Existence of Rotating Spots with Spatially Dependent Feedback in the Plane in a Wave Front Interaction Model
- 122 Downloads
Abstract
In this paper, we study the localized rotating spots created by the spatially inhomogeneous feedback. We adopt the wave front interaction model proposed by Zykov and Showalter in 2005. The existence of rotating spots in the plane are shown by choosing the spatially dependent feedbacks appropriately.
Keywords
Rotating spot Localized spiral wave pattern Angular speedNotes
Acknowledgments
The authors would like to thank the Mathematics Division of NCTS (Taipei Office) for the support of the third author’s visit to Taiwan for which this paper was inspired. We also thank the anonymous referees for their valuable comments and suggestions. This work was supported in part by the Grants MOST 102-2811-M-032-007 and MOST 102-2115-M-032-003-MY3 from the Ministry of Science and Technology of Taiwan (ROC), by a Research Project Grant (A) from the Institute of Science and Technology, Meiji University, and by JSPS KAKENHI Grant Numbers 25610036 and 26287024.
References
- 1.Chen, Y.-Y., Guo, J.-S., Ninomiya, H.: Existence and uniqueness of rigidly rotating spiral waves by a wave front interaction model. Physica D 241, 1758–1766 (2012)CrossRefMATHGoogle Scholar
- 2.Chen, Y.-Y., Kohsaka, Y., Ninomiya, H.: Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems. Discret. Contin. Dyn. Syst. B 19, 697–714 (2014)MathSciNetCrossRefMATHGoogle Scholar
- 3.Guo, J.-S., Nakamura, K.-I., Ogiwara, T., Tsai, J.-C.: On the steadily rotating spirals. Jpn. J. Ind. Appl. Math. 23, 1–19 (2006)MathSciNetCrossRefMATHGoogle Scholar
- 4.Guo, J.-S., Ninomiya, H., Tsai, J.-C.: Existence and uniqueness of stabilized propagating wave segments in wave front interaction model. Physica D 239, 230–239 (2010)MathSciNetCrossRefMATHGoogle Scholar
- 5.Guo, J.-S., Ninomiya, H., Wu, C.-C.: Existence of a rotating wave pattern in a disk for a wave front interaction model. Commun. Pure Appl. Anal. 12, 1049–1063 (2013)MathSciNetCrossRefMATHGoogle Scholar
- 6.Mihaliuk, E., Sakurai, T., Chirila, F., Showalter, K.: Experimental and theoretical studies of feedback stabilization of propagating wave segments. Faraday Discuss 120, 383–394 (2001)CrossRefGoogle Scholar
- 7.Mihaliuk, E., Sakurai, T., Chirila, F., Showalter, K.: Feedback stabilization of unstable propagating waves. Phys. Rev. E 65, 065602 (2002)CrossRefGoogle Scholar
- 8.Or-Guil, M., Bode, M., Schenk, C.P., Purwins, H.-G.: Spot bifurcations in three-component reaction-diffusion systems: the onset of propagation. Phys. Rev. E 57, 6432–6437 (1998)CrossRefGoogle Scholar
- 9.Pismen, L.M.: Perturbation theory for traveling droplets. Phys. Rev. E 74, 041605 (2006)CrossRefGoogle Scholar
- 10.Sakurai, T., Mihaliuk, E., Chirila, F., Showalter, K.: Design and control of wave propagation patterns in excitable media. Science 296, 2009–2012 (2002)CrossRefGoogle Scholar
- 11.Sumino, Y., Magome, N., Hamada, T., Yoshikawa, K.: Self-running droplet: emergence of regular motion from nonequilibrium noise. Phys. Rev. Lett. 94, 068301 (2005)CrossRefGoogle Scholar
- 12.Teramoto, T., Suzuki, K., Nishiura, Y.: Rotational motion of traveling spots in dissipative systems. Phys. Rev. E 80, 046208 (2009)CrossRefGoogle Scholar
- 13.Zykov, V.S., Showalter, K.: Wave front interaction model of stabilized propagating wave segments. Phys. Rev. Lett. 94, 068302 (2005)CrossRefGoogle Scholar
- 14.Zykov, V.S.: Selection mechanism for rotating patterns in weakly excitable media. Phys. Rev. E 75, 046203 (2007)MathSciNetCrossRefGoogle Scholar
- 15.Zykov, V.S.: Kinematics of rigidly rotating spiral waves. Physica D 238, 931–940 (2009)MathSciNetCrossRefMATHGoogle Scholar