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Many systems in physics, chemistry, and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example, linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here, we introduce a nonlinear transformation of stochastic oscillators to a complex-valued function Q 1 * ( x ) that greatly simplifies and unifies the mathematical description of the oscillator’s spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function Q 1 * ( x ) is the eigenfunction of the Kolmogorov backward operator with the least negative (but nonvanishing) eigenvalue λ 1 = μ 1 + iω 1 . The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency ω 1 and half-width μ 1 ; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around ω 1 ; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.
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On the Relation between Infinitesimal Shape Response Curves and Phase-Amplitude Reduction for Single and Coupled Limit-Cycle Oscillators
Phase reduction is a well-established method to study weakly driven and weakly perturbed oscillators. Traditional phase-reduction approaches characterize the perturbed system dynamics solely in terms of the timing of the oscillations. In the case of large perturbations, the introduction of amplitude (isostable) coordinates improves the accuracy of the phase description by providing a sense of distance from the underlying limit cycle. Importantly, phase-amplitude coordinates allow for the study of both the timing and shape of system oscillations. A parallel tool is the infinitesimal shape response curve (iSRC), a variational method that characterizes the shape change of a limit-cycle oscillator under sustained perturbation. Despite the importance of oscillation amplitude in a wide range of physical systems, systematic studies on the shape change of oscillations remain scarce. Both phase-amplitude coordinates and the iSRC represent methods to analyze oscillation shape change, yet a relationship between the two has not been previously explored. In this work, we establish the iSRC and phase-amplitude coordinates as complementary tools to study oscillation amplitude. We extend existing iSRC theory and specify conditions under which a general class of systems can be analyzed by the joint iSRC phase-amplitude approach. We show that the iSRC takes on a dramatically simple form in phase-amplitude coordinates, and directly relate the phase and isostable response curves to the iSRC. We apply our theory to weakly perturbed single oscillators, and to study the synchronization and entrainment of coupled oscillators.
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- Award ID(s):
- 2052109
- PAR ID:
- 10573477
- Publisher / Repository:
- Society for Industrial and Applied Mathematics
- Date Published:
- Journal Name:
- SIAM Journal on Applied Dynamical Systems
- Volume:
- 23
- Issue:
- 2
- ISSN:
- 1536-0040
- Page Range / eLocation ID:
- 1636 to 1676
- Subject(s) / Keyword(s):
- isostable coordinates, entrainment, synchronization, limit cycle, shape-response curve, phase-response curve
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1137/23M1575159
