December  2021, 29(6): 4199-4213. doi: 10.3934/era.2021079

Path-connectedness in global bifurcation theory

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

For Norman Dancer on his 75th Birthday

Received  April 2021 Revised  August 2021 Published  December 2021 Early access  October 2021

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. Electronic Research Archive, 2021, 29 (6) : 4199-4213. doi: 10.3934/era.2021079
References:
[1]

R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math., 1 (1951), 43-51.  doi: 10.2140/pjm.1951.1.43.  Google Scholar

[2]

R. H. Bing, Snake-like continua, Duke Math. J., 18 (1951), 653-663.   Google Scholar

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R. Böhme, Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme, Math. Zeit., 127 (1972), 105-126.  doi: 10.1007/BF01112603.  Google Scholar

[4] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation,, Princeton University Press, Princeton and Oxford, 2003.  doi: 10.1515/9781400884339.  Google Scholar
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C. E. Burgess, Chainable continua and indecomposability, Pacific J. Math., 9 (1959), 653-659.  doi: 10.2140/pjm.1959.9.653.  Google Scholar

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M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

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E. N. Dancer, Bifurcation theory for analytic operators, Proc. Lond. Math. Soc., 26 (1973), 359-384.  doi: 10.1112/plms/s3-26.2.359.  Google Scholar

[8]

E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. Lond. Math. Soc., 27 (1973), 747-765.  doi: 10.1112/plms/s3-27.4.747.  Google Scholar

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E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. Lond. Math. Soc., 34 (2002), 533-538.  doi: 10.1112/S002460930200108X.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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L. Fearnley, The pseudo-circle is unique,, Trans. Amer. Math. Soc., 149 (1970), 45-64.  doi: 10.1090/S0002-9947-1970-0261559-6.  Google Scholar

[13]

W. T. Ingram, A brief historical view of continuum theory, Topology Appl., 153 (2006), 1530-1539.  doi: 10.1016/j.topol.2004.08.024.  Google Scholar

[14]

B. Knaster, Un continu dont tout sous-continu est indécomposable, Fund. Math., 3 (1922), 247-286.  doi: 10.4064/fm-3-1-247-286.  Google Scholar

[15]

M. A. Krasnolsel'skiǐ, On a topological method in the problem of eigenfunctions of nonlinear operators, Dokl. Akad. Nauk SSSR (N.S.), 74 (1950), 5-7.   Google Scholar

[16]

M. A. Krasnolsel'skiǐ, Some problems in nonlinear analysis, Amer. Math. Soc. Trans. Ser., 10 (1958), 345-409.   Google Scholar

[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.) Google Scholar

[18]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[19]

P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202.  doi: 10.1216/RMJ-1973-3-2-161.  Google Scholar

[20]

H. L. Royden, Real Analysis, Third Edition, McMillan, New York, 1988.  Google Scholar

[21]

J. F. Toland, Global bifurcation for $k$-set contractions without multiplicity assumptions, Quart. J. Math. Oxford Ser., 27 (1976), 199-216.  doi: 10.1093/qmath/27.2.199.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math., 1 (1951), 43-51.  doi: 10.2140/pjm.1951.1.43.  Google Scholar

[2]

R. H. Bing, Snake-like continua, Duke Math. J., 18 (1951), 653-663.   Google Scholar

[3]

R. Böhme, Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme, Math. Zeit., 127 (1972), 105-126.  doi: 10.1007/BF01112603.  Google Scholar

[4] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation,, Princeton University Press, Princeton and Oxford, 2003.  doi: 10.1515/9781400884339.  Google Scholar
[5]

C. E. Burgess, Chainable continua and indecomposability, Pacific J. Math., 9 (1959), 653-659.  doi: 10.2140/pjm.1959.9.653.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[7]

E. N. Dancer, Bifurcation theory for analytic operators, Proc. Lond. Math. Soc., 26 (1973), 359-384.  doi: 10.1112/plms/s3-26.2.359.  Google Scholar

[8]

E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. Lond. Math. Soc., 27 (1973), 747-765.  doi: 10.1112/plms/s3-27.4.747.  Google Scholar

[9]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. Lond. Math. Soc., 34 (2002), 533-538.  doi: 10.1112/S002460930200108X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Fearnley, The pseudo-circle is unique,, Trans. Amer. Math. Soc., 149 (1970), 45-64.  doi: 10.1090/S0002-9947-1970-0261559-6.  Google Scholar

[13]

W. T. Ingram, A brief historical view of continuum theory, Topology Appl., 153 (2006), 1530-1539.  doi: 10.1016/j.topol.2004.08.024.  Google Scholar

[14]

B. Knaster, Un continu dont tout sous-continu est indécomposable, Fund. Math., 3 (1922), 247-286.  doi: 10.4064/fm-3-1-247-286.  Google Scholar

[15]

M. A. Krasnolsel'skiǐ, On a topological method in the problem of eigenfunctions of nonlinear operators, Dokl. Akad. Nauk SSSR (N.S.), 74 (1950), 5-7.   Google Scholar

[16]

M. A. Krasnolsel'skiǐ, Some problems in nonlinear analysis, Amer. Math. Soc. Trans. Ser., 10 (1958), 345-409.   Google Scholar

[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.) Google Scholar

[18]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[19]

P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202.  doi: 10.1216/RMJ-1973-3-2-161.  Google Scholar

[20]

H. L. Royden, Real Analysis, Third Edition, McMillan, New York, 1988.  Google Scholar

[21]

J. F. Toland, Global bifurcation for $k$-set contractions without multiplicity assumptions, Quart. J. Math. Oxford Ser., 27 (1976), 199-216.  doi: 10.1093/qmath/27.2.199.  Google Scholar

[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.  Google Scholar

Figure 1.  Schematic diagram of $ \widehat \Omega $
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