A02 NAKAO, Hiroya |Proposed Research Projects (2014-2015)

Paper | Original Paper


*Wataru Kurebayashi, Sho Shirasaka, and Hiroya Nakao,
A criterion for timescale decomposition of external inputs for generalized phase reduction of limit-cycle oscillators,
Nonlinear Theory and Its Applications (IEICE) 6, 171-180 (2015).

[Summary] The phase reduction method is a dimension reduction method for weakly driven limit-cycle oscillators, which has played an important role in the theoretical analysis of synchronization phenomena. Recently, we proposed a generalization of the phase reduction method [W. Kurebayashi et al., Phys. Rev. Lett. 111, 2013]. This generalized phase reduction method can robustly predict the dynamics of strongly driven oscillators, for which the conventional phase reduction method fails. In this generalized method, the external input to the oscillator should be properly decomposed into a slowly varying component and remaining weak fluctuations. In this paper, we propose a simple criterion for timescale decomposition of the external input, which gives accurate prediction of the phase dynamics and enables us to systematically apply the generalized phase reduction method to a general class of limit-cycle oscillators. The validity of the criterion is confirmed by numerical simulations.

*Yoji Kawamura, Hiroya Nakao,
Phase description of oscillatory convection with a spatially translational mode,
Physica D: Nonlinear Phenomena 295-296, 11–29 (2015).

[Summary] We formulate a theory for the phase description of oscillatory convection in a cylindrical Hele–Shaw cell that is laterally periodic. This system possesses spatial translational symmetry in the lateral direction owing to the cylindrical shape as well as temporal translational symmetry. Oscillatory convection in this system is described by a limit-torus solution that possesses two phase modes; one is a spatial phase and the other is a temporal phase. The spatial and temporal phases indicate the “position” and “oscillation” of the convection, respectively. The theory developed in this paper can be considered as a phase reduction method for limit-torus solutions in infinite-dimensional dynamical systems, namely, limit-torus solutions to partial differential equations representing oscillatory convection with a spatially translational mode. We derive the phase sensitivity functions for spatial and temporal phases; these functions quantify the phase responses of the oscillatory convection to weak perturbations applied at each spatial point. Using the phase sensitivity functions, we characterize the spatiotemporal phase responses of oscillatory convection to weak spatial stimuli and analyze the spatiotemporal phase synchronization between weakly coupled systems of oscillatory convection.


*Wataru Kurebayashi, Tsubasa Ishii, Mikio Hasegawa, and Hiroya Nakao,
Design and control of noise-induced synchronization patterns,
EPL (Europhysics Letters) 107, 10009/1-6 (2014).

[Summary] We propose a method for controlling synchronization patterns of limit-cycle oscillators by common noisy inputs, i.e., by utilizing noise-induced synchronization. Various synchronization patterns, including fully synchronized and clustered states, can be realized by using linear filters that generate appropriate common noisy signals from given noise. The optimal linear filter can be determined from the linear phase response property of the oscillators and the power spectrum of the given noise. The validity of the proposed method is confirmed by numerical simulations.

*Hiroya Nakao, Tatsuo Yanagita, Yoji Kawamura,
Phase-Reduction Approach to Synchronization of Spatiotemporal Rhythms in Reaction-Diffusion Systems,
Physical Review X 4, 021032/1-23 (2014).

[Summary] Reaction-diffusion systems can describe a wide class of rhythmic spatiotemporal patterns observed in chemical and biological systems, such as circulating pulses on a ring, oscillating spots, target waves, and rotating spirals. These rhythmic dynamics can be considered limit cycles of reaction-diffusion systems. However, the conventional phase-reduction theory, which provides a simple unified framework for analyzing synchronization properties of limit-cycle oscillators subjected to weak forcing, has mostly been restricted to low-dimensional dynamical systems. Here, we develop a phase-reduction theory for stable limit-cycle solutions of reaction-diffusion systems with infinite-dimensional state space. By generalizing the notion of isochrons to functional space, the phase-sensitivity function—a fundamental quantity for phase reduction—is derived. For illustration, several rhythmic dynamics of the FitzHugh-Nagumo model of excitable media are considered. Nontrivial phase-response properties and synchronization dynamics are revealed, reflecting their complex spatiotemporal organization. Our theory will provide a general basis for the analysis and control of spatiotemporal rhythms in various reaction-diffusion systems.

Masahiro Kazama, *Wataru Kurebayashi, Takahiro Tsuchida, Yuta Minoshima, Mikio Hasegawa, Koji Kimura, and Hiroya Nakao,
Enhancement of noise correlation for noise-induced synchronization of limit-cycle oscillators by threshold filtering,
NOLTA, IEICE 5, 157-171 (2014).

[Summary] Nonlinear oscillators driven by correlated noisy signals can synchronize without di- rect mutual interactions. Here we show that correlation between noisy signals can be enhanced by applying a threshold filter, and the filtered signals can be used to improve noise-induced synchronization. We derive analytical expressions for the correlation coefficient between the filtered signals, and, using simple examples, we demonstrate that the correlation can actually be enhanced and the synchronization can be improved by the threshold filtering in some cases.