American Institute of Mathematical Sciences

Advanced
November  2013, 12(6): 2923-2933. doi: 10.3934/cpaa.2013.12.2923

A double saddle-node bifurcation theorem

Ping Liu 1, , Junping Shi 2, and  Yuwen Wang 1,

1. 

Y.Y. Tseng Functional Analysis Research Center and School of Mathematics Science, Harbin Normal University, Harbin, Heilongjiang, 150025, China, China
2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187

Received  September 2010 Revised  July 2012 Published  May 2013

In this paper, we consider an abstract equation $F(\lambda,u)=0$ with one parameter $\lambda$, where $F\in C^p(\mathbb{R} \times X, Y)$, $p\geq 2$, is a nonlinear differentiable mapping, and $X,Y$ are Banach spaces. We apply Lyapunov-Schmidt procedure and Morse Lemma to obtain a "double" saddle-node bifurcation theorem with a two-dimensional kernel. Applications include a perturbed problem and a semilinear elliptic equation.
Keywords: two-mimensional kernel, Saddle-node bifurcation, Morse lemma..
Mathematics Subject Classification: Primary: 34C23, 35R15; Secondary: 47J1.
Citation: Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923
References:
[1]

E. L. Allgower, K. Böhmer and M. Zhen, A complete bifurcation scenario for the $2$-d nonlinear Laplacian with Neumann boundary conditions on the unit square, in "Bifurcation and Chaos: Analysis, Algorithms, Applications" (Würzburg, 1990), 1-18, Internat. Ser. Numer. Math., 97, Birkhäuser, Basel, 1991. doi: 10.1007/978-3-0348-7004-7_1.  Google Scholar

[2]

S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4.  Google Scholar

[3]

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

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.  Google Scholar

[5]

M. del Pino, J. García-Melián and M. Musso, Local bifurcation from the second eigenvalue of the Laplacian in a square, Proc. Amer. Math. Soc., 131 (2003), 3499-3505. doi: 10.1090/S0002-9939-03-06906-5.  Google Scholar

[6]

M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math., 32 (1979), 21-98. doi: 10.1002/cpa.3160320103.  Google Scholar

[7]

H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004.  Google Scholar

[8]

S. Krömer, T. J. Healey and H. Kielhöfer, Bifurcation with a two-dimensional kernel, J. Differential Equations, 220 (2006), 234-258. doi: 10.1016/j.jde.2005.02.008.  Google Scholar

[9]

P. Liu and Y. W. Wang, The generalized saddle-node bifurcation of degenerate solution, Comment. Math. Prace Mat., 45 (2005), 145-150.  Google Scholar

[10]

P. Liu, J. P. Shi and Y. W. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. doi: 10.1016/j.jfa.2007.06.015.  Google Scholar

[11]

M. Zhen, "Numerical Bifurcation Analysis for Reaction-diffusion Equations," Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 2000.  Google Scholar

[12]

P. Rabier, A generalization of the implicit function theorem for mappings from $R^{n+1}$ into $R^n$ and its applications, J. Funct. Anal., 56 (1984), 145-170. doi: 10.1016/0022-1236(84)90085-5.  Google Scholar

[13]

J. P. Shi, Saddle solutions of the balanced bistable diffusion equation, Comm. Pure Appl. Math., 55 (2002), 815-830. doi: 10.1002/cpa.3027.  Google Scholar

[14]

S. D. Taliaferro, Bifurcation at multiple eigenvalues and stability of bifurcating solutions, J. Funct. Anal., 55 (1984), 247-275. doi: 10.1016/0022-1236(84)90012-0.  Google Scholar

[15]

C. A. Tiahrt and A. B. Poore, A bifurcation analysis of the nonlinear parametric programming problem, Math. Programming (Ser. A), 47 (1990), 117-141. doi: 10.1007/BF01580856.  Google Scholar

[16]

J. F. Wang, J. P. Shi and Y. W. Wang, Bifurcation from the second eigenvalue of a class of semilinear elliptic equations, (Chinese) Natur. Sci. J. Harbin Normal Univ., 21 (2005), 1-4.  Google Scholar

show all references

References:
[1]

