Stability and Noise-induced Transitions in an Ensemble of Nonlocally Coupled Chaotic Maps

The influence of noise on chimera states arising in ensembles of nonlocally coupled chaotic maps is studied. There are two types of chimera structures that can be obtained in such ensembles: phase and amplitude chimera states. In this work, a series of numerical experiments is carried out to uncover the impact of noise on both types of chimeras. The noise influence on a chimera state in the regime of periodic dynamics results in the transition to chaotic dynamics. At the same time, the transformation of incoherence clusters of the phase chimera to incoherence clusters of the amplitude chimera occurs. Moreover, it is established that the noise impact may result in the appearance of a cluster with incoherent behavior in the middle of a coherence cluster.     

Nonadiabatic couplings and chaos in a dynamical potential well model

The mechanism of nonadiabatic couplings between quantum states of a potential well model with finite heights and a dynamical width coordinate is investigated in detail. The system is described in a mixed quantum-classical approach in which the oscillations of the classical width coordinate induce transitions between the quantum states of a particle trapped inside the well. The dynamics of the system is considered in detail for transitions between two quantum states and resulting coupled Bloch-oscillator equations. Poincaré sections showing a mixed phase space with chaotic and regular behaviour are found by a numerical investigation. In particular, chaos results for high energies of the well width oscillations when the mixing between the adiabatic reference states is strong. The inclusion of relaxation is considered and shown that in this case the regimes of chaotic and regular dynamics are not separated as in the relaxation free case. In particular, for some initial conditions chaos can become a transient phenomena placed in a time window between regular oscillations of the system.     

Entanglement,fidelity and quantum chaos in cavity QED

Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality microcavity is studied. We consider the strongly coupled atom–field system as a quantum–classical hybrid with dynamically coupled quantum and classical degrees of freedom. We show that, even in the absence of any other interaction with environment, the coupling of quantum and classical degrees of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics. Chaos manifests itself in the atomic external degree of freedom as a random walking of an atom inside a cavity with prominent fractal-like behavior and in the quantum atom–field degrees of freedom as a sensitive dependence of atomic inversion on small variations in initial conditions. It is shown that dependences of variance of quantum entanglement and of the maximum Lyapunov exponent on the detuning of the atom–field resonance correlate strongly. It is shown that the Jaynes–Cummings dynamics can be unstable in the regime of chaotic walking of an atom in the quantized field of a standing wave in the absence of any other interaction with environment. Quantum instability manifests itself in strong variations of quantum purity and entropy and in exponential sensitivity of fidelity of quantum states to small variations in the atom–field detuning. It is quantified in terms of the respective classical maximal Lyapunov exponent that can be estimated in appropriate in–out experiments. This result provides a quantum–classical correspondence in a closed physical system.     

Transiently chaotic neural networks with piecewise linear output functions

Admitting both transient chaotic phase and convergent phase, the transiently chaotic neural network (TCNN) provides superior performance than the classical networks in solving combinatorial optimization problems. We derive concrete parameter conditions for these two essential dynamic phases of the TCNN with piecewise linear output function. The confirmation for chaotic dynamics of the system results from a successful application of the Marotto theorem which was recently clarified. Numerical simulation on applying the TCNN with piecewise linear output function is carried out to find the optimal solution of a travelling salesman problem. It is demonstrated that the performance is even better than the previous TCNN model with logistic output function.     

On resonant interactions of gravity‐capillary waves without energy exchange

We consider resonant triad interactions of gravity‐capillary waves and investigate in detail special resonant triads that exchange no energy during their interactions so that the wave amplitudes remain constant in time. After writing the resonance conditions in terms of two parameters (or two angles of wave propagation), we first identify a region in the two‐dimensional parameter space, where resonant triads can be always found, and then describe the variations of resonant wavenumbers and wave frequencies over the resonance region. Using the amplitude equations recovered from a Hamiltonian formulation for water waves, it is shown that any resonant triad inside the resonance region can interact without energy exchange if the initial wave amplitudes and relative phase satisfy the two conditions for fixed point solutions of the amplitude equations. Furthermore, it is shown that the symmetric resonant triad exchanging no energy forms a transversely modulated traveling wave field, which can be considered a two‐dimensional generalization of Wilton ripples.     

