November  2019, 12(7): 1955-1975. doi: 10.3934/dcdss.2019127

Branching and bifurcation

1. 

Department of Mathematics, University of Maryland, 4176 Campus Dr, College Park, MD 20742, USA

2. 

INDAM, Dipartimento di Scienze Matematiche, Politecnico di Torino, Duca degli Abruzzi 24, 10129 Torino, Italy

* Corresponding author

Dedicated to Norman Dancer

Received  January 2018 Revised  August 2018 Published  December 2018

Fund Project: J. Pejsachowicz is supported by GNAMPA-INDAM.

By relating the set of branch points $ \mathcal{B} (f) $ of a Fredholm mapping $ f $ to linearized bifurcation, we show, among other things, that under mild local assumptions at a single point, the set $ \mathcal B(f) $ is sufficiently large to separate the domain of the mapping. In the variational case, we will also provide estimates from below for the number of connected components of the complement of $ \mathcal B(f). $

Citation: Patrick M. Fitzpatrick, Jacobo Pejsachowicz. Branching and bifurcation. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1955-1975. doi: 10.3934/dcdss.2019127
References:
[1]

R. Abraham and J. Robbin, Transversal Mappings and Flows, W. A. Benjamin, New York/Amsterdam, 1967.

[2]

J. C. Alexander, Bifurcation of zeroes of parametrized functions, J. of Funct. Anal, 29 (1978), 37-53. doi: 10.1016/0022-1236(78)90045-9.

[3] A. Ambrosetti and G. Prodi, A primer of Nonlinear Analysis, Cambridge U. Press, 1993. 
[4]

S. B. Angenent and R. Vandervorst, A priori bounds and renormalized Morse indices of solutions of an elliptic system, Ann. Inst. Poincaré, Analyse non linéaire, 17 (2000), 277-306.  doi: 10.1016/S0294-1449(00)00110-4.

[5]

S. B. Angenent and R. Vandervorst, A superquadratic indefinite elliptic system and its Morse-Conley-Floer homology, Math. Zeichr., 231 (1999), 203-248.  doi: 10.1007/PL00004731.

[6]

M. F. AtiyahV. K. Patodi. and I. M. Singer, Spectral Asymmetry and Riemannian Geometry Ⅲ, Proc. Cambridge Philos. Soc., 79 (1976), 71-99.  doi: 10.1017/S0305004100052105.

[7]

A. Bahri and P. L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math., 45 (1992), 1205-1215.  doi: 10.1002/cpa.3160450908.

[8]

A. Bahri and P. L. Lions, Morse-index of some min-max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math., 41 (1988), 1027-1037. doi: 10.1002/cpa.3160410803.

[9]

P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree, Ann. Sci. Math. Québec, 22 (1998), 131-148. 

[10]

M. S. Berger and R. A. Plastock, On the singularities of nonlinear Fredholm operators, Bull. Amer. Math. Soc., 83 (1977), 1316-1318.  doi: 10.1090/S0002-9904-1977-14431-5.

[11]

G. Borisovich, I. Sapronov and V. G. Zvyagin, Nonlinear Fredholm maps and Leray-Schauder theory, Russian Math. Surveys, 32 (1977), 3-54.

[12]

R. Caccioppoli, Sulle corrispondenze funzionali inverse diramate: Teoria generale e applicazioni ad alcune equazioni funzionali nonlineari e al problema di Plateau, Ⅰ, Ⅱ, Rend. Accad. Naz. Lincei, 24 (1936), 258-263,416-421, Opere Scelte, Vol 2, Edizioni Cremonese, Roma.

[13]

P. T. Church, E. N. Dancer and J. G. Timourian, The structure of a nonlinear elliptic operator, Trans. Amer. Math. Soc., 338 (1993), 1-42. doi: 10.1090/S0002-9947-1993-1124165-2.

[14]

P. T. Church and J. G. Timourian, Global fold maps in differential and integral equations, Nonlinear Analysis: Theory, Methods & Applications 18 (1992), 743-758. doi: 10.1016/0362-546X(92)90169-F.

[15]

K. D. Elworthy and A. J. Tromba, Degree theory on Banach manifolds, Nonlinear Functional Analysis, Proc. Sympos. Pure Math., ⅩⅧ (1970), 8-94. 

[16]

P. M. Fitzpatrick, Homotopy, linearization, and bifurcation, Nonlinear Anal., 12 (1988), 171-184.  doi: 10.1016/0362-546X(88)90033-8.

[17]

P. M. Fitzpatrick and J. Pejsachowicz, Parity and generalized multiplicity, Trans. Amer. Math. Soc., 326 (1991), 281-305. doi: 10.1090/S0002-9947-1991-1030507-7.

[18]

P. M. Fitzpatrick, J. Pejsachowicz and C. A. Stuart, Spectral flow for paths of unbounded operators and bifurcation of critical points, preprint, 2004.

