Dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity

In this paper, we consider the dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity. The amplitude equations are derived in the general case using the harmonic balance method. The model without discontinuity is first considered. The effects of the amplitude of the parametric modulation and some particular coefficients are found in the response curves. The transition to chaotic behavior is found using numerical simulations of the equations of motion. We find that chaos appears in the model between the quasi-periodic and periodic orbits when the amplitude of the external excitation E₀ vary. An adaptive Lyapunov control strategy enables us to drive the system from the chaotic states to a targeting periodic orbit. The effects of elasticity and damping on the dynamics of the self-sustained electromechanical system are also derived.     

Matter-wave chaos with a cold atom in a standing-wave laser field

Coherent motion of cold atoms in a standing-wave field is interpreted as a propagation in two optical potentials. It is shown that the wave-packet dynamics can be either regular or chaotic with transitions between these potentials after passing field nodes. Manifestations of de Broglie-wave chaos are found in the behavior of the momentum and position probabilities and the Wigner function. The probability of those transitions depends on the ratio of the squared detuning to the Doppler shift and is large in that range of the parameters where the classical motion is shown to be chaotic.     

Nonlinear vibration and characteristics of flexible plate-strips with non-symmetric boundary conditions

Regular and chaotic vibrations together with bifurcations of flexible plate-strips with non-symmetric boundary conditions, are investigated through the Bubnov–Galerkin method and a finite difference method of error O(h⁴). Particular attention is paid to non-symmetric boundary conditions. Lyapunov exponents are estimated via Bennetin’s method. Some new examples of routes from regular to chaotic dynamics, and within chaotic dynamics are illustrated and discussed. The phase transitions from chaos to hyperchaos, and a novel phenomenon of a shift from hyperchaos to hyperhyper chaos is also reported.     

Scaling features of multimode motions in coupled chaotic oscillators

Two different methods (the WTMM- and DFA-approaches) are applied to investigate the scaling properties in the return-time sequences generated by a system of two coupled chaotic oscillators. Transitions from twomode asynchronous dynamics (torus or torus–chaos) to different states of chaotic phase synchronization are found to significantly reduce the degree of multiscality. The influence of external noise on the possibility of distinguishing the various chaotic states is considered.     

Regular low frequency cardiac output oscillations observed in critically ill surgical patients

A serendipitous finding during development of an automated “electronic flow chart” system to gather data on ICU patients [1] was the observation of low frequency oscillations in blood pressure that were not explained by systematic variability in the environment. [2] This finding has since been confirmed by others. [3,4] In the present report, hemodynamic data for critically ill surgical patients was continuously collected and visualized on a computer workstation to search for patterns not noted by standard monitoring. With this system, we observed low-frequency periodic oscillations in the cardiac output of ten patients, with regular periodicities of 4 to 280 minutes (average = 34 minutes). The mortality rate in these patients was 40%, while the mortality was only 10.8% in 83 similarly monitored intensive care unit (ICU) patients who did not develop regular oscillations in cardiac output. Interestingly, these oscillatory patterns appear to be associated with inadequate resuscitation of increased metabolic rates. The mathematical definition of “chaos” refers to irregular behavior that appears to be random but is actually deterministic. [5] A surprising finding concerning transitions between states of apparent randomness and order in nonlinear systems is that many systems become more organized after being disturbed. Chaotic behavior in biological systems may represent a normal physiologic state, while the loss of chaotic behavior may herald a pathophysiologic state. [6] The mechanism of the regular low frequency oscillations in cardiac output remains to be determined, but the high mortality rate suggests that it is a pathophysiologic marker, perhaps due to inadequate oxygen delivery in under-resuscitated shock. © 1997 John Wiley & Sons, Inc.     

Chaos in temporarily destabilized regular systems with the slow passage effect

We provide evidences for chaotic behaviour in temporarily destabilized regular systems. In particular, we focus on time-continuous systems with the slow passage effect. The extreme sensitivity of the slow passage phase enables the existence of long chaotic transients induced by random pulsatile perturbations, thereby evoking chaotic behaviour in an initially regular system. We confirm the chaotic behaviour of the temporarily destabilized system by calculating the largest Lyapunov exponent. Moreover, we show that the newly obtained unstable periodic orbits can be easily controlled with conventional chaos control techniques, thereby guaranteeing a rich diversity of accessible dynamical states that is usually expected only in intrinsically chaotic systems. Additionally, we discuss the biological importance of presented results.     

