Effect of time delay on the upper bound of the time derivative of information entropy in a stochastic dynamical system

The effects of the time delay on the upper bound of the time derivative of information entropy are investigated in a time-delayed dynamical system driven by correlated noise. Using the Markov approximation of the stochastic delay differential equations and the Schwartz inequality principle, we obtain an analytical expression for the upper bound U_B(t) of the time derivative of the information entropy. The results show that there is a critical value of \tau (delay time), and U_B(t) presents opposite behaviours on difference sides of the critical value. For the case of the weak additive noise, \tau can induce a reentrance transition. Delay time \tau also causes a reversal behaviour in U_B(t)-\lambda plot, where \lambda denotes the degree of the correlation between the two noises.