E. L. Allgower, K. Böhmer and M. Zhen, A complete bifurcation scenario for the $2$-d nonlinear Laplacian with Neumann boundary conditions on the unit square, in "Bifurcation and Chaos: Analysis, Algorithms, Applications" (Würzburg, 1990), 1-18, Internat. Ser. Numer. Math., 97, Birkhäuser, Basel, 1991. doi: 10.1007/978-3-0348-7004-7_1.  Google Scholar

[2]

S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4.  Google Scholar

[3]

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

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.  Google Scholar

[5]

M. del Pino, J. García-Melián and M. Musso, Local bifurcation from the second eigenvalue of the Laplacian in a square, Proc. Amer. Math. Soc., 131 (2003), 3499-3505. doi: 10.1090/S0002-9939-03-06906-5.  Google Scholar

[6]

M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math., 32 (1979), 21-98. doi: 10.1002/cpa.3160320103.  Google Scholar

[7]

H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004.  Google Scholar

[8]

S. Krömer, T. J. Healey and H. Kielhöfer, Bifurcation with a two-dimensional kernel, J. Differential Equations, 220 (2006), 234-258. doi: 10.1016/j.jde.2005.02.008.  Google Scholar

[9]

P. Liu and Y. W. Wang, The generalized saddle-node bifurcation of degenerate solution, Comment. Math. Prace Mat., 45 (2005), 145-150.  Google Scholar

[10]

P. Liu, J. P. Shi and Y. W. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. doi: 10.1016/j.jfa.2007.06.015.  Google Scholar

[11]

M. Zhen, "Numerical Bifurcation Analysis for Reaction-diffusion Equations," Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 2000.  Google Scholar

[12]

P. Rabier, A generalization of the implicit function theorem for mappings from $R^{n+1}$ into $R^n$ and its applications, J. Funct. Anal., 56 (1984), 145-170. doi: 10.1016/0022-1236(84)90085-5.  Google Scholar

[13]

J. P. Shi, Saddle solutions of the balanced bistable diffusion equation, Comm. Pure Appl. Math., 55 (2002), 815-830. doi: 10.1002/cpa.3027.  Google Scholar

[14]

S. D. Taliaferro, Bifurcation at multiple eigenvalues and stability of bifurcating solutions, J. Funct. Anal., 55 (1984), 247-275. doi: 10.1016/0022-1236(84)90012-0.  Google Scholar

[15]

C. A. Tiahrt and A. B. Poore, A bifurcation analysis of the nonlinear parametric programming problem, Math. Programming (Ser. A), 47 (1990), 117-141. doi: 10.1007/BF01580856.  Google Scholar

[16]

J. F. Wang, J. P. Shi and Y. W. Wang, Bifurcation from the second eigenvalue of a class of semilinear elliptic equations, (Chinese) Natur. Sci. J. Harbin Normal Univ., 21 (2005), 1-4.  Google Scholar

[1]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
[2]

Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203
[3]

Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215
[4]

Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21
[5]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[6]

Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069
[7]

Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205
[8]

Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
[9]

Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911
[10]

Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047
[11]

Iliya D. Iliev, Chengzhi Li, Jiang Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Communications on Pure & Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583
[12]

Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 631-674. doi: 10.3934/dcds.2007.19.631
[13]

Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021019
[14]

Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[15]

Mauro Garavello, Paola Goatin. The Cauchy problem at a node with buffer. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 1915-1938. doi: 10.3934/dcds.2012.32.1915
[16]

Shaobo Gan. A generalized shadowing lemma. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 627-632. doi: 10.3934/dcds.2002.8.627
[17]

Simon Lloyd. On the Closing Lemma problem for the torus. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 951-962. doi: 10.3934/dcds.2009.25.951
[18]

Lan Wen. The selecting Lemma of Liao. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 159-175. doi: 10.3934/dcds.2008.20.159
[19]

Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473
[20]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences