Various types of interesting pattern dynamics such as self-replicating patterns and spiral patterns have been observed in reaction–diffusion (RD) systems. In recent years, periodically oscillating pulses called breathers have been found in several RD systems. In addition, the transient dynamics from traveling breathers to standing breathers have been numerically investigated, and the existence and stability of breathers have been studied by (semi-)rigorous approaches. However, the mechanism of transient dynamics has yet to be clarified, even using numerical approaches, since the global bifurcation diagram of breathers has not been obtained. In this article, we propose a numerical scheme that enables unstable breathers to be tracked. By using the global bifurcation diagram, we numerically investigate the global behavior of unstable manifolds emanating from the bifurcation point associated with the transient dynamics and clarify the onset mechanism of the transient dynamics.
Oscillatory pulse Reaction–diffusion system Transient dynamics Bifurcation
Mathematics Subject Classification (2000)
This is a preview of subscription content, log in to check access.
Mimura M., Nagayama M., Ikeda H., Ikeda T.: Dynamics of travelling breathers arising in reaction-diffusion systems—ODE modelling approach. Hiroshima Math. J. 30(2), 221–256 (2000)zbMATHMathSciNetGoogle Scholar
Aprille T.J., Trick T.N.: A computer algorithm to determine the steady-state response of nonlinear oscillators. Circuit Theory, IEEE Trans 19(4), 354–360 (1972)CrossRefMathSciNetGoogle Scholar
Doedel E., Keller H.B., Kernevez J.P.: Numerical analysis and control of bifurcation problems (I): bifurcation in finite dimensions. Int. J. Bifurcat. Chaos 1(4), 745–772 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
Doedel E., Keller H.B., Kernevez J.P.: Numerical analysis and control of bifurcation problems (II): bifurcation in infinite dimensions. Int. J. Bifurcat. Chaos 1(3), 493–520 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A.: Spectral methods in fluid dynamics. Springer, Berlin (1988)zbMATHGoogle Scholar