Applications of deformed oscillators

Some variants of the definitions and properties of deformed oscillators are reviewed. The q-analogs of coherent states are discussed. We also consider some applications of deformed oscillators, including q-oscillator representations of the simplest quantum algebras and superalgebras, q-coherent states of the su_q(2) and su_q(1,1) quantum algebras of the Jaynes-Cummings model. Generalizations are given for the case of several degrees of freedom.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 37–74, 1991.     

Synchronization dynamics in a ring of four mutually coupled biological systems

This paper considers the synchronization dynamics in a ring of four mutually coupled biological systems described by coupled Van der Pol oscillators. The coupling parameter are non-identical between oscillators. The stability boundaries of the process are first evaluated without the influence of the local injection using the eigenvalues properties and the fourth-order Runge–Kutta algorithm. The effects of a locally injected trajectory on the stability boundaries of the synchronized states are performed using numerical simulations. In both cases, the stability boundaries and the main dynamical states are reported on the stability maps in the (K₁, K₂) plane.     

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.     

Coherent states for the Hartmann potential

We obtain the coherent states for a particle in the noncentral Hartmann potential by transforming the problem into four isotropic oscillators evolving in a parametric time. We use path integration over the holomorphic coordinates to find the quantum states for these oscillators. The decomposition of the transition amplitudes gives the coherent states and their parametric-time evolution for the particle in the Hartmann potential. We also derive the coherent states in the parabolic coordinates by considering the transition amplitudes between the coherent states and eigenstates in the configuration space. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 439–452, June, 2008.     

Generalized coherent states for oscillators connected with Meixner and Meixner—Pollaczek polynomials

The authors continue to study generalized coherent states for oscillator-like systems connected with a given family of orthogonal polynomials. In this work, we consider oscillators connected with Meixner and Meixner— Pollaczek polynomials and define generalized coherent states for these oscillators. A completeness condition for these states is proved by solution of a related classical moment problem. The results are compared with the other authors ones. In particular, we show that the Hamiltonian of the relativistic model of a linear harmonic oscillator can be treated as the linearization of a quadratic Hamiltonian, which arises naturally in our formalism. Bibliography: 56 titles. The authors dedicate this work to their friend and colleague P. P. Kulish on the occasion of his 60th birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 317, 2004, pp. 66–93.     

Dynamics of three coupled limit cycle oscillators with application to artificial intelligence

We study a system of three limit cycle oscillators which exhibits two stable steady states. The system is modeled by both phase-only oscillators and by van der Pol oscillators. We obtain and compare the existence, stability and bifurcation of the steady states in these two models. This work is motivated by application to the design of a machine which can make decisions by identifying a given initial condition with its associated steady state.     

Periodic Gibbs states

Periodic Gibbs states for quantum lattice systems are investigated. We formulate the definition of the periodic Gibbs states and the measures associated with them. Theorems of existence are proved for these states. We also prove the existence of the critical temperature for the system of anharmonic quantum oscillators with pairwise interaction.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 451–458, April, 1993.     

Stability Switches and Hopf Bifurcations in a Pair of Delay-Coupled Oscillators

In this paper, we consider a pair of delay-coupled limit-cycle oscillators. Regarding the arithmetical average of two delays as a parameter, we investigate the effect of time delays on its dynamics. We show that there exist stability switches for time delays under certain conditions, which do not occur for the corresponding coupled system without time delays. A similar result has been reported for the same delay by Ramana Reddy et al. (Physica D, 129 [1999]), but in the present paper we give more detailed and specific conditions determining the amplitude death for different delays. On the other hand, we also investigate Hopf bifurcations induced by time delays using the normal form theory and center manifold reduction. In the region where the stability switches may occur, we not only specifically determine the direction of Hopf bifurcations but also show that the bifurcating periodic solutions are orbitally asymptotically stable. Numerical simulation results are also given to support the theoretical predictions.     

Vibration analysis and bifurcations in the self-sustained electromechanical system with multiple functions

We consider in this paper the dynamics of the self-sustained electromechanical system with multiple functions, consisting of an electrical Rayleigh–Duffing oscillator, magnetically coupled with linear mechanical oscillators. The averaging and the harmonic balance method are used to find the amplitudes of the oscillatory states respectively in the autonomous and nonautonomous cases, and analyze the condition in which the quenching of self-sustained oscillations appears. The influence of system parameters as well as the number of linear mechanical oscillators on the bifurcations in the response of this electromechanical system is investigated. Various bifurcation structures, the stability chart and the variation of the Lyapunov exponent are obtained, using numerical simulations of the equations of motion.     

