Effective long-time phase dynamics of limit-cycle oscillators driven by weak colored noise

An effective white-noise Langevin equation is derived that describes long-time phase dynamics of a limit-cycle oscillator driven by weak stationary colored noise. Effective drift and diffusion coefficients are given in terms of the phase sensitivity of the oscillator and the correlation function of the noise, and are explicitly calculated for oscillators with sinusoidal phase sensitivity functions driven by two typical colored Gaussian processes. The results are verified by numerical simulations using several types of stochastic or chaotic noise. The drift and diffusion coefficients of oscillators driven by chaotic noise exhibit anomalous dependence on the oscillator frequency, reflecting the peculiar power spectrum of the chaotic noise.     

An approach to chaotic synchronization

This paper deals with the chaotic oscillator synchronization. An approach to the synchronization of chaotic oscillators has been proposed. This approach is based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators. It has been shown that complete synchronization, phase synchronization, lag synchronization, and generalized synchronization are the particular cases of the synchronized behavior called "time-scale synchronization." The quantitative measure of chaotic oscillator synchronous behavior has been proposed. This approach has been applied for the coupled R?ssler systems and two coupled Chua's circuits.     

Transition to fully developed chaos in a system of two unidirectionally coupled backward-wave oscillators

The chaotic dynamics of a system of two unidirectionally coupled backward-wave oscillators (BWOs) is studied in the case when a signal from the driving BWO in (periodic or chaotic) self-modulation mode is applied to the driven oscillator, which exhibits strong periodic self-modulation in the autonomous case. The oscillation evolution with the amount of coupling is traced. The use of a chain of coupled BWOs is shown to significantly reduce the threshold of transition to the regime of wide-band chaotic oscillations with a uniform continuous spectrum (so-called fully developed chaos), which is of interest for applications.     

Self-organization of temporal structures in a multiequilibrium self-excited oscillator system with frequency control

An analysis is made of the nonlinear dynamics of a model of a self-excited oscillator system with automatic fine tuning of the frequency and possessing more than one equilibrium state. It is shown that, depending on the delay parameters of the control loop and the initial frequency detuning, various periodic and stochastic temporal structures may be formed, accompanied by the generation of various limit cycles and chaotic attractors in phase space. Reasons for the onset of self-modulation are set forth, and the position of the different oscillation regions is established. The main bifurcations are studied together with scenarios for the transformation of the oscillator self-modulation modes as a function of the parameters. Zh. Tekh. Fiz. 67, 1–8 (March 1997)     

Quantum dissipative chaos in the statistics of excitation numbers

A quantum manifestation of chaotic classical dynamics is found in the framework of oscillatory number statistics for the model of a nonlinear dissipative oscillator. It is shown that the probability distributions and variances of oscillatory number states are strongly transformed in the order-to-chaos transition. A nonclassical, sub-Poissonian statistics of oscillatory excitation numbers is established for chaotic dissipative dynamics in the framework of the Fano factor and Wigner functions. It is proposed to use these results in experimental studies of the quantum dissipative chaos.     

Network reconstruction from random phase resetting

We propose a novel method of reconstructing the topology and interaction functions for a general oscillator network. An ensemble of initial phases and the corresponding instantaneous frequencies is constructed by repeating random phase resets of the system dynamics. The desired details of network structure are then revealed by appropriately averaging over the ensemble. The method is applicable for a wide class of networks with arbitrary emergent dynamics, including full synchrony.     

Three-dimensional chaotic autonomous van der pol–duffing type oscillator and its fractional-order form

