TY - JOUR
T1 - Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems
AU - Kokubu, Hiroshi
AU - Nishiura, Yasumasa
AU - Oka, Hiroe
N1 - Funding Information:
Global bifurcation such as heteroclinic or homoclinic bifurcation has attracted much attention recently in dynamical system theory (see * This work was supported in part by grants from the Science Foundation of the Ministry of Education of Japan, 63790133 (H.K.) and 63540181 (Y.N.), and from DARPA (H.O.).
PY - 1990/8
Y1 - 1990/8
N2 - We apply a general heteroclinic and homoclinic bifurcation theory to the study of bifurcations of travelling waves of bistable reaction diffusion systems. Using the notion of separation, we first prove the existence of a cusp point of the set of travelling front solutions in the parameter space. This as well as the symmetry of the system yields a coexisting pair of front and back solutions which undergoes the homoclinic bifurcation producing a pulse solution. All the hypotheses imposed on the general heteroclinic and homoclinic bifurcation theorem are rigorously verified for a system of bistable reaction diffusion equations containing a small parameter ε by using singular perturbation techniques, especially the SLEP method. A relation between the stability of front (or back) solutions and the intersecting manner of the stable and unstable manifolds is also given by means of the separation.
AB - We apply a general heteroclinic and homoclinic bifurcation theory to the study of bifurcations of travelling waves of bistable reaction diffusion systems. Using the notion of separation, we first prove the existence of a cusp point of the set of travelling front solutions in the parameter space. This as well as the symmetry of the system yields a coexisting pair of front and back solutions which undergoes the homoclinic bifurcation producing a pulse solution. All the hypotheses imposed on the general heteroclinic and homoclinic bifurcation theorem are rigorously verified for a system of bistable reaction diffusion equations containing a small parameter ε by using singular perturbation techniques, especially the SLEP method. A relation between the stability of front (or back) solutions and the intersecting manner of the stable and unstable manifolds is also given by means of the separation.
UR - http://www.scopus.com/inward/record.url?scp=38249016389&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=38249016389&partnerID=8YFLogxK
U2 - 10.1016/0022-0396(90)90033-L
DO - 10.1016/0022-0396(90)90033-L
M3 - Article
AN - SCOPUS:38249016389
VL - 86
SP - 260
EP - 341
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 2
ER -