We consider the effect of external noise on the dynamics of limit cycle oscillators. The Lyapunov exponent becomes negative under influence of small white noise, what means synchronization of two or more identical systems subject to common noise. We analytically study the effect of small non-identities in the oscillators and in the noise, and derive statistical characteristics of deviations from the perfect synchrony. Large white noise can lead to desynchronization of oscillators, provided they are non-isochronous. This is demonstrated for the Van der Pol-Duffing system.

We develop an approach for the description of the dynamics of large populations of phase oscillators based on "circular cumulants" instead of the Kuramoto-Daido order parameters. In the thermodynamic limit, these variables yield a simple representation of the Ott-Antonsen invariant solution [E. Ott and T. M. Antonsen, Chaos 18, 037113 (2008)CHAOEH1054-150010.1063/1.2930766] and appear appropriate for constructing perturbation theory on top of the Ott-Antonsen ansatz. We employ this approach to study the impact of small intrinsic noise on the dynamics. As a result, a closed system of equations for the two leading cumulants, describing the dynamics of noisy ensembles, is derived. We exemplify the general theory by presenting the effect of noise on the Kuramoto system and on a chimera state in two symmetrically coupled populations.

The phase description is a powerful tool for analyzing noisy limit cycle oscillators. The method, however, has found only limited applications so far, because the present theory is applicable only to the Gaussian noise while noise in the real world often has non-Gaussian statistics. Here, we provide the phase reduction method for limit cycle oscillators subject to general, colored and non-Gaussian, noise including heavy-tailed one. We derive quantifiers like mean frequency, diffusion constant, and the Lyapunov exponent to confirm consistency of the results. Applying our results, we additionally study a resonance between the phase and noise. Limit cycle oscillators effectively model various sustained oscillations in many fields of science and technology including chemical reactions, biology, electric circuits, and lasers [1][2][3][4]. The phase reduction method is a powerful analytical tool which approximates highdimensional dynamics of limit cycle oscillators with single phase variable that characterizes timing of oscillation [1,5]. Since the phase is neutrally stable, phase perturbations persist in time and result in various remarkable phenomena where weak action leads to significant effects, such as those addressed in the theory of synchronization [6,7]. While the theory of the phase reduction had been developed for deterministic oscillators, recent studies successfully extended the theory to limit cycle oscillators subject to noise [4,8,9] and revealed that interplay between nonlinearity and noise results in fascinating noise-induced phenomena including frequency modulation and noise-induced synchronization [12,13]. This extended phase reduction method, however, has found limited applications so far, since the method is applicable only to Gaussian noise. While the noise in the real world often has non-Gaussian statistics, few theories have considered nonlinear systems subject to general non-Gaussian noise, which has forced people to use the Gaussian approximation. In particular, whether the phase description is still valid for oscillators subject to non-Gaussian noise and how quantifiers of the phase dynamics should be amended remains unknown. In this paper, we develop the phase reduction method for limit cycle oscillators subject to general, colored and nonGaussian noise. By correctly evaluating the influence of amplitude perturbations up to second order in the noise strength, we derive the stochastic differential equation of phase, which allows us to study nonlinear oscillations in the real world without the Gaussian approximation. To confirm consistency of the result, we derive closed expressions of quantifiers of the phase dynamics such as mean frequency, phase diffusion constant, and the Lyapunov exponent. The only limitation we impose is the weakness of the noise. Thus, the obtained results are applicable even when higher order moments of the noise diverge as long as the second order moment is finite and we confirm this fact numerically. As an application of the results, we study a limit cycle oscillato...