Golden mean relevance for chaos inhibition in a system of two coupled modified van der Pol oscillators

In this work, we present a novel evidence of the importance of the golden mean criticality of a system of oscillators in agreement with El Naschie’s E-infinity theory. We focus on chaos inhibition in a system of two coupled modified van der Pol oscillators. Depending on the coupling between the two oscillators, the system shows chaotic behavior for different ranges of the coupling parameter. Chaos suppression, as a transition from irregular behavior to a periodical one, is induced by perturbing the system with a harmonic signal with amplitude considerably lower than the value which causes entrainment. The frequency of the perturbation is related to the main frequencies in the spectrum of the freely running system (without perturbation) by the golden mean. We demonstrate that this effect is also obtained for a perturbation with frequency such that the ratio of half the frequency of the first main component in the freely running chaotic spectrum over the frequency of the perturbation is very close (five digits coincidence) to the golden mean. This result is shown to hold for arbitrary values of the coupling parameter in the various ranges of chaotic dynamics of the free running system.     

Bistability of rotational modes in a system of coupled pendulums

The main goal of this research is to examine any peculiarities and special modes observed in the dynamics of a system of two nonlinearly coupled pendulums. In addition to steady states, an in-phase rotation limit cycle is proved to exist in the system with both damping and constant external force. This rotation mode is numerically shown to become unstable for certain values of the coupling strength. We also present an asymptotic theory developed for an infinitely small dissipation, which explains why the in-phase rotation limit cycle loses its stability. Boundaries of the instability domain mentioned above are found analytically. As a result of numerical studies, a whole range of the coupling parameter values is found for the case where the system has more than one rotation limit cycle. There exist not only a stable in-phase cycle, but also two out-of phase ones: a stable rotation limit cycle and an unstable one. Bistability of the limit periodic mode is, therefore, established for the system of two nonlinearly coupled pendulums. Bifurcations that lead to the appearance and disappearance of the out-ofphase limit regimes are discussed as well.     

Dynamics of four coupled phase-only oscillators

We study the dynamics of a system of four coupled phase-only oscillators. This system is analyzed using phase difference variables in a phase space that has the topology of a three-dimensional torus. The system is shown to exhibit numerous phase-locked motions. The qualitative dynamics are shown to depend upon a parameter representing coupling strength. This work has application to MEMS artificial intelligence decision-making devices.     

Multistability in a system of two coupled oscillators with delayed feedback

In this paper, nonlocal dynamics of a system of two differential equations with a compactly supported nonlinearity and delay is studied. For some set of initial conditions asymptotics of solutions of considered system is constructed. By this asymptotics we build a special mapping. Dynamics of this mapping describes dynamics of initial system in general: it is proved that stable cycles of this mapping correspond to exponentially orbitally stable relaxation periodic solutions of initial system of delay differential equations. It is shown that amplitude, period of solutions of initial system, and number of coexisting stable solutions depend crucially on coupling parameter. Algorithm for constructing many coexisting stable solutions is described.     

Mean‐Field Evolution of Fermionic Mixed States

In this paper we study the dynamics of fermionic mixed states in the mean‐field regime. We consider initial states that are close to quasi‐free states and prove that, under suitable assumptions on the initial data and on the many‐body interaction, the quantum evolution of such initial data is well approximated by a suitable quasi‐free state. In particular, we prove that the evolution of the reduced one‐particle density matrix converges, as the number of particles goes to infinity, to the solution of the time‐dependent Hartree‐Fock equation. Our result holds for all times and gives effective estimates on the rate of convergence of the many‐body dynamics towards the Hartree‐Fock evolution.© 2015 Wiley Periodicals, Inc.     

