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Bifurcation from a homoclinic orbit in parabolic differential equations. (English) Zbl 0636.35010

The author considers the equation \(\dot z+Az=f(z)\) (*) where A is sectorial on a Banach space X and f is C 1 with \(f(0)=0\). He supposes that for \(f=g\) there is a homoclinic orbit p(t) asymptotic to zero. He proves that, under natural hypotheses, \(\{\) f: (*) has a homoclinic orbit near \(p\}\) is a codimension-one submanifold \(\Gamma\) near g and that for f’s on one side of \(\Gamma\) (in the appropriate space) (*) has a unique stable periodic orbit near p(t). This generalizes work of L. P. Sil’nikov [Mat. Sb., Nov. Ser. 61 (103), 443-466 (1963; Zbl 0121.075)] to some infinite dimensional cases.
Reviewer: E.Dancer

MSC:

35B32 Bifurcations in context of PDEs
35B20 Perturbations in context of PDEs
37G99 Local and nonlocal bifurcation theory for dynamical systems
35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations

Citations:

Zbl 0121.075
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References:

[1] DOI: 10.1016/0022-0396(80)90104-7 · Zbl 0439.34035 · doi:10.1016/0022-0396(80)90104-7
[2] Blázquez, Nonlinear Anal. 6 (1986)
[3] Andronov, Theory of Bifurcations in the Plane (1973)
[4] Smoller, Shock Waves and Reaction-Diffusion Equations (1983) · Zbl 0508.35002 · doi:10.1007/978-1-4684-0152-3
[5] Sil’nikov, Mat. Sb. N.S. 74 pp 378– (1967)
[6] Sil’nikov, Dokl. Akad. Nauk SSSR 160 pp 558– (1965)
[7] Conley, Lectures Notes in Physics 38 (1975)
[8] Sil’nikov, Mat. Sb. (N.S.) 61 pp 443– (1963)
[9] Lin, Exponential dichotomies and homoclinic orbits in functional differential equations (1984)
[10] DOI: 10.1007/BF00251249 · Zbl 0507.58031 · doi:10.1007/BF00251249
[11] Henry, Lecture Note in Mathematics 840 (1981)
[12] Friedman, Partial Differential Equations (1969) · Zbl 0224.35002
[13] DOI: 10.1007/978-1-4612-5561-1 · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1
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