Phase sensitivity for coherence resonance oscillators
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ORIGINAL PAPER
Phase sensitivity for coherence resonance oscillators Jinjie Zhu
Received: 14 September 2020 / Accepted: 11 November 2020 © Springer Nature B.V. 2020
Abstract The phase reduction approach has manifested its power in analyzing the rhythmic behaviors for limit cycle oscillators. For coherent oscillation purely induced by noise, e.g., the coherence resonance oscillator, the stochastic dynamics exhibit almost deterministic limit cycle phenomenon which inspires the application of the phase reduction approach to this kind of systems. In this paper, the FitzHugh–Nagumo system in coherence resonance is modeled as a jump process. The phase sensitivity is obtained by applying the phase reduction approach for the hybrid system. A modified direct method is proposed to compare the theoretical results with those of Monte Carlo simulation for the stochastic system, which shows a relatively good agreement for the perturbation not being too small. The phase reduction results of the coherent oscillators are applied to two coupled FitzHugh–Nagumo neurons. It is interesting that the phase difference of the coupled coherent oscillators does not converge as the deterministic oscillators, but forms a distribution where the peaks and valleys correspond to the stable and unstable synchronizations predicted by the phase coupling functions. The idea in this paper could be applied to other coherent oscillators where the stochastic dynamics follow almost deterministic paths.
J. Zhu (B) School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China e-mail: [email protected]
Keywords Phase reduction · Coherence resonance · Hybrid system · Phase coupling function
1 Introduction Limit cycle (LC) behaviors are universal in nature and human society, which may refer to oscillations in chemistry, vibrations in engineering, rhythmicity in biology, cycling in ecology, etc. For limit cycle oscillators, responses can differ for perturbations on different timings on it. The phase reduction approach [1–5] is a useful tool to reduce the complexity of the original system and characterize these differences. Due to the simplicity and effectivity, the phase reduction approach has been extended to time-delayed systems [6], strongly perturbed oscillators [7], globally coupled oscillators [8], reaction-diffusion systems [9], quantum synchronization [10], etc. When there is noise present in the system, researches have also shown the power of this approach. For example, Teramae and Tanaka [11] utilized the phase reduction approach to analyze the noise-induced synchronization in uncoupled limit cycle oscillators. Arai and Nakao investigated phase coherence induced by the common Poisson impulses both theoretically and experimentally [12]. Teramae et al. developed a general reduction method for noisy limit cycle oscillators [13]. The previous researches require the system to possess a deterministic limit cycle in advance, whether the system is noisy or noise-free. For oscillations purely
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