Kamyar Tavakoli S, Longtin André
Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario K1N 6N5, Canada.
Chaos. 2021 Oct;31(10):103129. doi: 10.1063/5.0056097.
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.
我们研究了具有多个延迟的一阶非线性微分方程向简单动力学的转变。通过适当选择参数，单个延迟可以使一个定点失稳。相比之下，多个延迟既能使定点失稳并促进高维混沌，也能诱导系统稳定到更简单的动力学状态。我们表明，这些系统的动力学取决于延迟的精确分布。单个延迟之间的间距较窄会导致混沌行为，而延迟密度较低则会产生稳定的周期或定点行为。随着动力学变得更简单，定点周围特征方程的不稳定根的数量会减少。事实上，这些根的行为与这些多延迟系统的李雅普诺夫指数和柯尔莫哥洛夫-西奈熵的行为表现出惊人的相似性。理论分析表明，随着延迟数量的增加，这些根是如何向稳定性移动的。我们的结果基于对这些多延迟系统的李雅普诺夫谱的数值确定以及排列熵计算。最后，我们报告了通过逆倍周期序列增加更多延迟时如何实现复杂度降低。
