-
Previous Article
Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition
- CPAA Home
- This Issue
-
Next Article
Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems
A double saddle-node bifurcation theorem
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004. |
[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. |
[9] |
P. Liu and Y. W. Wang, The generalized saddle-node bifurcation of degenerate solution, Comment. Math. Prace Mat., 45 (2005), 145-150. |
[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. |
[11] |
M. Zhen, "Numerical Bifurcation Analysis for Reaction-diffusion Equations," Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 2000. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004. |
[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. |
[9] |
P. Liu and Y. W. Wang, The generalized saddle-node bifurcation of degenerate solution, Comment. Math. Prace Mat., 45 (2005), 145-150. |
[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. |
[11] |
M. Zhen, "Numerical Bifurcation Analysis for Reaction-diffusion Equations," Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 2000. |
[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. |
[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. |
[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. |
[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. |
[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. |
[1] |
Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Related Fields, 2022, 12 (1) : 225-243. 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 and 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 and Continuous Dynamical Systems, 2012, 32 (6) : 1915-1938. doi: 10.3934/dcds.2012.32.1915 |
[16] |
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 |
[17] |
Lixia Duan, Zhuoqin Yang, Shenquan Liu, Dunwei Gong. Bursting and two-parameter bifurcation in the Chay neuronal model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 445-456. doi: 10.3934/dcdsb.2011.16.445 |
[18] |
Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 |
[19] |
Li Ma, Shangjiang Guo. Bifurcation and stability of a two-species diffusive Lotka-Volterra model. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1205-1232. doi: 10.3934/cpaa.2020056 |
[20] |
Chun-Hsiung Hsia, Tian Ma, Shouhong Wang. Bifurcation and stability of two-dimensional double-diffusive convection. Communications on Pure and Applied Analysis, 2008, 7 (1) : 23-48. doi: 10.3934/cpaa.2008.7.23 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]