[19]

P. M. Fitzpatrick, J. Pejsachowicz and P. Rabier, Degree theory for proper C2-Fredholm maps, J. Reine Angew. Math., 427 (1992), 1-33.

[20]

P. M. FitzpatrickJ. Pejsachowicz and L. Recht, Spectral flow and bifurcation of critical points of strongly-indefinite functionals-part Ⅰ : General theory, J. Funct. Anal., 162 (1999), 52-95.  doi: 10.1006/jfan.1998.3366.

[21]

A. Floer, An instanton invariant for 3-manifolds, Comun. Math. Physics, 118 (1988), 215-240.  doi: 10.1007/BF01218578.

[22]

D. Henderson, Infinite dimensional manifolds are open subsets of Hilbert space, Bull. Amer. Math. Soc., 75 (1969), 759-762.  doi: 10.1090/S0002-9904-1969-12276-7.

[23]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonial Systems, Springer-Verlag, 1989. doi: 10.1007/978-1-4757-2061-7.

[24] J. Milnor, Topology from Differentiable View-Point, Princeton University Press, Princeton, NJ, 1997. 
[25]

L. Nicolaescu, The Maslov index, the spectral flow and decompositions of manifolds, C.R.Acad.Sci.Paris, 317 (1993), 515-519. 

[26]

J. Pejsachowicz and P. J. Rabier, Degree theory for C1-Fredholm mappings of index 0, Journal d'Analyse Mathematique, 76 (1998), 289-319. doi: 10.1007/BF02786939.

[27]

J. Pejsachowicz and N. Waterstraat, Bifurcation of critical points for continuous families of C2-functionals of Fredholm type, Journal of Fixed Point Theory and Applications, 13 (2013), 537-560.  doi: 10.1007/s11784-013-0137-0.

[28]

J. Phillips, Self-adjoint fredholm operators and spectral flow, Canad. Math. Bull., 39 (1996), 460-467.  doi: 10.4153/CMB-1996-054-4.

[29]

J. Robbin and D. Salamon, The spectral flow and the maslov index, Bull. London Math. Soc., 27 (1995), 1-33.  doi: 10.1112/blms/27.1.1.

[30]

H. L. Smith and C. A. Stuart, A uniqueness theorem for fixed points, Proc. Amer. Math. Soc., 79 (1980), 237-240.  doi: 10.1090/S0002-9939-1980-0565346-2.

[31]

A. Szulkin, Bifurcation for strongly indefinite functionals and a Liapunov type theorem for Hamiltonian systems, J. Differential Integral Equations, 7 (1994), 217-234. 

[32]

K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Commun. Partial Differ. Equ., 14 (1989), 99-128. doi: 10.1080/03605308908820592.

[33]

A. J. Tromba, Some theorems on Fredholm maps, Proc. Amer. Math. Soc., 34 (1972), 578-585.  doi: 10.1090/S0002-9939-1972-0298713-0.

[34]

N. Waterstraat, Spectral flow and bifurcation for a class of strongly indefinite elliptic systems, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), 1097-1113, arXiv: 1512.04109 [math. AP] doi: 10.1017/S0308210517000324.

show all references

Dedicated to Norman Dancer

References:
[1]

R. Abraham and J. Robbin, Transversal Mappings and Flows, W. A. Benjamin, New York/Amsterdam, 1967.

[2]

J. C. Alexander, Bifurcation of zeroes of parametrized functions, J. of Funct. Anal, 29 (1978), 37-53. doi: 10.1016/0022-1236(78)90045-9.

[3] A. Ambrosetti and G. Prodi, A primer of Nonlinear Analysis, Cambridge U. Press, 1993. 
[4]

S. B. Angenent and R. Vandervorst, A priori bounds and renormalized Morse indices of solutions of an elliptic system, Ann. Inst. Poincaré, Analyse non linéaire, 17 (2000), 277-306.  doi: 10.1016/S0294-1449(00)00110-4.

[5]

S. B. Angenent and R. Vandervorst, A superquadratic indefinite elliptic system and its Morse-Conley-Floer homology, Math. Zeichr., 231 (1999), 203-248.  doi: 10.1007/PL00004731.

[6]

M. F. AtiyahV. K. Patodi. and I. M. Singer, Spectral Asymmetry and Riemannian Geometry Ⅲ, Proc. Cambridge Philos. Soc., 79 (1976), 71-99.  doi: 10.1017/S0305004100052105.

[7]

A. Bahri and P. L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math., 45 (1992), 1205-1215.  doi: 10.1002/cpa.3160450908.

[8]

A. Bahri and P. L. Lions, Morse-index of some min-max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math., 41 (1988), 1027-1037. doi: 10.1002/cpa.3160410803.

[9]

P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree, Ann. Sci. Math. Québec, 22 (1998), 131-148. 

[10]

M. S. Berger and R. A. Plastock, On the singularities of nonlinear Fredholm operators, Bull. Amer. Math. Soc., 83 (1977), 1316-1318.  doi: 10.1090/S0002-9904-1977-14431-5.

