Correlation Functions of an Autonomous Stochastic System with Nonlinear Time Delays

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1 Commun. Theor. Phys. 63 (2015) Vol. 63, No. 2, February 1, 2015 Correlation Functions of an Autonomous Stochastic System with Nonlinear Time Delays ZHU Ping ( ) 1, and ZHU Yi-Jie ( ) 2 1 School of Science and Technology, Puer University, Puer , China 2 Faculty of Construction Management and Real Estate, Chongqing University, Chongqing , China (Received August 21, 2014; revised manuscript received November 15, 2014) Abstract The auto-correlation function and the cross-correlation of an autonomous stochastic system with nonlinear time-delayed feedback are investigated by using the stochastic simulation method. There are prominent differences between the roles of quadratic time-delayed feedback and cubic time-delayed feedback on the correlations of an autonomous stochastic system. Under quadratic time-delayed feedback, the nonlinear time delay fails to improve the noisy state of the autonomous stochastic system, the auto-correlation decreases monotonously to zero, and the cross-correlation increases monotonously to zero with the decay time. Under cubic time-delayed feedback, the nonlinear time delay can improve the noisy state of the autonomous stochastic system; the auto-correlation and the cross-correlation show periodical oscillation and attenuation, finally tending to zero with the decay time. Comparing the correlations of the system between with nonlinear time-delayed feedback and linear time-delayed feedback, we find that nonlinear time-delayed feedback lowers the correlation strength of the autonomous stochastic system. PACS numbers: a, r, Cb Key words: autonomous system, noise, nonlinear time delay, correlation functions, stochastic simulation 1 Introduction In a stochastic dynamics process, the analysis of steady-state correlation function is a very useful tool in the study of a wide range of a nonlinear stochastic system. The correlation function characterizes the related activity of the dynamic variables between two states at different times. The correlation function can be classified into the auto-correlation function which shows the related activity of the same variable at different times and the cross-correlation function which shows the related activity between different variables at different times. Over the last decades, the research of the correlation function of the stochastic system has attracted a great deal of attentions, where many interesting and important results have been given in a series of experimental and theoretical papers. [1 21] In the world of nature, many realistic systems can be regarded as to obey auto-regulating control, affect, and optimize mechanisms in terms of feedback loops by the output signal or other key quantities to the input, particularly, which can make the optimal and the effective output in the stochastic system driver by noise. [22 30] In the meanwhile, usually, the transmission takes some time, that is to say, the input signals are related to the output signals at earlier time, and the feedback effect of the output to the input possesses the time delay, [31 37] including the population system, the neural network system, the motor control system, the laser system, and the chemical reaction system. [38 50] So investigating the dynamic properties of the practical stochastic systems, we should take into account the effects of time-delay feedback. In recent years, many significant studying results on stochastic systems with time-delayed feedback have been obtained. In a fully synchronized state the time delay induces the clustering, synchronization, and the multistability. [51] The experimental phenomenon of time-delay-induced amplitude death in two coupled nonlinear electronic circuits are observed. [52] The time delay may generate the perfect synchrony in the Kuramoto model [53] and the phenomenon of slow switching in globally coupled oscillators. [54] In the stochastic system with an external driving force, the time delay induces positive motion of the particle, [55] and restrains the unbounded reproducibility of species. [56] For the stochastic system without an external signal, through interaction of noise with delay time, the regularity of the oscillatory states is maximal for a certain noise level, [57] and interesting coherence resonance is induced by the time delay in an autonomous system. [58] In these problems, the time-delayed feedback term is linear. However, for certain complex systems, the time-delayed feedback term is nonlinear, which changes the dynamical behavior of the stochastic system and makes the system show the incoherence maximization at certain noise strength. [59] Adding Supported by the National Natural Science Foundation of China under Grant No and Yunnan Province Open Key Laboratory of Mechanics in Colleges and Universities zhuupp@163.com c 2015 Chinese Physical Society and IOP Publishing Ltd

