Embedding phase reduction for fast-slow systems with noise-induced stochastic quasiperiodic orbits

Noise is ubiquitous in natural systems and can induce coherent stochastic oscillations which cannot occur in the absence of noise, or stochastic quasiperiodic orbits which are markedly different from the deterministic limit cycle. For fast-slow systems, noise acting on the fast subsystem can induce quasi-periodic orbits tuned by the slow variable, which were termed as self-induced stochastic resonance (SISR). In our previous work [Zhu et al., 2022 [46]], we proposed a phase reduction framework on the SISR-oscillator by the hybrid approximation on the stochastic quasiperiodic orbit and the estimation of the effective noise. In this work, we remove the restriction of the additive Gaussian form assumption on the effective noise, and propose a more natural dimensionality-reduction method which we refer to as embedding phase reduction. Different SISR-oscillators induced by varying noise intensities, which can be embedded into a limit cycle, can have a unified form of the reduced phase equation. The efficiency of the embedding phase reduction is validated by Monte Carlo simulations of the case of periodic forcing, where excellent agreement is achieved for the entrainment and phase drifting dynamics. The simplicity and generality of the proposed embedding phase reduction approach indicate its potential applications in practical experiments and relevance in biological systems.