Hybrid Oscillators: Phase and Amplitude in a Class of Non-Smooth Systems

M Kvalheim, Shai Revzen

2019 SIAM Conference on Dynamical Systems, 2019

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Abstract

In the last few years there has been a growing interest in composition operator representations of dynamics, also known as ‘Koopman Theory’, where dynamics are encoded in a semigroup Kt which acts on ‘observables’ ψby advancing them through time. The spectrum of the ‘Koopman Operator’ K and its eigen-observables lead to many useful applications when the spectrum is discrete. The case of ‘principal’ eigen-observables with discrete spectrum generalizes many known results from classical dynamical systems theory. We present some early results on Bouligand differentiable observables: functions which have a directional derivative which is also a first-order approximation. These functions arise more often than most realize – the flows of many hybrid systems [Burden, et.al. SIADS 2016], and most commonly used norms are in this class. Of particular interest to the workshop attendees is the latter observation, which is closely related to action variables in action-angle coordinates. We will show some cases where action-angle coordinates can be obtained for systems described in terms of Bouligand differentiable observables, and discuss some of the related Koopman theory.

BibTeX

@conference{Kvalheim-2019-hybosc-SIAM,
  author = {Kvalheim, M and Revzen, Shai},
  title = {Hybrid Oscillators: Phase and Amplitude in a Class of Non-Smooth Systems},
  booktitle = {2019 SIAM Conference on Dynamical Systems},
  year = {2019},
  address = {Snowbird, UT},
  organization = {SIAM},
  url = {https://meetings.siam.org/sess/dsp_talk.cfm?p=98071}
}