2 182 Communications in Theoretical Physics Vol. 63 the Bernoulli variable and using stochastic analysis techniques, Tang et al. studied the synchronization problem of stochastic coupled neural networks. [60 62] By constructing appropriate Lyapunov functions and employing the comparison principle, Zhang et al. investigated further the synchronization problem of a class of nonlinear delayed dynamical networks. [63 64] In the study about an autonomous stochastic system, many interesting results are obtained in applications of physics, chemistry, biology, and other fields, such as the forced pendulum, the Duffing equation, the Van der Pol oscillator, and so on. [65 75] Investigating the autocorrelation function and the cross-correlation function of a stochastic autonomous system with nonlinear timedelayed feedback, we can further understand the dynamic properties of the autonomous stochastic system. In this paper we investigate the effects of the nonlinear time delay on correlations of the autonomous stochastic system. The paper is organized as follows: In Sec. 2, we introduce the an autonomous stochastic system with nonlinear time-delayed feedback, discussing dynamical properties of the autonomous stochastic delay system. In Sec. 3, we show the auto-correlation of the autonomous stochastic system with nonlinear time-delay feedback. In Sec. 4, we show the cross-correlation of the autonomous stochastic system with nonlinear time-delay feedback. Finally, discussions and conclusions of the results conclude the paper in Sec Dynamic Properties of an Autonomous Stochastic System with Nonlinear Time-Delayed Feedback An autonomous stochastic system of a two-dimensional model with the n-th power of time-delayed feedback follows the stochastic delay differential equations [75 80] dx dt = x(1 x2 y 2 )+y(x 2 y 2 b) ɛx n (t τ)+q 1 (t), (1) dy dt = y(1 x2 y 2 ) x(x 2 y 2 b) ɛy n (t τ)+q 2 (t), (2) where x and y are the dynamic variables of the system, ɛx n (t τ) and ɛy n (t τ) are the n-th power of timedelayed feedback terms, τ is the delay time, ɛ is the feedback strength in which when ɛ > 0, the system is the negative feedback and when ɛ < 0, the system is the positive feedback. In the autonomous stochastic system, the highest powers of the dynamical variables x and y are the third power, generally, in the practical problem, the feedback terms to be weaker than other terms of the system, where the exponent of time-delayed feedback n takes n = 0, 1, 2, and 3, which is also the so-called the delay exponent. When n = 0, the system is without the time-delayed feedback; when n = 1, the system is with the linear time-delayed feedback; when n = 2 and 3, the system is with the nonlinear time-delayed feedback. Q 1 (t) and Q 2 (t) are zero-mean Gaussian white noise, whose statistical properties are Q i (t) = 0, (3) Q i (t)q j (t ) = Dδ ij δ(t t ) = 0, i, j = 1, 2. (4) To discuss the statistical properties of the autonomous stochastic system, we need to solve the delay Fokker Planck equation corresponding to Eqs. (1) and (2), by which we can attain the analytic distribution functions of the probability density of the dynamic variables x and y as well as other relative results. However, it is extremely difficult or is even impossible for us to obtain an analytic solution of the delay Fokker Planck equation corresponding Eqs. (1) and (2) for arbitrary values of the noise strength D and the delay time τ by using the analytic method. Here, we discuss relative statistical properties employing the stochastic simulation method. The data quoted come from the numerical simulation of Eqs. (1) and (2) by means of the Euler method forward procedure, in which the Box Mueller algorithm is used to generate Gaussian noise. For the initial values under the condition with timedelayed feedback, it is rational to let x(t τ) = x(0), and y(t τ) = y(0), respectively, as t < τ. The correctness of all the following results are confirmed by changing