Müller-Bender D, Otto A, Radons G
Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany.
Philos Trans A Math Phys Eng Sci. 2019 Sep 9;377(2153):20180119. doi: 10.1098/rsta.2018.0119. Epub 2019 Jul 22.
We demonstrate that a time-varying delay in nonlinear systems leads to a rich variety of dynamical behaviour, which cannot be observed in systems with constant delay. We show that the effect of the delay variation is similar to the Doppler effect with self-feedback. We distinguish between the non-resonant and the resonant Doppler effect corresponding to the dichotomy between conservative delays and dissipative delays. The non-resonant Doppler effect leads to a quasi-periodic frequency modulation of the signal, but the qualitative properties of the solution are the same as for constant delays. By contrast, the resonant Doppler effect leads to fundamentally different solutions characterized by low- and high-frequency phases with a clear separation between them. This is equivalent to time-multiplexed dynamics and can be used to design systems with well-defined multistable solutions or temporal switching between different chaotic and periodic dynamics. We systematically study chaotic dynamics in systems with large dissipative delay, which we call generalized laminar chaos. We derive a criterion for the occurrence of different orders of generalized laminar chaos, where the order is related to the dimension of the chaotic attractor. The recently found laminar chaos with constant plateaus in the low-frequency phases is the zeroth-order case with a very low dimension compared to the known high dimension of turbulent chaos in systems with conservative delay. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.
我们证明，非线性系统中的时变延迟会导致丰富多样的动力学行为，而在具有恒定延迟的系统中则无法观察到这种行为。我们表明，延迟变化的效应类似于具有自反馈的多普勒效应。我们区分了与保守延迟和耗散延迟二分法相对应的非共振多普勒效应和共振多普勒效应。非共振多普勒效应会导致信号的准周期频率调制，但解的定性性质与恒定延迟时相同。相比之下，共振多普勒效应会导致本质上不同的解，其特征是低频和高频相位之间有明显的分离。这等同于时分复用动力学，可用于设计具有明确多稳态解或在不同混沌和周期动力学之间进行时间切换的系统。我们系统地研究了具有大耗散延迟的系统中的混沌动力学，我们将其称为广义层流混沌。我们推导了不同阶广义层流混沌出现的判据，其中阶数与混沌吸引子的维度有关。最近发现的在低频相位具有恒定平台的层流混沌是零阶情况，与具有保守延迟的系统中已知的高维湍流混沌相比，其维度非常低。本文是主题为“延迟系统的非线性动力学”的一部分。