Synchronization of Colpitts oscillators with different orders

In this paper, we consider chaos synchronization between chaotic Colpitts oscillators with different orders, consisting of standard and improved version of Colpitts oscillators. Firstly, the normalized state equation of the improved version of the Colpitts oscillator designed to operate in the ultrahigh frequency range are presented. It is found that this version is described by fourth-order nonlinear differential equations. The equations of motion are solved numerically using the Runge–Kutta algorithm and simulations demonstrate chaos in the microwave frequencies range. Secondly, the problem of synchronization dynamics of third and fourth orders systems in the chaotic states is also investigated, and a controller is proposed based on stability theory by constructing the Lyapunov function, to ensure synchronization between both oscillators. Computer experiments demonstrate the effectiveness and feasibility of the proposed technique for these oscillators.     

Limit Cycle-Strange Attractor Competition

To understand the competition between what are known as limit cycle and strange attractor dynamics, the classical oscillators that display such features were coupled and studied with and without external forcing. Numerical simulations show that, when the Duffing equation (the strange attractor prototype) forces the van der Pol oscillator (the limit cycle prototype), the limit cycle is destroyed. However, when the van der Pol oscillator is coupled to the Duffing equation as linear forcing, the two traditionally stable steady states are destabilized and a quasi-periodic orbit is born. In turn, this limit cycle is eventually destroyed because the coupling strength is increased and eventually gives way to strange attractor or chaotic dynamics. When two van der Pol oscillators are coupled in the absence of external periodic forcing, the system approaches a stable, nonzero steady state when the coupling strengths are both unity; trajectories approach a limit cycle if coupling strengths are equal and less than 1. Solutions grow unbounded if the coupling strengths are equal and greater than 1. Quasi-periodic solutions give way to chaos as the coupling strength increases and one oscillator is strongly coupled to the other. Finally, increasing the nonlinearity in both the oscillators is stabilizing whereas increasing the nonlinearity in a single oscillator results in subcritical instability.     

Synchronization and anti-synchronization of chaotic oscillators under input saturation

This paper addresses the design of simple state feedback controllers for synchronization and anti-synchronization of chaotic oscillators under input saturation and disturbance. By employing sector condition, linear matrix inequality (LMI)-based sufficient conditions are derived to design (global or local) controllers for chaos synchronization. The proposed local synchronization strategy guarantees a region of stability in terms of difference between states of the master–slave systems. This region of stability can be enlarged by means of an LMI-based optimization algorithm, through which asymptotic synchronization of chaotic oscillators can be ensured for a large difference in their initial conditions. Further, a novel LMI-based robust control strategy is developed, for local synchronization of input-constrained chaotic oscillators, by providing an upper bound on synchronization error in terms of disturbance and initial conditions of chaotic systems. Moreover, the proposed robust state feedback control methodology is modified to provide an inaugural treatment for robust anti-synchronization of chaotic systems under input saturation and disturbance. The results of the proposed methodologies are verified through numerical simulations for synchronization and anti-synchronization of the master–slave chaotic Chua’s circuits under input saturation.     

Landau Damping to Partially Locked States in the Kuramoto Model

In the Kuramoto model of globally coupled oscillators, partially locked states (PLS) are stationary solutions that capture the emergence of partial synchrony when the interaction strength increases. While PLS have long been considered, existing results on their stability are limited to neutral stability of the linearized dynamics in strong topology or to specific invariant subspaces (obtained via the so‐called Ott‐Antonsen (OA) ansatz) with specific frequency distributions for the oscillators. In the mean‐field limit, the Kuramoto model shows various ingredients of the Landau damping mechanism in the Vlasov equation. This analogy has been a source of inspiration for stability proofs of regular Kuramoto equilibria. In addition, the major mathematical issue with PLS asymptotic stability is that these states consist of heterogeneous and singular measures. Here we establish an explicit criterion for their spectral stability and prove their local asymptotic stability in weak topology for a large class of analytic frequency marginals. The proof strongly relies on a suitable functional space that contains (Fourier transforms of) singular measures, and for which the linearized dynamics is well under control. For illustration, the stability criterion is evaluated in some standard examples. We confirm in particular that no loss of generality results in assuming the OA ansatz. To the best of our knowledge, our result provides the first proof of Landau damping to heterogeneous and irregular equilibria in the absence of dissipation. © 2018 Wiley Periodicals, Inc.     

