%0 Book Section %1 statphys23_0582 %A Tsukamoto, N.T. %A Fujisaka, H.F. %A Ouchi, K.O. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K complex dynamics equation ginzburg-landau media oscillatory phase statphys23 topic-5 %T Analysis of renormalized phase dynamics in oscillatory media %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=582 %X The complex Ginzburg-Landau equation (CGLE), equation \psi=\psi+(1+ic_1)\nabla^2\psi-(1+ic_2)|\psi|^2\psi, equation is a universal equation which describes slow variation in systems near the supercritical Hopf bifurcation. Depending on the paramter values, Eq.~(1) exhibits several dynamics; homogeneous oscillation, plane wave, spiral and spatio-temporal chaos (defect-mediated and phase turbulences). It is known that the phase reduction method is a powerful tool to investigate some types of dynamical state like the phase turbulence. However, when the amplitude plays an important role (e.g., defect-mediated turbulence), the method is invalid and the dynamics cannot be described by only the phase variable. The aim of this study is to show that in spite of this fact, the behaviors observed in CGLE can be described by only the phase dynamics appropriately constructed. We construct a mapping model based on CGLE, equation \psi_n+1($x$)=LF(\psi_n($x$)), equation where $L=e^(1+ic_1)\nabla^2$ is a linear operator, and $F(\psi)=|\psi|^-(1+ic_2)\psi$ for $\psi0$ and $F(\psi)=0$ for $\psi=0$. This mapping model has the same features in both the spatial coupling and the isochron structure as CGLE. In this presentation, we will show the results of analysis of the dynamics (2) by both numerical simulation and analytical treatment. It is found that Eq.~(2) reproduces the dynamics of CGLE. Furthermore, it is proved that the dynamics (2) can be described by only the renormalized phase $þeta_n=\arg\psi_n-c_2|\psi_n|$ and the phase singular point ($\psi_n=0$). These results suggest that the CGLE dynamics can be described by the renormalized phase variable even if the amplitude component plays an important role in the dynamics.

@incollection{statphys23_0582, abstract = {The complex Ginzburg-Landau equation (CGLE), \begin{equation} \dot{\psi}=\psi+(1+ic_1)\nabla^2\psi-(1+ic_2)|\psi|^2\psi, \end{equation} is a universal equation which describes slow variation in systems near the supercritical Hopf bifurcation. Depending on the paramter values, Eq.~(1) exhibits several dynamics; homogeneous oscillation, plane wave, spiral and spatio-temporal chaos (defect-mediated and phase turbulences). It is known that the phase reduction method is a powerful tool to investigate some types of dynamical state like the phase turbulence. However, when the amplitude plays an important role (e.g., defect-mediated turbulence), the method is invalid and the dynamics cannot be described by only the phase variable. The aim of this study is to show that in spite of this fact, the behaviors observed in CGLE can be described by only the phase dynamics appropriately constructed. We construct a mapping model based on CGLE, \begin{equation} \psi_{n+1}(\mbox{\boldmath $x$})=\mathcal{L}F(\psi_n(\mbox{\boldmath $x$})), \end{equation} where $\mathcal{L}=e^{(1+ic_1)\nabla^2}$ is a linear operator, and $F(\psi)=|\psi|^{-(1+ic_2)}\psi$ for $\psi\ne 0$ and $F(\psi)=0$ for $\psi=0$. This mapping model has the same features in both the spatial coupling and the isochron structure as CGLE. In this presentation, we will show the results of analysis of the dynamics (2) by both numerical simulation and analytical treatment. It is found that Eq.~(2) reproduces the dynamics of CGLE. Furthermore, it is proved that the dynamics (2) can be described by only the renormalized phase $\theta_n=\arg\psi_n-c_2\ln |\psi_n|$ and the phase singular point ($\psi_n=0$). These results suggest that the CGLE dynamics can be described by the renormalized phase variable even if the amplitude component plays an important role in the dynamics.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Tsukamoto, N.T. and Fujisaka, H.F. and Ouchi, K.O.}, biburl = {https://www.bibsonomy.org/bibtex/2fd5c062707bd73e82959c2d190470639/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {0815f04805601e1080a96a7f668c8806}, intrahash = {fd5c062707bd73e82959c2d190470639}, keywords = {complex dynamics equation ginzburg-landau media oscillatory phase statphys23 topic-5}, month = {9-13 July}, timestamp = {2007-06-20T10:16:24.000+0200}, title = {Analysis of renormalized phase dynamics in oscillatory media}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=582}, year = 2007 }