Delayed differential equations (DDEs) are class of dynamical systems in which the current states strongly depend on their previous states. They received considerable attention due to their rich dynamical properties such as hyperchaotic attractors. One of the most useful models is the Lang-Kobayashi (LK) equations for a semiconductor laser with optical feedback. The chaotic dynamics of semiconductor laser have been exploited for applications such as random bit generation [1]. The performance of this time delayed system can be improved by destroying the time delay signature, a feature which can be extracted from the time series by various methods such as autocorrelation function. It has been shown experimentally and numerically that using multiple delays in this model makes the signature vanish [2]. However, we have found out that increasing the number of delays up to a very large number may result in much simpler dynamics [3]. This is due to the increased averaging of the feedback fluctuations as the number of delays increases, thus simpler dynamics can be observed. This complexity collapse in multi-delayed LK model is similar to distributed delay systems with an infinite number of delays. It has been shown for different dynamical models with distributed delays that broadening the delay kernel simplifies the dynamics, and that consequently, limit cycle and stable fixed point behavior can be expected. We found that this transition from hyperchaotic behavior to stable oscillatory and fixed point behavior occurs in other nonlinear delayed differential equations as well, such as the Mackey-Glass system, when the number of delays increases.
Here we studied the complexity of these nonlinear DDE models with multiple delays by estimation of dynamical invariants, namely Kolmogorov-Sinai entropy (KS entropy) and Permutation entropy. We approximated KS entropy for a multi-delayed system by extending a recent method introduced in [4] for estimation of Lyapunov exponent in nonlinear DDEs from single delay to multiple delays. We found that the complexity, as measured through KS entropy and permutation entropy, decreases for a large number of delays.
References
[1] Uchida, A., Amano, K., Inoue, M., Hirano, K., Naito, S., Someya, H., Oowada, I., Kurashige, T., Shiki, M., Yoshimori, S., Yoshimura, K. & Davis, P. Nat. Photonics 2, 728-732 (2008).
[2] Y. Xu, M. Zhang, L. Zhang, P. Lu, S. Mihailov, and X. Bao, Opt. Lett., vol. 42, no. 20, p. 4107, Oct. 2017.
[3]- S. K. Tavakoli and A. Longtin, Phys. Rev. Research 2, 033485 (2020).
[4]- D. Breda and S. D. Schiava, Discrete Continuous Dynamical Systems - B 23, 2727 (2018).