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Applied equivariant degree. (English) Zbl 1123.47043

AIMS Series on Differential Equations and Dynamical Systems 1. Springfield, MO: American Institute of Mathematical Sciences (ISBN 978-1-60133-001-7/hbk). xxiii, 552 p. (2006).
This book is devoted to the theory and applications of the so-called equivariant degree, a more sophisticated tool than the classical degree, to study the existence, multiplicity and bifurcations of solutions to nonlinear equations with symmetries. Equations of this type, such as systems of ODEs, FODEs, PDEs and FPDEs, arise from the mathematical modeling of natural phenomena, in physics, chemistry, biology and engineering, due to geometrical and physical regularities: symmetric domains, mechanical symmetries, phase transition in crystals, breaking spherical symmetry of atoms, etc. Historically, the equivariant degree was defined by J. Ize, I. Massabó and A. Vignoli [Trans. Am. Math. Soc. 315, No. 2, 433–510 (1989; Zbl 0695.58006)] and rigorously studied by [J. Ize and A. Vignoli, “Equivariant degree theory” (de Gruyter Series in Nonlinear Analysis and Applications 8) (2003; Zbl 1033.47001)] for abelian groups.
The aim of this monograph is to present the state of art in equivalent degree theory including the non-abelian case, and to provide practical routines for the computation of the degree for several classes of problems.
The book is organized as follows: Introduction (chapter 1); Part I: Equivariant degree theory: technical tools (chapters 2–8, theoretical part); Part II: Applications of equivariant degree to differential equations and bifurcation problems (ODEs with symmetries, symmetric Hopf bifurcation for FODEs, symmetric bifurcation for parabolic systems, symmetric Van Der Pol equations, variational problems with symmetries); Part III: Appendices, References, and Index.
Undoubtedly, the book is a valuable contribution to the theory and applications of the topological methods of nonlinear analysis and will mainly interest researchers and graduate students dealing with nonlinear problems with symmetries.

MSC:

47Jxx Equations and inequalities involving nonlinear operators
47H11 Degree theory for nonlinear operators
47J25 Iterative procedures involving nonlinear operators
37C80 Symmetries, equivariant dynamical systems (MSC2010)
34B15 Nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34K10 Boundary value problems for functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34G20 Nonlinear differential equations in abstract spaces
35A15 Variational methods applied to PDEs
35J20 Variational methods for second-order elliptic equations
35K10 Second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K55 Nonlinear parabolic equations
47H10 Fixed-point theorems
47J05 Equations involving nonlinear operators (general)
47J30 Variational methods involving nonlinear operators
54H20 Topological dynamics (MSC2010)
54H25 Fixed-point and coincidence theorems (topological aspects)
55M20 Fixed points and coincidences in algebraic topology
55M25 Degree, winding number
55M35 Finite groups of transformations in algebraic topology (including Smith theory)
55Q10 Stable homotopy groups
55Q91 Equivariant homotopy groups
57S17 Finite transformation groups
57S25 Groups acting on specific manifolds
58C30 Fixed-point theorems on manifolds
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
58E09 Group-invariant bifurcation theory in infinite-dimensional spaces
58E40 Variational aspects of group actions in infinite-dimensional spaces
92D25 Population dynamics (general)
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
34A34 Nonlinear ordinary differential equations and systems
34C14 Symmetries, invariants of ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
17A60 Structure theory for nonassociative algebras
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
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