the time step and the total time. Figures 1 plots the evolutions of the dynamic variables x(t) and y(t) of the autonomous nonlinear delay stochastic system for various delay exponents. Figure 1(a) shows the evolutions of the dynamic variable x(t) for b = 0.9, D = 0.8, ɛ=1, τ = 5, and the delay exponent taking n = 0, n = 1, n = 2, and n = 3, respectively. Figure 1(b) shows the case of y(t) for b = 1.2, D = 0.8, ɛ=1, τ = 3, and the delay exponent taking n = 0, n = 1, n = 2, and n = 3, respectively. From Fig. 1 we see clearly when n = 0, the stochastic system without delay feedback, the Gaussian noise induces a noisy state. When n = 1, the stochastic system with linear delay feedback, delay time improves effectively the noisy state of the autonomous system. However, when n = 2, the system shows a more noisy state, implying that the second power of nonlinear delay can not improve the noisy state of the system. When n = 3, the evolutions of the variables x(t) and y(t) are periodically ordered, i.e., the third power of nonlinear delay feedback further improves the spatiotemporal order. In Eqs. (1) and (2), when ɛ < 0, the autonomous stochastic system with nonlinear time-delayed feedback is the positive feedback system. Figure 2 exhibits under the positive feedback the evolutions of the variables x(t) and y(t) of the autonomous nonlinear delay stochastic system for various delay exponents. Regardless of the delay exponent is n = 0, n = 1, n = 2, or n = 3, the dynamical variables of the system show noisy states. Here the delay time fails to possess the role improving the noisy state of the autonomous stochastic system.

3 No. 2 Communications in Theoretical Physics 183 Fig. 1 The evolutions of dynamical variables with the time for different delay exponents. (a) The evolution of x(t) for b = 0.9, D = 0.8, ɛ=1, τ = 5, and the delay exponent taking n = 0, n = 1, n = 2, and n = 3, respectively. (b) The evolutions of y(t) for b = 1.2, D = 0.8, ɛ=1, τ = 3, and the delay exponent taking n = 0, n = 1, n = 2, and n = 3, respectively. Fig. 2 Under positive feedback of the system, the evolution of dynamical variables with the time for different delay exponents. (a) The evolution of x(t) for b = 0.9, D = 0.5, ɛ=-1, τ = 2, and the delay exponent taking n = 0, n = 1, n = 2, and n = 3, respectively. (b) The evolution of y(t) for b = 0.9, D = 0.5, ɛ=-1, τ = 4, and the delay exponent taking n = 0, n = 1, n = 2, and n = 3, respectively. 3 The Auto-Correlation of the Autonomous Stochastic System with Nonlinear Time- Delay Feedback For a general stochastic process of a two-dimensional model for which a stationary state exists, the normalized auto-correlation function of the dynamic variable X i (i = 1, 2) is given by [8 9] C Xi (s) = δx i(t + s)δx i (t) st (δx i (t)) 2 st

4 184 Communications in Theoretical Physics Vol. 63 where X i (t + s)x i (t ) X i (t) 2 = lim t (δx i (t)) 2, (5) δx i (t + s) = X i (t + s) X i (t + s), (6) δx i (t) = X i (t) X i (t). (7) initial value, and exhibits alternate change from positive correlation to negative correlation, again to positive correlation as the decay time increases, periodically, in which the auto-correlation function periodically oscillates, the peak value decreases gradually, and finally tends to zero. The larger the delay time of the cubic nonlinear feedback is, the larger the evolution period and the oscillation amplitude will be. Thus the total time of C x (s) attenuating to zero is lengthened as the delay time increases. Fig. 4 The comparison of the auto-correlation function C y(s) of the system under the quadratic time-delayed feedback (n = 2) with the case without time-delayed feedback (n = 0). Parameters chosen are b = 0.8, D = 2, ɛ = 1, and τ = 0.5. Fig. 3 The auto-correlation C x(s) as a function of the decay time s for nonlinear time-delayed feedback. Parameters chosen are b = 1.2, D = 0.5, ɛ = 1, and τ taking different values. (a) The system with quadratic time-delayed feedback (n = 2). (b) The system with cubic time-delayed feedback (n = 3). Figure 3 shows the evolutions of the auto-correlation function of the dynamical variable x(t) of the system with quadratic nonlinear feedback (n = 2) (a) and cubic nonlinear feedback (n = 3) (b), where the parameters chosen are b = 1.2, D = 0.5, ɛ = 1, and the delay time taking various values. Under the quadratic nonlinear feedback, the autocorrelation function decreases monotonously from the initial value 1 to zero with the