian of the system. Quantitative measures of quantum chaos that, in the classical limit, go over to the Lyapunov exponent and the classical stability parameter are proposed. The proposed criteria of quantum chaos are applied to solving standard problems of modern dynamical chaos theory.     

Gaussian Multiplicative Chaos and KPZ Duality

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ ² beyond the transition phase (i.e. γ ² > 2d) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and to understand the duality relation in Liouville quantum gravity.     

Nonlinear dynamos: A complex generalization of the Lorenz equations

Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite.     

Controlling chaos by a modified straight-line stabilization method

By adjusting external control signal, rather than some available parameters of the system, we modify the straight-line stabilization method for stabilizing an unstable periodic orbit in a neighborhood of an unstable fixed point formulated by Ling Yang et al., and derive a more simple analytical expression of the external control signal adjustment. Our technique solves the problem that the unstable fixed point is independent of the system parameters, for which the original straight-line stabilization method is not suitable. The method is valid for controlling dissipative chaos, Hamiltonian chaos and hyperchaos, and may be most useful for the systems in which it may be difficult to find an accessible system parameter in some cases. The method is robust under the presence of weak external noise. Received 10 January 2001     

The Lorenz model for single-mode homogeneously broadened laser: analytical determination of the unpredictable zone

We have applied harmonic expansion to derive an analytical solution for the Lorenz-Haken equations. This method is used to describe the regular and periodic self-pulsing regime of the single mode homogeneously broadened laser. These periodic solutions emerge when the ratio of the population decay rate ? is smaller than 0:11. We have also demonstrated the tendency of the Lorenz-Haken dissipative system to behave periodic for a characteristic pumping rate “2C _P ”[7], close to the second laser threshold “2C _2th ”(threshold of instability). When the pumping parameter “2C” increases, the laser undergoes a period doubling sequence. This cascade of period doubling leads towards chaos. We study this type of solutions and indicate the zone of the control parameters for which the system undergoes irregular pulsing solutions. We had previously applied this analytical procedure to derive the amplitude of the first, third and fifth order harmonics for the laser-field expansion [7, 17]. In this work, we extend this method in the aim of obtaining the higher harmonics. We show that this iterative method is indeed limited to the fifth order, and that above, the obtained analytical solution diverges from the numerical direct resolution of the equations.     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.