| In an excitable or oscillatory system, noise may induce an oscillation and its SNR (signal tonoise ratio) shows a bell-shape dependence on the noise intensity, which is well-known as thecoherence resonance (CR), originally termed as “autonomous SR”. Generally, there are two kinds ofcoherence resonance behavior: one is that in an excitable system, for instance of a neural system,noise induces large amplitude relaxation oscillations (known as spikes) in the sub-threshold region,for which the maximum is reached at an optimal noise level; the other is the one that in theneighbourhood of the supercritical hopf bifurcation (SHB) point, noise induces some small-amplitudeharmonic oscillations. The second case for CR is mainly studied in the present thesis.In some nonlinear systems that are perturbed by both weak low-frequency and high-frequencysignals, the response amplitude at the low-frequency is a nonlinear function of the amplitude of thehigh-frequency signal. By adjusting some system parameters, the response amplitude at the low-frequency can achieve its maximum, the phenomenon of which is termed as vibration resonance.Based on the principle of VR, the weak low-frequency signal may be amplified or suppressed in thesystem.In this presente thesis, beside the coherence resonance that occurs within the neighbourhood ofthe Hopf bifurcation point for a nonlinear Van Der Pol system perturbed by the recycled noise, thevibration resonance that is observed within a quintic system is also investigated. The main work andthe results are as follows:1. The phenomenon of CR for the nonlinear system that is within the neighbourhood of the Hopfbifurcation point and is driven by a recycled white noise or recycled color noise is studied, for which,both the effect of the time delay quantity of the recycled noise and the correlated time of the colornoise of the recycled color noise upon the coherence are mainly investigated.Firstly, by applying the numerical simulation based on the stochastic Runge-Kutta method,several response results, which characterize the CR phenomenon, such as the most probableamplitude, the SNR and the power spectrum density that presents itself as a the function of the timedelay quantity within the recycled noise are obtained.Secondly, the stochastic normal form theory is applied to simplify the nonlinear system.Thirdly, via the stochastic averaging, the non-Markov process which is the solution to thesimplified equation is transformed to its Markov process approximation. Then, the Stratonovich and Ito stochastic differential equations are derived and based on which the FPK equation is obtained.The solution to the FPK equation gives the analytical results.Fouthly, through a comparison between the numerical and the analytical results, several facts arediscovered:(ⅰ) No matter what the kind of the recycled noise is, the white or the color, the variation of theresponse with respect to the time delay quantity within in the recycled noise won’t change, whichmeans that they share the same frequency as the natural frequency of the system.(ⅱ) The response of the system changes monotonously with the time correlation of color noise.2. The vibration resonance in the quintic system with time delay is studied.By using the method of direct separation of motions, the initial differential equations of the motionof the system are transformed to a system of integro-differential equations of the fast variables and theslow variables. Via a series approximate method, the approximate solutions to integro-differentialequations are obtained, and then, the analytical expression of the response at the low-frequency isobtained. By comparing the numerical simulation of the under-damped system with the analyticalresults, the correctness of the method is validated and the conditions and characteristics of vibrationresonance are given... |
PDF Full Text Request