decay time, in which the larger the delay time is, the stronger the auto-correlation will be, as shown in Fig. 3(a). Under the cubic nonlinear feedback, as shown Fig. 3(b), the auto-correlation function C x (s) of the dynamic variable x begins the positive Fig. 5 The comparison of the auto-correlation function C x(s) of the system under the cubic time-delayed feedback (n = 3) with the case under linear time-delayed feedback (n = 1). Parameters chosen are b = 1.2, D = 0.5, ɛ = 1, and τ = 1.5. To compare the auto-correlations of the dynamical variable between the cases with the quadratic time delay (n = 2) and without the time delay (n = 0), we plot the auto-correlation function of the variable y(t) versus the decay time s in Fig. 4. From Fig. 4, we see that the autocorrelation of the system with the quadratic time delay

5 No. 2 Communications in Theoretical Physics 185 resembles that of without the time delay, which totally decrease monotonously with the decay time, but on the other hand, there is difference between them, i.e., the autocorrelation strength of the system under the quadratic time delay is larger than that of without time delay. In Fig. 5, we give the comparison of the auto-correlations of the variable x(t) between the cases under linear timedelayed feedback (n = 1) and cubic time-delayed feedback (n = 3). The oscillation amplitude, the oscillation period, and the total time decaying to zero of the auto-correlation function C x (s) under the cubic delay time is smaller than that under the linear time delay. 4 The Cross-Correlation of the Autonomous Stochastic System with Nonlinear Time- Delay Feedback The normalized stationary cross-correlation functions between dynamic variables X i and X j (i, j = 1, 2 and i j) are given by [83] C XiX j (s) = δx i (t + s)δx j (t) st (δxi (t + s)) 2 st (δxj (t)) 2 st δx i (t + s)δx j (t) = lim t (δxi (t + s)) 2 (δx j (t)) 2. (8) Figure 6 shows the evolutions of the cross-correlation function C xy (s) of the dynamical variables x(t) and y(t) under quadratic time-delayed feedback (n = 2) (a) and cubic time-delayed feedback (n = 3) (b), where the parameters chosen are b = 1.2, D = 0.5, ɛ = 1, and the delay time taking various values. Under the quadratic time-delayed feedback, the cross-correlation function C xy (s) begins a negative initial value, reduces to one minimum, and again increases monotonously to zero with the decay time. The larger the delay time is, the larger the absolute value of the C xy (s), i.e., the stronger the cross-correlation of the variables x(t) and y(t). For the cubic time-delayed feedback, as shown Fig. 6(b), the cross-correlation function C xy (s) begins a negative initial value, and exhibits alternate change from negative correlation to positive correlation, again to negative correlation as the decay time increases, periodically; the cross-correlation function periodically oscillates and decreases with decay time; finally, the cross-correlation tends to zero. The evolution period and the oscillation amplitude of the cross-correlation C xy (s) increase, and the total time C xy (s) tending to zero is lengthened as the delay time increases. Fig. 6 The cross-correlation C xy(s) as a function of the decay time s for nonlinear time-delayed feedback. Parameters chosen are b = 1.2, D = 0.5, ɛ = 1, and τ taking different values. (a) The system with quadratic time-delayed feedback (n = 2). (b) The system with cubic time-delayed feedback (n = 3). To compare the cross-correlations of the dynamical variable of the system between the cases with the quadratic time delay (n = 2) and without the time delay (n = 0), we plot the cross-correlation function C yx (s) of the variable y(t) and x(t) versus the decay time s in Fig. 7. From Fig. 7, it has shown that the cross-correlation with the quadratic time delay resembles that of without time delay, which increase monotonously with the decay time, but on the other hand, there is difference between them, i.e., the cross-correlation strength of the system with the quadratic time delay is larger than that of without time delay. In Fig. 8, we give the comparison of the crosscorrelations of the variables x(t) and y(t) between under linear time-delayed feedback (n = 1) and cubic timedelayed feedback (n = 3). The oscillation amplitude, the oscillation period, and the total time decaying to zero of the cross-correlation function C xy (s) under the cubic delay time is smaller than that under linear time delay.

