Research On Effects Of Time Delay On Autonomous System Driven By Noise

Based on the theory of stochastic dynamics in the statistical physics, we anal-ysis systematically an autonomous nonlinear systems driven by noise, summarize and discuss the means and methods dealing with the stochastic time-delayed dy-namics problem. Then we investigate further the effects of time-delayed feedback on an autonomous nonlinear system driven by noise without external periodic sig-nals. Taking into account an autonomous nonlinear system of a two-dimensional model which possesses the widespread applications and the typical system features, and employing the method obtaining relative data from stochastic simulation and then making analysis of theory, we carry out some studies as follows:1. The correlation between stochastic dynamics variables of an autonomous nonlinear time-delayed system driven by noise is studied. The self-correlation function of the dynamic variable of the autonomous delay stochastic system starts the positive initial value, and exhibits alternate change from positive correlation to negative correlation, again to positive correlation as the decay time increase. The peak value of the self-correlation function periodically oscillates and decreases with decay time; finally, the self-correlation tends to zero. The cross-correlation function of the dynamic variable of the autonomous delay stochastic system starts the negative initial value, and exhibits alternate change from negative correlation to positive correlation, again to negative correlation as the decay time increase. The peak value of the cross-correlation function periodically oscillates and decreases with decay time; finally, the cross-correlation tends to zero. The correlation of the autonomous time-delayed system is obviously different with the correlations of the autonomous system without time-delayed the correlation of which change monotonously, and finally tends to zero as the decay time increases. The time delay can generate the important effects on the correlations of the system. Meanwhile, the effects of other parameters of the system on the correlations of the autonomous time-delayed feedback system driven by noise are given.2. The effects of the time delay on the power spectrum of dynamic variables of an autonomous stochastic system are investigated. The time delay possesses the effects adjusting and optimizing the power spectrum of the system output, and makes the power spectrum change from the noisy state to the relative order state, again to the more ordered state. The modest delay time can make the peak frequency of the power spectrum of the system displays maximum structure and achieve the most optimal and effective output of the system. According to the criterion of coherence resonance, we further discuss the effects of the time-delayed feedback on the coherence of the autonomous stochastic system, and reveal the coherence resonance phenomenon induced by the time delay without external periodic force, before this work the coherence resonance is not reported, which is a new kind of coherence resonance induced by the time delay under interaction of noise with delay time. Meanwhile, the effects of other parameters of the system on the coherence resonance induced by the time delay are given.3. The effects of the time delay of the second power and third power of nonlinear time-delayed feedback on an autonomous stochastic system are explored. The evolutions of the correlations between dynamic variables of the autonomous stochastic system with the square of nonlinear time-delayed feedback with the decay time are similar to the correlations of the autonomous stochastic system without time-delayed feedback, i.e., the self-correlation decreases monotonously to zero with the decay time and the cross-correlation increases monotonously to zero with the decay time; however, the absolute values are larger. The time delay of the second power of nonlinear time-delayed feedback enhances the strengths of the self-correlation and the cross-correlation the system. The evolutions of the correlations between dynamic variables of the autonomous stochastic system with of the third power of nonlinear time-delayed feedback with the decay time are similar to the correlations of the autonomous stochastic system with the first power of time-delayed feedback, i.e., the self-correlation and the cross-correlation of the autonomous stochastic system exhibit periodically oscillations and decrease to zero with decay time; however, the oscillation periods are smaller, the oscillation amplitudes decrease, the velocities of decay are enhanced, and the periods decaying to zero became smaller compared to one of the first power of time-delayed feedback.4. The effects of the time delay of nonlinear feedback on coherence and antico-herence of the autonomous stochastic system are given. First, we discuss the effects of the time delay of the square of nonlinear feedback on the power spectrum of the system. The time delay of the square of nonlinear feedback intensifies turbulence of noise. The dynamic variables display a disorder and noisy state, and the values of the power spectrum of the system output are almost zero. The time delay of the square of nonlinear feedback can not generate the effects adjusting and optimizing the output of the autonomous stochastic system. Again, we discuss the case of the time delay of the third power of nonlinear feedback. The time delay of the square of nonlinear feedback makes the power spectrum change from the peak structure of the relative order state to the disorder and noisy state, again to the peak struc-ture of the more ordered state, i.e., incoherence maximized by the nonlinear delay time. According to the criterion of the coherence resonance, we further discuss the effects of the time delay of the cube of time-delayed feedback on the signal-to-noise ratio of the autonomous stochastic system, and reveal the anticoherenc resonance and the coherence resonance in a stochastic dynamic process of the an autonomous system induced by the time delay of the cube of nonlinear feedback, which is a new kind of stochastic resonance phenomenon. The coherence and the anticoherenc are two states possessing different physic features, however, under modest conditions, the coherence state and the anticoherence state can transform each other. The time delay of the cube of nonlinear feedback is an effective and important condition achieving the transform. The effects of other parameters on coherence resonance and anticoherence resonance of the system induced by the time delay are shown.Research on effects of time delay on an autonomous system driven by noise and the results obtained further develop the new realm of the investigation on the autonomous system and stochastic resonance, give some prospects of new appli-cation, enrich the theory of stochastic dynamics, and promote the development of stochastic dynamics. Meanwhile, the means and methods can be used in research on complex nonlinear stochastic dynamic systems.