In this paper, a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator is proposed. The three-dimensional autonomous VdPD oscillator is obtained by replacing the external periodic drive source of two-dimensional chaotic nonautonomous VdPD type oscillator by a direct positive feedback loop. By analyzing the stability of the equilibrium points, the existence of Hopf bifurcation is established. The dynamical properties of proposed three-dimensional autonomous VdPD type oscillator is investigated showing that for a suitable choice of the parameters, it can exhibit periodic behaviors, chaotic behaviors and coexistence between periodic and chaotic attractors. Moreover, the physical existence of the chaotic behavior and coexisting attractors found in three-dimensional proposed autonomous VdPD type oscillator is verified by using Orcard-PSpice software. A good qualitative agreement is shown between the numerical simulations and Orcard-PSpice results. In addition, fractional-order chaotic three-dimensional proposed autonomous VdPD type oscillator is studied. The lowest order of the commensurate form of this oscillator to exhibit chaotic behavior is found to be 2.979. The stability analysis of the controlled fractional-order-form of proposed three-dimensional autonomous VdPD type oscillator at its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Finally, an example of observer-based synchronization applied to unidirectional coupled identical proposed chaotic fractional-order oscillator is illustrated. It is shown that synchronization can be achieved for appropriate coupling strength.     

Integrator-based circuit-independent chaotic oscillator structure

An integrator-based chaotic oscillator structure composed of three cascaded inverting, noninverting, and differential integrators is presented. The nonlinearity responsible for folding the trajectories is introduced by a single switching diode which is controlled by the output of the first integrator in the cascade. Chaotic behavior is verified on the functional level of the structure rendering it circuit-independent. A possible circuit realization is given and a canonical single-parameter-controlled ordinary differential equation capturing the qualitative dynamics of similar fourth-order integrator-based chaotic oscillators is proposed.     

From flow to map in an experimental high-dimensional electro-optic nonlinear delay oscillator

An optoelectronic nonlinear delay oscillator seeded by a pulsed laser source is used to experimentally demonstrate a new transition scenario for the general class of delay differential dynamics, from continuous to discrete time behavior. This transition scenario differs from the singular limit map, or adiabatic approximation model that is usually considered. The transition from the map to the flow is observed when increasing the pulse repetition rate. The mechanism of this transition opens the way to new interpretations of the general properties of delay differential dynamics, which are universal features of many other scientific domains. We anticipate that the nonlinear delay oscillator architecture presented here will have significant applications in chaotic communication systems.     

Hopping of chaotic dynamics in a doubly resonant parametric oscillator

We study a doubly resonant optical parametric oscillator where the pump can feed two pairs of signal-idler modes. We assume the presence of gain at the pump frequency. We investigate the various oscillation states of interest, namely, when only the first pair oscillates with the other pair having null amplitudes and vice versa. We demonstrate the exchange of dynamics between the mode pairs when the relevant parameters of the cavity, namely, the phase mismatch factors or the decay rates switch because of fluctuations. The exchange of dynamics is shown to be independent of the nature of dynamics, i.e. independent of whether the motion isn-periodic or chaotic. We also investigate the case where both the pairs can exhibit chaotic dynamics though these states are difficult to realize because of fluctuations.     

Classical dissipation and asymptotic equilibrium via interaction with chaotic systems

We study the energy flow between a one-dimensional oscillator and a chaotic system with two degrees of freedom in the weak coupling limit. The oscillator's observables are averaged over an initially microcanonical ensemble of trajectories of the chaotic system, which plays the role of an environment for the oscillator. We show numerically that the oscillator's average energy exhibits irreversible dynamics and ‘thermal’ equilibrium at long times. We use linear response theory to describe the dynamics at short times and we derive a condition for the absorption or dissipation of energy by the oscillator from the chaotic system. The equilibrium properties at long times, including the average equilibrium energies and the energy distributions, are explained with the help of statistical arguments. We also check that the concept of temperature defined in terms of the ‘volume entropy’ agrees very well with these energy distributions.     

Coupling of the relaxation and resonant elements in the autonomous chaotic relaxation oscillator (ACRO)

If a harmonic oscillator is embedded in a relaxation oscillator, the resulting system may behave like an autonomous chaotic relaxation oscillator (ACRO). The discharge transient of the relaxation oscillator excites sinusoidal oscillations in the harmonic oscillator and these sinusoids affect when the next discharge occurs. This can lead to chaotic intervals in the oscillator periods. A simple electronic model of the ACRO is studied over a wide range of parameters using numerical, analytic, and experimental techniques. The dynamics of the ACRO is found to be determined by three parameters: (1) tuning, (2) coupling, and (3) damping. Complex, intermittent outputs can always be inhibited by increasing the damping of the harmonic oscillator. For weak damping, strong coupling yields chaotic periods. With weak damping and weak coupling, complex behavior only occurs if the relaxation oscillator is tuned near a resonance of the harmonic oscillator. A new path to chaos, called a disruption bifurcation, is the source for intermittency in the ACRO. This bifurcation occurs when the amplitude of internal resonances is excited to the degree that existing limit cycles are disrupted.     

