Kaszás Bálint, Feudel Ulrike, Tél Tamás
Institute for Theoretical Physics, Eötvös Loránd University, Pázmány Péter Sétány 1/A, H-1117, Budapest, Hungary.
Theoretical Physics/Complex Systems, ICBM, University of Oldenburg, 26129, Oldenburg, Germany.
Sci Rep. 2019 Jun 17;9(1):8654. doi: 10.1038/s41598-019-44863-3.
Tipping phenomena, i.e. dramatic changes in the possible long-term performance of deterministic systems subjected to parameter drift, are of current interest but have not yet been explored in cases with chaotic internal dynamics. Based on the example of a paradigmatic low-dimensional dissipative system subjected to different scenarios of parameter drifts of non-negligible rates, we show that a number of novel types of tippings can be observed due to the topological complexity underlying general systems. Tippings from and into several coexisting attractors are possible, and one can find fractality-induced tipping, the consequence of the fractality of the scenario-dependent basins of attractions, as well as tipping into a chaotic attractor. Tipping from or through an extended chaotic attractor might lead to random tipping into coexisting regular attractors, and rate-induced tippings appear not abruptly as phase transitions, rather they show up gradually when the rate of the parameter drift is increased. Since chaotic systems of arbitrary time-dependence call for ensemble methods, we argue for a probabilistic approach and propose the use of tipping probabilities as a measure of tipping. We numerically determine these quantities and their parameter dependence for all tipping forms discussed.
倾斜现象，即确定性系统在参数漂移作用下可能出现的长期性能的剧烈变化，是当前研究的热点，但在具有混沌内部动力学的情况下尚未得到探讨。基于一个典型的低维耗散系统在不可忽略速率的不同参数漂移情形下的例子，我们表明，由于一般系统所具有的拓扑复杂性，能够观察到多种新型的倾斜现象。可以出现从多个共存吸引子进入和离开的倾斜，并且能够发现分形性诱导倾斜，这是依赖于情形的吸引域分形性的结果，以及进入混沌吸引子的倾斜。从扩展混沌吸引子或通过其发生的倾斜可能导致随机倾斜进入共存的规则吸引子，并且速率诱导倾斜并非像相变那样突然出现，而是当参数漂移速率增加时逐渐显现。由于具有任意时间依赖性的混沌系统需要系综方法，我们主张采用概率方法，并建议使用倾斜概率作为倾斜的一种度量。我们通过数值方法确定了所讨论的所有倾斜形式的这些量及其参数依赖性。
