Oscillators have two main limitations: their synchronization properties are limited (i.e they have a finite synchronization region) and they have no memory of past interactions (i.e. they return to their intrinsic frequency whenever the entraining signal disappears). We previously proposed a general mechanism to transform an oscillator into an adaptive frequency oscillator which adapts its parameters to learn the frequency of any input signal. The synchronization region then becomes infinite and the oscillator retains the entrainment frequency when the driving signal disappears. While this mechanism has been successfully used in various applications, such as robot control or observer design for active prosthesis, a formal understanding of its properties is still missing. In this paper, we study the adaptation mechanism in the case of strongly coupled phase oscillators and show that non-trivial slow-fast dynamics is at the origin of the adaptation. We show the existence of a layered structure of stable and unstable invariant slow manifolds and demonstrate how the input signal forces the dynamics to jump between these manifolds at regular intervals, leading to exponential convergence of the frequency adaptation. We extend the idea to a network of oscillators with amplitude adaptation and show that the slow invariant manifolds structure persists. Numerical simulations validate our analysis and extend the discussion to more complex cases.
Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling Revisited
Wirkus和Rand在2002年研究了速度延迟项耦合的两个van der Pol振子问题。小的分析导致在一个缓慢的流动，其中包含延迟条款。为了简化分析，Wirkus和Rand遵循了一个共同的程序，即用非延迟项替换延迟项，这一步骤据说对小扰动有效，导致慢流是ODE而不是DDE（延迟-微分方程）。在本文中，我们考虑同样的问题，但在慢流中留下延迟项，从而提供了一个评估的近似简化以前。
Frequency bursts in adaptive delay-coupled oscillators
Humboldt-Universität zu Berlin（柏林洪堡大学）；Potsdam Institute for Climate Impact Research（波茨坦气候影响研究所）；University of Exeter（埃克塞特大学）；Southeast University（东南大学）；University College Cork（科克大学学院）
Emergence of Subcritical Bifurcations in a System of Randomly Coupled Supercritical Andronov-Hopf Oscillators: A Potential Mechanism for Neural Network Type Switching
Compatibility of Higher-Order Slow-Manifold Reduction and Continuum Limits in Adaptive Networks
Technical University of Munich（慕尼黑工业大学）；Munich Data Science Institute (MDSI)（慕尼黑数据科学研究所）；Complexity Science Hub Vienna (CSH)（维也纳复杂性科学中心）；Leiden University（莱顿大学）
