Mackey-Glass System Resonance Study

In this paper, we investigate resonance phenomena in a Mackey-Glass sys-tem. In chapter1, the study background and status of stochastic resonance, resonance suppression, resonance activation and relaxation time are briefly in-troduced, respectively; in chapter2, we have analytically described the study means of resonance in nonlinear systems, i.e., the study methods of stochastic resonance,resonance suppression, resonance activation and the calculative way of relaxation time; in chapter3, the dynamic behavior of Mackey-Glass system in the deterministic case is simply presented; the content of Chapter4is my research work:in the stochastic case, the dynamic behavior of Mackey-Glass system. The Mackey-Glass system is a typically time-delayed bistable system, By means of stochastic simulations, we have respectively calculated the power spectrum, the quality factor of the power spectrum, the mean first-passage time (MFPT) and relaxation time of the system. The calculative results indicate that:(1)We investigated stochastic resonance (SR)-like and resonance suppres-sion (RS)-like phenomena in a Mackey-Glass system with time delay and ad-ditive white noise.(i) as the system is driven by a small periodic signal, the quality factor as a function delay time exhibits a maximal value at smaller noise intensities, i.e., an SR-like phenomenon. With the increment in additive noise intensity, the extremum gradually disappears and the quality factor decreases monotonously with delay time.(ii) As the additive noise intensity is smaller, the curve of the mean first-passage time with respect to delay time displays a peak, i.e., an RS-like phenomenon. At higher levels of noise, however, the non-monotonic behavior is lost.(2)By means of calculating numerically the mean first-passage time (MFPT) of a particle from one stable state to the other state, a time-delayed Mackey-Glass system driven by correlated multiplicative and additive noises are stud-ied,(i) The MFPT as a function of multiplicative noise intensity exhibits both one vale and and one peak as the noise correlation strength (λ) between the two noises is positive, while decreases monotonically as λ<0.(ii) At fixed multiplicative and additive noise intensities, the MFPT displays a maximum with respect to delay time whether λ is positive or negative, i.e., a resonance suppression-like phenomenon.(iii) There is a minimum in the curve of the MFPT vs.A as the other noise parameters and delay time are unchanged, i.e., a resonance activation phenomenon.(3)By dint of stochastic simulations, we study the relaxation time Tc in a time-delayed Mackey-Glass system driven by correlated multiplicative and additive noises. The calculative results indicate that:(i) As the multiplicative strength D is equal to additive noise intensity a, it appears a critical phe-nomenon with the noise correlation intensity in the system. At some fixed correlation strengths λ, there is a critical phenomenon in the curve of the re-laxation time Tc with respect to delay time and there are two symmetrical critical points λc. Namely, when λ=λc, the curve of the relaxation time Tc is almost level, and as λ≠λc, there are two monotonically adverse curves.(ii) In the case of D≠a, the critical phenomenon is lost.(iii) Under the presence of D=α,λc changes with the change of D or α, and there are two symmetrical curves. In other words, when D and a increase at the same time, the critical value is close to λ=±1；conversely,is approach to λ=0.