Imperfect transcritical and pitchfork bifurcations

Journal of Functional Analysis

Volumn 251, Issue 2, 2007, Pages 573-600

  Liu, Ping ^a,b   Shi, Junping ^a,c   Wang, Yuwen ^a  

^a HARBIN NORMAL UNIVERSITY   (China)

^b NORTHEAST NORMAL UNIVERSITY   (China)

^c The College of William and Mary   (United States)

Author keywords

Bifurcation in Banach spaces; Imperfect bifurcation; Secondary bifurcation

Indexed keywords

EID: 34548415794     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2007.06.015     Document Type: Article

References (28)

1. Antman S.S. Nonlinear Problems of Elasticity. Appl. Math. Sci. second ed. vol. 107 (2005), Springer, New York
2. Chang K.-C. Infinite-dimensional Morse Theory and Multiple Solution Problems. Progr. Nonlinear Differential Equations Appl. vol. 6 (1993), Birkhäuser Boston, Boston, MA
3. Chow S.N., and Hale J.K. Methods of Bifurcation Theory (1982), Springer, New York
4. Church P.T., and Timourian J.G. Global structure for nonlinear operators in differential and integral equations. I. Folds. II. Cusps. Topological Nonlinear Analysis, II. Frascati, 1995. Progr. Nonlinear Differential Equations Appl. vol. 27 (1997), Birkhäuser Boston, Boston, MA 109-160 161-245
5. Crandall M.G., and Rabinowitz P.H. Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971) 321-340
6. Crandall M.G., and Rabinowitz P.H. Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52 (1973) 161-180
7. Deimling K. Nonlinear Functional Analysis (1985), Springer, Berlin
8. Fisher R.A. The wave of advance of advantageous genes. Ann. Eugenics 7 (1937) 353-369
9. Golubitsky M. An introduction to catastrophe theory and its applications. SIAM Rev. 20 2 (1978) 352-387
10. Golubitsky M., and Schaeffer D. A theory for imperfect bifurcation via singularity theory. Comm. Pure Appl. Math. 32 1 (1979) 21-98
11. Kielhöfer H. Bifurcation Theory. An Introduction with Applications to PDEs. Appl. Math. Sci. vol. 156 (2004), Springer, New York
12. Kolmogoroff A., Petrovsky I., and Piscounoff N. Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem. Moscow Univ. Bull. Math. 1 (1937) 1-25 (in French)
13. 1-equivalence of functions near isolated critical points. Symposium on Infinite-Dimensional Topology. Ann. of Math. Stud. vol. 69 (1972), Princeton Univ. Press, Princeton, NJ 199-218
14. Matkowsky B.J., and Reiss E.L. Singular perturbations of bifurcations. SIAM J. Appl. Math. 33 2 (1977) 230-255
15. Nirenberg L. Variational and topological methods in nonlinear problems. Bull. Amer. Math. Soc. (N.S.) 4 3 (1981) 267-302
16. Nirenberg L. Topics in Nonlinear Functional Analysis. Courant Lect. Notes Math. vol. 6 (2001), New York Univ., Courant Inst. Math. Sci./Amer. Math. Soc., Providence, RI
17. Liu P., and Wang Y.-W. A generalized saddle-node bifurcation for degenerate solutions. Commen. Math. XLV 2 (2005) 145-150
18. López-Gómez J. Spectral Theory and Nonlinear Functional Analysis. Chapman & Hall/CRC Res. Notes Math. vol. 426 (2001), Chapman & Hall/CRC, Boca Raton, FL
19. Oruganti S., Shi J., and Shivaji R. Diffusive logistic equation with constant effort harvesting, I: Steady states. Trans. Amer. Math. Soc. 354 9 (2002) 3601-3619
20. Ouyang T., and Shi J. Exact multiplicity of positive solutions for a class of semilinear problem. J. Differential Equations 146 1 (1998) 121-156
21. Ouyang T., and Shi J. Exact multiplicity of positive solutions for a class of semilinear problem: II. J. Differential Equations 158 1 (1999) 94-151
22. Perko L. Differential Equations and Dynamical Systems. Texts Appl. Math. first ed. vol. 7 (1991), Springer, New York
23. Reiss E.L. Imperfect bifurcation. Applications of Bifurcation Theory, Proc. Advanced Sem. Univ. Wisconsin, Madison, WI, 1976. Publ. Math. Res. Center Univ. Wisconsin vol. 38 (1977), Academic Press, New York 37-71
24. Shi J. Persistence and bifurcation of degenerate solutions. J. Funct. Anal. 169 2 (1999) 494-531
25. Shi J. Multi-parameter bifurcation and applications. In: Brezis H., Chang K.C., Li S.J., and Rabinowitz P. (Eds). ICM 2002 Satellite Conference on Nonlinear Functional Analysis: Topological Methods, Variational Methods and Their Applications (2003), World Scientific, Singapore 211-222
26. Shi J., and Shivaji R. Global bifurcation for concave semipositon problems. In: Goldstein G.R., Nagel R., and Romanelli S. (Eds). Advances in Evolution Equations: Proceedings in Honor of J.A. Goldstein's 60th Birthday (2003), Dekker, New York 385-398
27. Shi J., and Shivaji R. Persistence in reaction-diffusion models with weak Allee effect. J. Math. Biol. 52 6 (2006) 807-829
28. J. Shi, Z. Xie, Smoothness of crossing curves, 2007, submitted for publication