Influences of time delay and noise on the chaotic motion of a bistable system

Influences of time delay and noise on the chaotic motion of a prototypical bistable system with delayed state feedback under additive bounded noise excitation are studied theoretically and numerically in this Letter. A random Melnikov's technique is employed to analyze the chaotic behavior of the system. Analytical analyses reveal that for negative feedback the presence of time delay lowers the threshold and enlarges the possible chaotic domain in parameter space, while the presence of noise enhances the threshold and reduces the possible chaotic domain in parameter space; for positive feedback the presence of time delay enhances the threshold and reduces the possible chaotic domain in parameter space, while the presence of noise lowers the threshold and enhances the possible chaotic domain when the noise intensity is less than a certain threshold or enhances the threshold and lowers the possible chaotic domain when the noise intensity is greater than the threshold in parameter space. The analytical results are further verified numerically through the top Lyapunov exponents and Poincaré maps of the system. Both the two methods lead to fully consistent results.