Dynamics of a driven magneto-martensitic ribbon

We investigate the dynamics of a sinusoidally driven ferromagnetic martensitic ribbon by adopting a recently introduced model that involves strain and magnetization as order parameters. Retaining only the dominant mode of excitation we reduce the coupled set of partial differential equations for strain and magnetization to a set of coupled ordinary nonlinear equations for the strain and magnetization amplitudes. The equation for the strain amplitude takes the form of parametrically driven oscillator. Finite strain amplitude can only be induced beyond a critical value of the strength of the magnetic field. Chaotic response is seen for a range of values of all the physically interesting parameters. The nature of the bifurcations depends on the choice of temperature relative to the ordering of the Curie and the martensite transformation temperatures. We have studied the nature of response as a function of the strength and frequency of the magnetic field, and magneto-elastic coupling. In general, the bifurcation diagrams with respect to these parameters do not follow any standard route. The rich dynamics exhibited by the model is further illustrated by the presence of mixed mode oscillations seen for low frequencies. The geometric structure of the mixed mode oscillations in the phase space has an unusual deep crater structure with an outer and inner cone on which the orbits circulate. We suggest that these features should be seen in experiments on driven magneto-martensitic ribbons.     

On the Caginalp system with dynamic boundary conditions and singular potentials

This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in H ², the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Lojasiewicz inequality in order to prove the convergence of solutions to steady states.      

Well‐posedness and asymptotic analysis for a Penrose–Fife type phase field system

In this paper, an asymptotic analysis of the (non‐conserved) Penrose–Fife phase field system for two vanishing time relaxation parameters ε and δ is developed, in analogy with the similar analyses for the phase field model proposed by G. Caginalp (Arch. Rational Mech. Anal. 1986; 92 :205–245), which were carried out by Rossi and Stoth (Adv. Math. Sci. Appl. 2003; 13 :249–271; Quart. Appl. Math. 1995; 53 :695–700). Although formally the singular limits for ε ↓ 0 and for ε and δ ↓ 0 are, respectively, the viscous Cahn–Hilliard equation and the Cahn–Hilliard equation, it turns out that the Penrose–Fife system is indeed a bad approximation for these equations. Therefore, we consider an alternative approximating phase field system, which could be viewed as a generalization of the classical Penrose–Fife phase field system, featuring a double non‐linearity given by two maximal monotone graphs. A well‐posedness result is proved for such a system, and it is shown that the solutions converge to the unique solution of the viscous Cahn–Hilliard equation as ε ↓ 0, and of the Cahn–Hilliard equation as ε ↓ 0 and δ ↓ 0. Copyright © 2004 John Wiley & Sons, Ltd.     

Transient modelling of flow distribution in automotive catalytic converters

The transient catalytic converter performance is governed by complex interactions between exhaust gas flow and the monolithic structure of the catalytic converter. Therefore, during typical operating conditions of interest, one has to take into account the effect of the inlet diffuser on the flow field at the entrance. Computational fluid dynamics (CFD) is a powerful tool for calculating the flow field inside the catalytic converter. Radial velocity profiles, obtained by a commercial CFD code, present very good agreement with respective experimental results published in the literature. However the applicability of CFD for transient simulations is limited by the high CPU demands.The present study proposes an alternative computational method for the prediction of transient flow fields in axi-symmetric converters time-efficiently. The method is based on the use of equivalent flow resistances to simulate the flow paths in the inlet and outlet catalyst sections. The proposed flow resistance modelling (FRM) method is validated against the results of CFD predictions over a wide range of operating conditions. Apart from the apparent CPU advantages, the proposed methodology can be readily coupled with already available transient models for the chemical reactions in the catalyst. A transient model for heat transfer inside the monolith is presented. An example of coupling between FRM and transient heat transfer inside the converter is included. This example illustrates the effect of flow distribution in the thermal response of a catalytic converter, during the critical phase of catalytic converter warm-up.     