Chaos synchronization between single and double wells Duffing–Van der Pol oscillators using active control

This paper presents chaos synchronization between single and double wells Duffing–Van der Pol (DVP) oscillators with Φ⁴ potential based on the active control technique. The technique is applied to achieve global synchronization between identical double-well DVP oscillators, identical single-well DVP oscillators and non-identical DVP oscillators, consisting of the double-well and the single-well DVP oscillators, respectively. Numerical simulations are also presented to verify the analytical results.     

Synchronization of a chaotic electronic circuit system with cubic term via adaptive feedback control

This paper presents an adaptive feedback control scheme for the synchronization of the chaotic system consisting of Van der Pol oscillators coupled to linear oscillators with cubic term when the parameters of the master system are unknown and different with the those of the slave system. Based on the Lyapunov stability theory, an adaptive control law is derived to make the states of two slightly mismatched chaotic systems asymptotically synchronized. This method is efficient and easy to implement. Numerical simulations results confirming the analytical predictions are shown and pspice simulations are also performed to confirm the efficiency of the proposed control scheme.     

Reliability of repairable multi-state two-phase mission systems with finite number of phase switches

In this article, a repairable multi-state two-phase mission system with finite number of phase switches is considered. System states can be divided into a subset of working states and a subset of failed states, and the system can be repaired from the failed subset to the working subset. However, when the number of switches between two phases exceeds a specified integer, the system will be permanently damaged and cannot be repaired any more. The closed-form formulas of the probability density function of the time to first failure of the system, the point-wise availability, the interval availability and other reliability indexes are obtained by using the theory of aggregated stochastic processes. Finally, a detailed example of lithium-ion batteries is given to illustrate the proposed model and obtained results.     

Isotropy of Angular Frequencies and Weak Chimeras with Broken Symmetry

The notion of a weak chimeras provides a tractable definition for chimera states in networks of finitely many phase oscillators. Here, we generalize the definition of a weak chimera to a more general class of equivariant dynamical systems by characterizing solutions in terms of the isotropy of their angular frequency vector—for coupled phase oscillators the angular frequency vector is given by the average of the vector field along a trajectory. Symmetries of solutions automatically imply angular frequency synchronization. We show that the presence of such symmetries is not necessary by giving a result for the existence of weak chimeras without instantaneous or setwise symmetries for coupled phase oscillators. Moreover, we construct a coupling function that gives rise to chaotic weak chimeras without symmetry in weakly coupled populations of phase oscillators with generalized coupling.     

Adaptive control and synchronization of chaotic systems consisting of Van der Pol oscillators coupled to linear oscillators

This paper deals with the problem of control and synchronization of coupled second-order oscillators showing a chaotic behavior. A classical feedback controller is first used to stabilize the system at its equilibrium. An adaptive observer is then designed to synchronize the states of the master and slave oscillators using a single scalar signal corresponding to an observable state variable of the driving oscillator. An interesting feature of the proposed approach is that it can be used for chaos control as well as synchronization purposes. Numerical simulations results confirming the analytical predictions are shown and pspice simulations are also performed to confirm the efficiency of the proposed control scheme.     

Fractional dynamics of systems with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction defined by a term proportional to 1/∣n − m∣^α+1. Continuous medium equation for this system can be obtained in the so-called infrared limit when the wave number tends to zero. We construct a transform operator that maps the system of large number of ordinary differential equations of motion of the particles into a partial differential equation with the Riesz fractional derivative of order α, when 0 < α < 2. Few models of coupled oscillators are considered and their synchronized states and localized structures are discussed in details. Particularly, we discuss some solutions of time-dependent fractional Ginzburg–Landau (or nonlinear Schrodinger) equation.     

A continuous model for the theory of organizational structure

We investigate a general class of linear models of dyadic interactions with a constant discrete time delay. We prove that the changes in stability of the stationary states occur for various intervals of the parameters that determine the strength and nature of emotional interactions between the partners. The dynamics of interactions depend on both reactivity of partners to their own emotional states as well as to the partner's states. The results suggest that reactivity to the partner's states has greater impact on the dynamics of the relationship than the reactivity to one's own states. Moreover, the results underscore the importance of deliberation in maintaining the stability of the relationship. Moreover, we have found that multiple stability switches are only possible when one of the partners reacts with delay to their own emotional states. We also propose a generalization to triadic interactions.