[11]

G. Borisovich, I. Sapronov and V. G. Zvyagin, Nonlinear Fredholm maps and Leray-Schauder theory, Russian Math. Surveys, 32 (1977), 3-54.

[12]

R. Caccioppoli, Sulle corrispondenze funzionali inverse diramate: Teoria generale e applicazioni ad alcune equazioni funzionali nonlineari e al problema di Plateau, Ⅰ, Ⅱ, Rend. Accad. Naz. Lincei, 24 (1936), 258-263,416-421, Opere Scelte, Vol 2, Edizioni Cremonese, Roma.

[13]

P. T. Church, E. N. Dancer and J. G. Timourian, The structure of a nonlinear elliptic operator, Trans. Amer. Math. Soc., 338 (1993), 1-42. doi: 10.1090/S0002-9947-1993-1124165-2.

[14]

P. T. Church and J. G. Timourian, Global fold maps in differential and integral equations, Nonlinear Analysis: Theory, Methods & Applications 18 (1992), 743-758. doi: 10.1016/0362-546X(92)90169-F.

[15]

K. D. Elworthy and A. J. Tromba, Degree theory on Banach manifolds, Nonlinear Functional Analysis, Proc. Sympos. Pure Math., ⅩⅧ (1970), 8-94. 

[16]

P. M. Fitzpatrick, Homotopy, linearization, and bifurcation, Nonlinear Anal., 12 (1988), 171-184.  doi: 10.1016/0362-546X(88)90033-8.

[17]

P. M. Fitzpatrick and J. Pejsachowicz, Parity and generalized multiplicity, Trans. Amer. Math. Soc., 326 (1991), 281-305. doi: 10.1090/S0002-9947-1991-1030507-7.

[18]

P. M. Fitzpatrick, J. Pejsachowicz and C. A. Stuart, Spectral flow for paths of unbounded operators and bifurcation of critical points, preprint, 2004.

[19]

P. M. Fitzpatrick, J. Pejsachowicz and P. Rabier, Degree theory for proper C2-Fredholm maps, J. Reine Angew. Math., 427 (1992), 1-33.

[20]

P. M. FitzpatrickJ. Pejsachowicz and L. Recht, Spectral flow and bifurcation of critical points of strongly-indefinite functionals-part Ⅰ : General theory, J. Funct. Anal., 162 (1999), 52-95.  doi: 10.1006/jfan.1998.3366.

[21]

A. Floer, An instanton invariant for 3-manifolds, Comun. Math. Physics, 118 (1988), 215-240.  doi: 10.1007/BF01218578.

[22]

D. Henderson, Infinite dimensional manifolds are open subsets of Hilbert space, Bull. Amer. Math. Soc., 75 (1969), 759-762.  doi: 10.1090/S0002-9904-1969-12276-7.

[23]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonial Systems, Springer-Verlag, 1989. doi: 10.1007/978-1-4757-2061-7.

[24] J. Milnor, Topology from Differentiable View-Point, Princeton University Press, Princeton, NJ, 1997. 
[25]

L. Nicolaescu, The Maslov index, the spectral flow and decompositions of manifolds, C.R.Acad.Sci.Paris, 317 (1993), 515-519. 

[26]

J. Pejsachowicz and P. J. Rabier, Degree theory for C1-Fredholm mappings of index 0, Journal d'Analyse Mathematique, 76 (1998), 289-319. doi: 10.1007/BF02786939.

[27]

J. Pejsachowicz and N. Waterstraat, Bifurcation of critical points for continuous families of C2-functionals of Fredholm type, Journal of Fixed Point Theory and Applications, 13 (2013), 537-560.  doi: 10.1007/s11784-013-0137-0.

[28]

J. Phillips, Self-adjoint fredholm operators and spectral flow, Canad. Math. Bull., 39 (1996), 460-467.  doi: 10.4153/CMB-1996-054-4.

[29]

J. Robbin and D. Salamon, The spectral flow and the maslov index, Bull. London Math. Soc., 27 (1995), 1-33.  doi: 10.1112/blms/27.1.1.

[30]

H. L. Smith and C. A. Stuart, A uniqueness theorem for fixed points, Proc. Amer. Math. Soc., 79 (1980), 237-240.  doi: 10.1090/S0002-9939-1980-0565346-2.

[31]

A. Szulkin, Bifurcation for strongly indefinite functionals and a Liapunov type theorem for Hamiltonian systems, J. Differential Integral Equations, 7 (1994), 217-234. 

[32]

K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Commun. Partial Differ. Equ., 14 (1989), 99-128. doi: 10.1080/03605308908820592.

[33]

A. J. Tromba, Some theorems on Fredholm maps, Proc. Amer. Math. Soc., 34 (1972), 578-585.  doi: 10.1090/S0002-9939-1972-0298713-0.

[34]

N. Waterstraat, Spectral flow and bifurcation for a class of strongly indefinite elliptic systems, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), 1097-1113, arXiv: 1512.04109 [math. AP] doi: 10.1017/S0308210517000324.

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