A celebrated result in bifurcation theory is that, when the operators involved are compact, global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem. This paper presents a simple example in which the hypotheses of the global bifurcation theorem are satisfied, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differentiable and classical bifurcation occurs locally at a simple eigenvalue, the global continua may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which bifurcate at non-zero eigenvalues, irrespective of multiplicity when the problem has gradient structure, may not be connected and may not contain any paths except singletons.
Citation: J. F. Toland. Path-connectedness in global bifurcation theory[J]. Electronic Research Archive, 2021, 29(6): 4199-4213. doi: 10.3934/era.2021079
A celebrated result in bifurcation theory is that, when the operators involved are compact, global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem. This paper presents a simple example in which the hypotheses of the global bifurcation theorem are satisfied, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differentiable and classical bifurcation occurs locally at a simple eigenvalue, the global continua may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which bifurcate at non-zero eigenvalues, irrespective of multiplicity when the problem has gradient structure, may not be connected and may not contain any paths except singletons.
[1] |
Concerning hereditarily indecomposable continua. Pacific J. Math. (1951) 1: 43-51. ![]() |
[2] | Snake-like continua. Duke Math. J. (1951) 18: 653-663. |
[3] |
Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme. Math. Zeit. (1972) 127: 105-126. ![]() |
[4] |
(2003) Analytic Theory of Global Bifurcation,. Princeton and Oxford: Princeton University Press. ![]() |
[5] |
Chainable continua and indecomposability. Pacific J. Math. (1959) 9: 653-659. ![]() |
[6] |
Bifurcation from simple eigenvalues. J. Funct. Anal. (1971) 8: 321-340. ![]() |
[7] |
Bifurcation theory for analytic operators. Proc. Lond. Math. Soc. (1973) 26: 359-384. ![]() |
[8] |
Global structure of the solutions of non-linear real analytic eigenvalue problems. Proc. Lond. Math. Soc. (1973) 27: 747-765. ![]() |
[9] |
Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one. Bull. Lond. Math. Soc. (2002) 34: 533-538. ![]() |
[10] |
J. J. Duistermaat and J. A. C. Kolk, Multidimensional Real Analysis I., Cambridge Studies in Advanced Mathematics 86. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511616716
![]() |
[11] | N. Dunford and J. T. Schwartz, Linear Operators, Part Ⅰ: General Theory, , With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7 Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. |
[12] |
The pseudo-circle is unique,. Trans. Amer. Math. Soc. (1970) 149: 45-64. ![]() |
[13] |
A brief historical view of continuum theory. Topology Appl. (2006) 153: 1530-1539. ![]() |
[14] |
Un continu dont tout sous-continu est indécomposable. Fund. Math. (1922) 3: 247-286. ![]() |
[15] | On a topological method in the problem of eigenfunctions of nonlinear operators. Dokl. Akad. Nauk SSSR (N.S.) (1950) 74: 5-7. |
[16] | Some problems in nonlinear analysis. Amer. Math. Soc. Trans. Ser. (1958) 10: 345-409. |
[17] | . A. Krasnolsel'skii, Topological Methods in the Theory of Nonlinear Eigenvalue Problems, Pergamon Press, Oxford, 1963. (Original in Russian: Topologicheskiye Metody v Teorii Nelineinykh Integral'nykh Uravnenii., Gostekhteoretizdat, Moscow, 1956.) |
[18] |
Some global results for nonlinear eigenvalue problems. J. Funct. Anal. (1971) 7: 487-513. ![]() |
[19] |
Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. (1973) 3: 161-202. ![]() |
[20] | H. L. Royden, Real Analysis, Third Edition, McMillan, New York, 1988. |
[21] |
Global bifurcation for $k$-set contractions without multiplicity assumptions. Quart. J. Math. Oxford Ser. (1976) 27: 199-216. ![]() |
[22] | M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, With a chapter on Newton's method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by Amiel Feinstein Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. |