6 186 Communications in Theoretical Physics Vol. 63 Fig. 7 The comparison of the auto-correlation function C yx(s) of the system under the quadratic time-delayed feedback (n = 2) with the case without time-delayed feedback (n = 0). Parameters chosen are b = 0.8, D = 2, ɛ = 1, and τ = Discussions and Conclusions From foregoing results, we can further understand the effects of nonlinear time-delayed feedback on the autocorrelation and the cross-correlation of an autonomous stochastic system. (i) There are prominent differences between the roles of quadratic time-delayed feedback and cubic time-delayed feedback on the correlations of an autonomous stochastic system. Under quadratic time-delayed feedback, the nonlinear time delay fails to improve the noisy state of the dynamical variables of the autonomous stochastic system; the interacts of the same dynamical variable at different times are positive relative and the interacts of the different dynamical variable at different times are negative relative. Under cubic time-delayed feedback, the nonlinear time delay can improve the noisy state of the dynamical variables of the autonomous stochastic system, and the interacts of the dynamical variables at different times are alternate change between the positive relative and the negative relative with the decay time, where the larger the delay time is, the more obvious the role of the nonlinear time delay will be. Fig. 8 The comparison of the auto-correlation functions C xy(s) of the system under the cubic time-delayed feedback (n = 3) with the case under linear time-delayed feedback (n = 1). Parameters chosen are b = 1.2, D = 0.5, ɛ = 1, and τ = 1.5. (ii) The nonlinear delay time has the important impacts for the auto-correlation and the cross-correlation of the autonomous stochastic system. The nonlinear delay time enhances the strength of the auto-correlation and the cross-correlation of the dynamical variables. (iii) The auto-correlation and the cross-correlation of the autonomous stochastic system with quadratic timedelayed feedback resembles the case without time-delayed feedback, and the correlations of the autonomous stochastic system with cubic time-delayed feedback resembles the case with linear time-delayed feedback. However, the strength of correlations of the autonomous stochastic system with the quadratic time delay is smaller than the case without the time delay, and the correlation strength of the system with the cubic time delay is also lower than the case with the linear time delay. The nonlinear delay exponent lowers the correlation strength of the an autonomous stochastic system. Acknowledgments The author wishes to express his most sincere thanks to Editor and Referee, who will read the manuscript carefully and give valuable advice and help. References [1] K. Kaminishi, R. Roy, R. Short, and L. Mandel, Phys. Rev. A 24 (1981) 370. [2] R. Short, L. Mandel, and R. Roy, Phys. Rev. Lett. 49 (1982) 647. [3] P.D. Lett and E.C. Gage, Phys. Rev. A 39 (1989) [4] P. Hänggi, Z. Phys. B 31 (1978) 407. [5] S.T. Dembinski, A. Kossakowski, L. Wolniewicz, et al., Z. Phys. B 32 (1978) 107. [6] A. Hernandez Machado, M. San Miguel, and S. Katz, Phys. Rev. A 31 (1985) [7] J. Casademunt and A. Hernandez Machado, Z. Phys. B 76 (1989) 403. [8] A. Hernandez Machado, J. Casademunt, M.A. Rodriguez, et al., Phys. Rev. A 43 (1991) 1744.

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