Effective approaches to explore rich dynamics of delay-differential equations

Discrete models are proposed to delve into the rich dynamics of nonlinear delayed systems under Euler discretization, such as backwards bifurcations, stable limit cycles, multiple limit-cycle bifurcations and chaotic behavior. The effect of breaking the special symmetry of the system is to create a wide complex operating conditions which would not otherwise be seen. These include multiple steady states, complex periodic oscillations, chaos by period doubling bifurcations. Effective computation of multiple bifurcations, stable limit cycles, symmetrical breaking bifurcations and chaotic behavior in nonlinear delayed equations is developed.     

Passive aerodynamics control of plasma instabilities

The possibility of using a smart-damping scheme to modify the dynamic responses of plasma oscillations governed by a two-fluid model is considered. The passive aerodynamics control strategy is used to address this issue. The control efficiency is found by analyzing the conditions satisfied by the control gain parameters for which, the amplitude of oscillations is reduced both in the harmonic and chaotic states. In the regular state, the analytical stability analysis uses for linear oscillations the Routh-Hurwitz criterion while the Whittaker method and Floquet theory are utilized for nonlinear harmonic oscillations. The stability boundaries in the control gain parameter space is derived. The agreement between the analytical and numerical results is good. In the chaotic states, numerical simulations are used to perform quenching of chaotic oscillations for an appropriate set of control parameters.     

Local chaos induced by spatial oscillations of a perturbation

We study motion of an one-dimensional Hamiltonian oscillator driven by an external force which is periodic in time and in coordinate as well. It is shown that dynamics of the oscillator is strongly affected by the resonance between spatial and temporal oscillations of the perturbation imposed. In particular, this resonance can induce strong but bounded chaotic diffusion in certain areas of phase space. The model of the Duffing oscillator is used as an example for the numerical simulation.     

Effects of higher nonlinearity on the dynamics and synchronization of two coupled electromechanical devices

The dynamics and synchronization of coupled electromechanical systems with both cubic and quintic nonlinearities are analyzed. A detail attention is carried out to the study of the effects of the introduced quintic nonlinearity on the amplitudes of the harmonic oscillatory states, the stability boundaries of the harmonic oscillations, and on the bifurcation structures. We examine the synchronization phenomena on the unidirectional capacitive and resistive coupled such electromechanical systems both in their regular and chaotic states. The stability of synchronization process is studied follows the Floquet theory and Hill infinite determinant. Numerical simulations confirm and complement the results obtained by the analytical approach.     

Chaos in a generalized van der Pol system and in its fractional order system

In this paper, chaos of a generalized van der Pol system with fractional orders is studied. Both nonautonomous and autonomous systems are considered in detail. Chaos in the nonautonomous generalized van der Pol system excited by a sinusoidal time function with fractional orders is studied. Next, chaos in the autonomous generalized van der Pol system with fractional orders is considered. By numerical analyses, such as phase portraits, Poincaré maps and bifurcation diagrams, periodic, and chaotic motions are observed. Finally, it is found that chaos exists in the fractional order system with the order both less than and more than the number of the states of the integer order generalized van der Pol system.     

Hamiltonian chaos in the interaction of moving two-level atoms with cavity vacuum

We study nonlinear dynamics of the fundamental cavity quantum-electrodynamical system consisting of a point-like collection of identical two-level atoms moving through a lossless single-mode cavity. Taking into account the interatomic and the atom-field quantum correlations of the first order, we go beyond the semiclassical model and derive a dynamical system that is able to describe the vacuum Rabi oscillations with atoms moving in a spatially inhomogeneous cavity field. A simple expression for the equilibrium points of this system provides a class of initial conditions for atoms and a cavity mode under which the atomic population and radiation may be trapped. In the strong-coupling limit and the rotating-wave approximation, the model is shown to be integrable with atoms moving through a resonant cavity with an arbitrary spatial profile of the mode along the propagation axis. The general exact solution is derived in an explicit form in terms of Jacobian elliptic functions. Numerical simulation confirms that perturbations, that are produced by a modulation of the coupling between moving atoms and a cavity mode, provide, out of resonance, a mechanism responsible for Hamiltonian chaos in the interaction of two-level atoms with cavity vacuum. These chaotic vacuum Rabi oscillations may be considered as a new kind of reversible spontaneous emission.     

Dynamical properties of discrete Lotka–Volterra equations

A discrete version of the Lotka–Volterra differential equations for competing population species is analyzed in detail in much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. It is found that in addition to the logistic dynamics – ranging from very simple to manifestly chaotic regimes in terms of governing parameters – the discrete Lotka–Volterra equations exhibit their own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is shown that the system exhibits “twisted horseshoe” dynamics associated with a strange invariant set for certain parameter ranges.     

Nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions

This paper deals with the nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions, described by an electrical Duffing oscillator magnetically coupled to linear mechanical oscillators. Firstly, the amplitudes of the sub- and super-harmonic oscillations for the resonant states are obtained and discussed using the multiple time scales method. The equations of motion are solved numerically using the Runge–Kutta algorithm. It is found that chaotic and periodic orbit coexist in the electromechanical system depending on the set of initial conditions. Secondly, the problem of synchronization dynamics of two coupled electromechanical systems both in the regular and chaotic states is also investigated, and estimation of the coupling coefficient under which synchronization and no-synchronization take place is made.     

Fractals in the open quantum kicked top model

Fractals are one of the most important features of the classically chaotic systems. We analyze the fractal phenomena in a quantum chaos system in terms of its fidelity and dynamical localization properties in the paper. We show that, even in the open and dissipative quantum kicked top model, the fidelity displays fractal fluctuations if the underlying dynamics is in the classically chaotic regime. Moreover, the fluctuations of the inverse participation ratio which characterize the dynamical localization behavior also exhibit fractality. The relations between the fractal dimensions and the decoherence rates are explored.     

Transition to chaos in the self-excited system with a cubic double well potential and parametric forcing

We examine the Melnikov criterion for a global homoclinic bifurcation and a possible transition to chaos in case of a single degree of freedom nonlinear oscillator with a symmetric double well nonlinear potential. The system was subjected simultaneously to parametric periodic forcing and self-excitation via negative damping term. Detailed numerical studies confirm the analytical predictions and show that transitions from regular to chaotic types of motion are often associated with increasing the energy of an oscillator and its escape from a single well.     

Cycles,chaos, and noise in predator–prey dynamics

In contrast to the single species models that were extensively studied in the 1970s and 1980s, predator–prey models give rise to long-period oscillations, and even systems with stable equilibria can display oscillatory transients with a regular frequency. Many fluctuating populations appear to be governed by such interactions. However, predator–prey models have been poorly studied with respect to the interaction of nonlinear dynamics, noise, and system identification. I use simulated data from a simple host–parasitoid model to investigate these issues. The addition of even a modest amount of noise to a stable equilibrium produces enough structured variation to allow reasonably accurate parameter estimation. Despite the fact that more-or-less regular cycles are generated by adding noise to any of the classes of deterministic attractor (stable equilibrium, periodic and quasiperiodic orbits, and chaos), the underlying dynamics can usually be distinguished, especially with the aid of the mechanistic model. However, many of the time series can also be fit quite well by a wrong model, and the fitted wrong model usually misidentifies the underlying attractor. Only the chaotic time series convincingly rejected the wrong model in favor of the true one. Thus chaotic population dynamics offer the best chance for successfully identifying underlying regulatory mechanisms and attractors.     

Using chaos to reduce oscillations: Experimental results

Chaotic dynamic systems are usually controlled in a way, which allows the replacement of chaotic behavior by the desired periodic motion. We give the example in which an originally regular (periodic) system is controlled in such a way as to make it chaotic. This approach based on the idea of dynamical absorber allows the significant reduction of the amplitude of the oscillations in the neighborhood of the resonance. We present experimental results, which confirm our previous numerical studies [D?browski A, Kapitaniak T. Using chaos to reduce oscillations. Nonlinear Phenomen Complex Syst 2001;4(2):206–11].     

Generalized synchronization of strictly different systems: Partial-state synchrony

Generalized synchronization (GS) occurs when the states of one system, through a functional mapping are equal to states of another. Since for many physical systems only some state variables are observable, it seems convenient to extend the theoretical framework of synchronization to consider such situations. In this contribution, we investigate two variants of GS which appear between strictly different chaotic systems. We consider that for both the drive and response systems only one observable is available. For the case when both systems can be taken to a complete triangular form, a GS can be achieved where the functional mapping between drive and response is found directly from their Lie-algebra based transformations. Then, for systems that have dynamics associated to uncontrolled and unobservable states, called internal dynamics, where only a partial triangular form is possible via coordinate transformations, for this situation, a GS is achieved for which the coordinate transformations describe the functional mapping of only a few state variables. As such, we propose definitions for complete and partial-state GS. These particular forms of GS are illustrated with numerical simulations of well-known chaotic benchmark systems.     

Global multifractal relation between topological entropies and fractal dimensions

In one-dimensional chaotic dynamics, a global multifractal relation between topological entropies and fractal dimensions of arbitrary period-p-tupling attractors is analyzed on all critical (accumulation) points of transitions to chaos, where the Lyapunov characteristic exponent is zero. The global metric regularity of topological entropies versus fractal dimensions is well characterized by the self-similarity. By the fractal interpolation based on the iterated function system, the fractal dimensions of the curves of topological entropies versus capacity dimensions and versus information dimensions are both found to be 1.82.