Bifurcation threshold of the delayed van der Pol oscillator under stochastic modulation

Transient response of the time-delay system excited by Gaussian noise based on complex fractional moments

In this paper, the transient response of the time-delay system under additive and multiplicative Gaussian white noise is investigated. Based on the approximate transformation method, we convert the time-delay system into an equivalent system without time delay. The one-dimensional Ito stochastic differential equation with respect to the amplitude response is derived by the stochastic averaging method, and Mellin transformation is utilized to transform the related Fokker-Planck-Kolmogorov equation in the real numbers field into a first-order ordinary differential equation (ODE) of complex fractional moments (CFM) in the complex number field. By solving the ODE of CFM, the transient probability density function can be constructed. Numerical methods are used to ascertain the effectiveness of the CFM method, the effects of system parameters on system response and the level of error vary with time as well as noise intensity are investigated. In addition, the CFM method is first implemented to analyze transient bifurcation, and the relation between CFM and bifurcation is discussed for the first time. Furthermore, the imperfect symmetry property appear on the projection map of joint probability density function.

Perturbations of linear delay differential equations at the verge of instability

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This paper considers linear DDEs that are on the verge of instability, i.e., a pair of roots of the characteristic equation lies on the imaginary axis of the complex plane and all other roots have negative real parts. It is shown that when small noise perturbations are present, the probability distribution of the dynamics can be approximated by the probability distribution of a certain one-dimensional stochastic differential equation (SDE) without delay. This is advantageous because equations without delay are easier to simulate and one-dimensional SDEs are analytically tractable. When the perturbations are also linear, it is shown that the stability depends on a specific complex number. The theory is applied to study oscillators with delayed feedback. Some errors in other articles that use multiscale approach are pointed out.

Nonlinear and additive white noise perturbations of linear delay differential equations at the verge of instability: An averaging approach

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.

Delay-induced stochastic bifurcations in a bistable system under white noise

In this paper, the effects of noise and time delay on stochastic bifurcations are investigated theoretically and numerically in a time-delayed Duffing-Van der Pol oscillator subjected to white noise. Due to the time delay, the random response is not Markovian. Thereby, approximate methods have been adopted to obtain the Fokker-Planck-Kolmogorov equation and the stationary probability density function for amplitude of the response. Based on the knowledge that stochastic bifurcation is characterized by the qualitative properties of the steady-state probability distribution, it is found that time delay and feedback intensity as well as noise intensity will induce the appearance of stochastic P-bifurcation. Besides, results demonstrated that the effects of the strength of the delayed displacement feedback on stochastic bifurcation are accompanied by the sensitive dependence on time delay. Furthermore, the results from numerical simulations best confirm the effectiveness of the theoretical analyses.

Time-delayed feedback control of coherence resonance near subcritical Hopf bifurcation: theory versus experiment

Using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation, we investigate the influence of time delay on noise-induced oscillations. It is shown that for appropriate choices of time delay, either suppression or enhancement of coherence resonance can be achieved. Analytical calculations are combined with numerical simulations and experiments on an electronic circuit.

On noise induced Poincaré-Andronov-Hopf bifurcation

It has been numerically seen that noise introduces stable well-defined oscillatory state in a system with unstable limit cycles resulting from subcritical Poincaré-Andronov-Hopf (or simply Hopf) bifurcation. This phenomenon is analogous to the well known stochastic resonance in the sense that it effectively converts noise into useful energy. Herein, we clearly explain how noise induced imperfection in the bifurcation is a generic reason for such a phenomenon to occur and provide explicit analytical calculations in order to explain the typical square-root dependence of the oscillations' amplitude on the noise level below a certain threshold value. Also, we argue that the noise can bring forth oscillations in average sense even in the absence of a limit cycle. Thus, we bring forward the inherent general mechanism of the noise induced Hopf bifurcation naturally realisable across disciplines.

Effect of autaptic activity on the response of a Hodgkin-Huxley neuron

An autapse is a special synapse that connects a neuron to itself. In this study, we investigated the effect of an autapse on the responses of a Hodgkin-Huxley neuron to different forms of external stimuli. When the neuron was subjected to a DC stimulus, the firing frequencies and the interspike interval distributions of the output spike trains showed periodic behaviors as the autaptic delay time increased. When the input was a synaptic pulse-like train with random interspike intervals, we observed low-pass and band-pass filtering behaviors. Moreover, the region over which the output ISIs are distributed and the mean firing frequency display periodic behaviors with increasing autaptic delay time. When specific autaptic parameters were chosen, most of the input ISIs could be filtered, and the response spike trains were nearly regular, even with a highly random input. The background mechanism of these observed dynamics has been analyzed based on the phase response curve method. We also found that the information entropy of the output spike train could be modified by the autapse. These results also suggest that the autapse can serve as a regulator of information response in the nervous system.