多尺度系统中的奇异盆地：在稳定状态间隧道效应
Singular basins in multiscale systems: tunneling between stable states
AI总结：
本文研究了多尺度系统中奇异盆地的形成机制，揭示了奇异漏斗对系统稳定性的影响，并展示了其在不同系统中的普遍性。
摘要：
现实中的系统往往在不同的时间尺度上演化，并具有多个共存的稳定状态。系统在受到扰动后是否返回给定的稳定状态，取决于其吸引盆地的形状和范围。我们证明，在多尺度系统中，吸引盆地可以表现出特殊几何特性，即奇异漏斗。尽管奇异漏斗很窄，但它们可以延伸到相空间的不同区域，并意外地影响系统对扰动的鲁棒性。因此，奇异漏斗可能在大时间尺度分离极限下阻止常见的维度缩减，如准静态近似、绝热消除和快变量的时间平均。我们将具有奇异漏斗的吸引盆地称为奇异盆地。我们证明奇异盆地是普遍存在的，并在多种多尺度系统中稳健地出现：具有缓慢适应参数的pitchfork分岔的正常形式、自适应主动旋转器，以及自适应相旋转器网络。
英文摘要：
Real-world systems often evolve on different timescales and possess multiple coexisting stable states. Whether or not a system returns to a given stable state after being perturbed away from it depends on the shape and extent of its basin of attraction. We show that basins of attraction in multiscale systems can exhibit special geometric properties in the form of singular funnels. Although singular funnels are narrow, they can extend to different regions of the phase space and, unexpectedly, impact the system's resilience to perturbations. Consequently, singular funnels may prevent common dimensionality reductions in the limit of large timescale separation, such as the quasi-static approximation, adiabatic elimination and time-averaging of the fast variables. We refer to basins of attraction with singular funnels as singular basins. We show that singular basins are universal and occur robustly in a range of multiscale systems: the normal form of a pitchfork bifurcation with a slowly adapting parameter, an adaptive active rotator, and an adaptive network of phase rotators.
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