Bifurcation theory. An introduction with applications of PDEs
This book considerably differs from other monographs on bifurcation theory in the following features: Firstly, all bifurcations are described in a general Banach space setting, which lays the groundwork for the applications to PDEs. Secondly, the research is consequently limited to stationary and periodic solutions, no initial value problems, no homoclinic and heteroclinic solutions, no invariant tori bifurcating from periodic solutions etc. Thirdly, the methods of the proofs are Lyapunov-Schmidt reduction, implicit function theorem, Newton polygon (for local bifurcations), and degree theory (for global bifurcations), no invariant manifold reduction, no normal form techniques. In particular, the periodic solutions are described by working in Banach spaces of periodic functions, no Poincaré map techniques. And finally, the bifurcation parameter is always one-dimensional, no unfoldings and codimensions of the bifurcations. The book is divided into three parts. Part I concerns local bifurcations: saddle node, Hopf, transcritical, pitchfork, constrained Hopf (conservative and reversible systems), saddle node for periodic solutions, period doubling, transcritical and Hopf bifurcations with degenerated transversality condition. In all cases exchange of stability and the principle of reduced stability are verified, and the so-called bifurcation formulae are explicitly calculated. Part II deals with global bifurcations for abstract equations with proper Fredholm maps and odd crossing numbers. In Part III the author shows the power of the abstract theory in applications to PDEs: local bifurcations for fully nonlinear elliptic equations on bounded domains and on lattices; local bifurcations of periodic-Dirichlet problems for semilinear wave equations; Hopf bifurcations for fully nonlinear parabolic equations. Last but not least the author presents his and I. J. Healey's results on preservation of nodal structure, smoothness and uniqueness of global positive solution branches to quasilinear elliptic boundary value problems.
scientific article; zbMATH DE number 2012272