Abstract
In this note it will be shown how a theorem of Alexander [1] and Ize [9] together with computational results of Alexander and Yorke [4] and Alexander and Fitzpatrick [2] may be used to generalize the existence theorem for, and to prove some global results about, certain wave-like solutions of nonlinear systems of partial differential equations.
The equations to be studied are weakly coupled parabolic systems of equations defined on a bounded axisymmetric domain. Such equations are often called reaction-diffusion equations (or interaction-diffusion equations) and arise in many parts of biology and chemistry. The question as to how wave-like solutions of these equations may bifurcate from a family of trivial solutions was studied by Auchmuty [5] and the results will be considerably extended here.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
ALEXANDER, J.C.: Bifurcation of zeroes of parameterized functions, J. Func. Anal.29 (1978) 37–54
ALEXANDER, J.C. and FITZPATRICK, P.M.: Bifurcation of fixed points of parameterized condensing operators, to be published. (J. Func. Anal.)
ALEXANDER, J.C. and YORKE, J.A.: Global bifurcation of periodic orbits, Am. J. Math.,100 (1978), 263–292
ALEXANDER, J.C. and YORKE, J.A.: Calculating bifurcation invariants as elements of the homotopy of the general linear group, J. Pure and Appl. Alg.13 (1978), 1–8
AUCHMUTY, J.F.G.: Bifurcating waves, Proc. Conf. on Bifurcation Theory and its Applications, N. Y. Acad. of Sciences (to appear)
CHOW, S. -N. and MALLET-PARET, J.: Fuller's index and global Hopf's bifurcation, to be published
CHOW, S. -N., MALLET-PARET, J. and YORKE, J.A.: Global Hopf bifurcation from a multiple eigenvalue, to be published
DANCER, E.N.: Global solution branches for positive mappings, Arch. Rat. Mech. and Anal.52 (1973), 181–192
IZE, J.: Bifurcation theory for Fredholm operators, Memoirs Am. Math. Soc., vol.7, Nr. 174 (1976)
KURATOWSKI, C.: Topologie II, Monografic Matematyzne, Warsaw, Poland 1950
NUSSBAUM, R.: A Hopf global bifurcation theorem for retarded functional differential equations, Trans A.M.S.238 (1978), 139–164
RABINOWITZ, P.H.: Some global results for nonlinear eigenvalue problems, J. Func. Anal.7 (1971), 483–513
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Alexander, J.C., Auchmuty, J.F.G. Global bifurcation of waves. Manuscripta Math 27, 159–166 (1979). https://doi.org/10.1007/BF01299293
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01299293