Blázquez, C. Miguel Bifurcation from a homoclinic orbit in parabolic differential equations. (English) Zbl 0636.35010 Proc. R. Soc. Edinb., Sect. A 103, 265-274 (1986). 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 Cited in 11 Documents 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 Keywords:sectorial operators; homoclinic orbit; periodic orbit; infinite dimensional Citations:Zbl 0121.075 PDFBibTeX XMLCite \textit{C. M. Blázquez}, Proc. R. Soc. Edinb., Sect. A, Math. 103, 265--274 (1986; Zbl 0636.35010) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.