Induced simplicity by multiple delays in delayed differential equations

A highly studied topic in dynamical systems theory has been delay-differential equations (DDEs). It has been shown in many studies that a single delay can induce different dynamical properties such as oscillatory and chaotic behavior, which makes DDEs appropriate to be exploited in a vast range of algorithms and devices. These demand different degrees of complexity, so one of the key challenges in the study of these infinite-dimensional dynamical systems is enhancing or decreasing the degree of chaos. It has been shown that multiple delays can affect the complexity of these systems. For example, time-series obtained experimentally and numerically for the semiconductor laser with multiple delayed feedback show much higher complexity than the single delayed feedback. However, we have found out that when the number of delays becomes large, a simple periodic attractor or stable fixed point can be expected. The transition from chaotic behavior to simplified dynamics upon adding delays is found to be similar to that seen in distributed delay systems (infinite number of delays) as the memory kernel broadens. We explain complexity reduction with increasing number of delays through the calculation of Permutation entropy and Kolmogorov-Sinai entropy estimation in multi-delay dynamics.