Variants of the Nosé–Hoover oscillator

The Nosé–Hoover oscillator is a well-studied chaotic system originally proposed to model a harmonic oscillator in equilibrium with a heat bath at constant temperature. Although it is a simple three-dimensional system with five terms and two quadratic nonlinearities, it displays a rich variety of unusual dynamics, but it falls considerably short of its original purpose. This review describes two simple variants of the Nosé–Hoover oscillator, the first of which satisfies the original goal exactly, and the second of which exhibits a hidden global chaotic attractor that fills all of its three-dimensional state space.

     

Complex dynamics of a double-cavity delayed feedback Klystron oscillator

The complex dynamics of a model double-cavity delayed feedback klystron oscillator is considered. The self-oscillation and stationary oscillation conditions are analyzed theoretically. The results of numerical simulation of the self-modulation and chaotic regimes are presented, and routes to chaos at the center and boundaries of the oscillation zone are studied in detail. The effect of space charge forces on the oscillator dynamics is discussed.     

The simplest case of chaotic wave scattering

We consider scattering of nondispersive linear waves on a discrete nonlinear element. The problem reduces to the dynamics of a forced damped nonlinear oscillator. Chaotic motions of the oscillator produce chaotic reflected and transmitted waves.     

Probing quantum dissipative chaos using purity

In this Letter, the purity of quantum states is applied to probe chaotic dissipative dynamics. To achieve this goal, a comparative analysis of regular and chaotic regimes of nonlinear dissipative oscillator (NDO) are performed on the base of excitation number and the purity of oscillatory states. While the chaotic regime is identified in our semiclassical approach by means of strange attractors in Poincaré section and with the Lyapunov exponent, the state in the quantum regime is treated via the Wigner function. Specifically, interesting quantum purity effects that accompany the chaotic dynamics are elucidated in this Letter for NDO system driven by either: (i) a time-modulated field, or (ii) a sequence of pulses with Gaussian time-dependent envelopes.     

Stochastic and chaotic relaxation oscillations

For relaxation oscillators stochastic and chaotic dynamics are investigated. The effect of random perturbations upon the period is computed. For an extended system with additional state variables chaotic behavior can be expected. As an example, the Van der Pol oscillator is changed into a third-order system admitting period doubling and chaos in a certain parameter range. The distinction between chaotic oscillation and oscillation with noise is explored. Return maps, power spectra, and Lyapunov exponents are analyzed for that purpose.     

Some peculiarities of stochastic layer and stochastic web formation

We discuss linear and non-linear oscillators perturbed by a plane-wave. It is shown that in the linear case under resonant relations between the frequencies the phase space of a system is covered by a stochastic web-like network within which the particle dynamics is chaotic. The existence of the stochastic network causes universal diffusion similar to the Arnold diffusion in the multidimensional case. A similar effect occurs for an unharmonic oscillator in a low velocity region where non-perturbed oscillations are very close to the linear ones.     

Stability and chaos in coupled two-dimensional maps on gene regulatory network of bacterium E. coli

The collective dynamics of coupled two-dimensional chaotic maps on complex networks is known to exhibit a rich variety of emergent properties which crucially depend on the underlying network topology. We investigate the collective motion of Chirikov standard maps interacting with time delay through directed links of gene regulatory network of bacterium Escherichia coli. Departures from strongly chaotic behavior of the isolated maps are studied in relation to different coupling forms and strengths. At smaller coupling intensities the network induces stable and coherent emergent dynamics. The unstable behavior appearing with increase of coupling strength remains confined within a connected subnetwork. For the appropriate coupling, network exhibits statistically robust self-organized dynamics in a weakly chaotic regime.