Abstract
We investigate transitions to simple dynamics in first-order nonlinear differential equations with multiple delays. With a proper choice of parameters, a single delay can destabilize a fixed point. In contrast, multiple delays can both destabilize fixed points and promote high-dimensional chaos but also induce stabilization onto simpler dynamics. We show that the dynamics of these systems depend on the precise distribution of the delays. Narrow spacing between individual delays induces chaotic behavior, while a lower density of delays enables stable periodic or fixed point behavior. As the dynamics become simpler, the number of unstable roots of the characteristic equation around the fixed point decreases. In fact, the behavior of these roots exhibits an astonishing parallel with that of the Lyapunov exponents and the Kolmogorov-Sinai entropy for these multi-delay systems. A theoretical analysis shows how these roots move back toward stability as the number of delays increases. Our results are based on numerical determination of the Lyapunov spectrum for these multi-delay systems as well as on permutation entropy computations. Finally, we report how complexity reduction upon adding more delays can occur through an inverse period-doubling sequence.
Highlights
In the past few decades, delay-differential equations (DDEs) have gained much attention in many real-world systems in optics, biology, neuroscience, network science, and beyond.[1,2,3,4] The presence of delays provides a more realistic understanding of a spatially extended dynamical system in which effects propagate with a nonnegligible speed back to the same system or to another system to which it is coupled
The second scalar nonlinear DDE with multiple delayed feedbacks that we study is the electro-optic oscillator (EOO) model defined as dx dt
We have investigated the role of the distribution of the discrete delays on the dynamical behavior of nonlinear first-order delaydifferential equations
Summary
In the past few decades, delay-differential equations (DDEs) have gained much attention in many real-world systems in optics, biology, neuroscience, network science, and beyond.[1,2,3,4] The presence of delays provides a more realistic understanding of a spatially extended dynamical system in which effects propagate with a nonnegligible speed back to the same system or to another system to which it is coupled. The most studied technique for chaos control works by adding a perturbation made of the weighted difference between the current state and the past state to the original dynamical equation, which results in a stabilized periodic attractor or stable fixed point. This method is enhanced by using multiple delayed feedback control.[11].
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