Modelling and nanoscale force spectroscopy of frequency modulation atomic force microscopy

In this paper, a simulation model for frequency modulation atomic force microscopy (FM-AFM) operating in constant amplitude dynamic mode is presented. The model is based on the slow time varying function theory. The mathematical principles to derive the dynamical equations for the amplitude and phase of the FM-AFM cantilever-tip motion are explained and the stability and performance of its closed-loop controller to keep the amplitude at constant value and phase at 90° is analysed. Then, the performance of the theoretical model is supported by comparison of numerical simulations and experiments. Furthermore, the transient behaviour of amplitude, phase and frequency shift of FM-AFM is investigated and the effect of controller gains on the transient motion is analysed. Finally, the derived FM-AFM model is used to simulate the single molecule/nanoscale force spectroscopy and study the effect of sample viscosity, stiffness and Hamaker constant on the response of FM-AFM.     

Synchronization in time-discrete model of two electrically coupled spike-bursting neurons

Dynamics of the ensemble of two model neurons interacting through electrical synapse is investigated. Both neurons are described by two-dimensional discontinuous map. It is shown that in four-dimensional phase space a chaotic attractor of relaxation type exists corresponding to spike-bursting chaotic oscillations. A new effect of recurrent transitory chaotic oscillations underlies a dynamical mechanism of chaotic bursts formation. It is shown that, under coupling, the transient from chaotic bursts generation into rest state occurs with a time delay. A new characteristic estimating the degree of spike-bursting synchronization is introduced. Dependence of the synchronism degree on the coupling strength is shown for some coupling interval where only activity synchronization occurs. A probabilistic study provides a dynamical explanation of these phenomena.     

Non-stationary nonlinear analysis of a composite rotating shaft passing through critical speed

In this paper, nonlinear non-stationary dynamics of a nonlinear composite shaft passing through critical speed is studied. The nonlinearity is due to the large amplitude of shaft vibration. The equations of motion are obtained by three-dimensional constitutive relationships of composite materials. The gyroscopic effect, rotary inertia and coupling caused by material anisotropy are considered but shear deformation is neglected. Without any simplification, axial-flexural-flexural-torsional equations of motion (EOM) for the elastic composite shaft with variable rotational speed are obtained. The approximate analytical method namely asymptotic method is applied to analyze the nonstationary behavior of the composite shaft with constant acceleration. First, the EOMs are discretized using one and two-term Galerkin method. Then, the resulted equations are transformed to normal coordinates. Finally, the asymptotic method is applied to equations described in normal coordinates. Analytical expressions governing the amplitude and phase of motion during passage through critical speeds are obtained. By comparing the results obtained from analytical solutions, it is shown that discretization by one mode is not enough due to the existence of coupling in the equations and at least two modes are necessary for this purpose. Effects of damping, eccentricity, initial angular velocity and fiber angle on response amplitude are investigated. For verification, the results of perturbation theory are compared with numerical simulations and it is shown that there is good agreement between both methods.     

“Coherence–incoherence” transition in ensembles of nonlocally coupled chaotic oscillators with nonhyperbolic and hyperbolic attractors

We consider in detail similarities and differences of the “coherence–incoherence” transition in ensembles of nonlocally coupled chaotic discrete-time systems with nonhyperbolic and hyperbolic attractors. As basic models we employ the Hénon map and the Lozi map. We show that phase and amplitude chimera states appear in a ring of coupled Hénon maps, while no chimeras are observed in an ensemble of coupled Lozi maps. In the latter, the transition to spatio-temporal chaos occurs via solitary states. We present numerical results for the coupling function which describes the impact of neighboring oscillators on each partial element of an ensemble with nonlocal coupling. Varying the coupling strength we analyze the evolution of the coupling function and discuss in detail its role in the “coherence–incoherence” transition in the ensembles of Hénon and Lozi maps.     

Modulation of synchronization dynamics in a network of self-sustained systems

This paper addresses the combined modulatory effects of non-nearest neighbor oscillators and local injection on synchronized states dynamics with their corresponding stability boundaries in a network of self-sustained systems. The Whittaker method and Floquet theory are used to predict analytically the stability of these states for identical and non-identical coupling parameters. Charts revealing the modulation of synchronized states and their stability boundaries at the second order of interaction in the cases of identical and non-identical coupling parameters are constructed with and without an external signal locally injected in the network. Numerical simulations validate and complement the results of analytical surveys. The limits of the stability regions are numerically explored when a small amount of Gaussian white noise is also injected in the network.