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Numerical bifurcation analysis for reaction-diffusion equations. (English) Zbl 0952.65105

Springer Series in Computational Mathematics. 28. Berlin: Springer. xiv, 414 p. (2000).
Reaction-diffusion equations \[ \frac{\partial u} {\partial t}=D\Delta u+f(u,\lambda), \tag{RDEqs} \] where \(u=(u_{1},\cdots, u_{k})\); \(\lambda\in \mathbb{R}^p\) is a vector of parameters; \(D\in \mathbb{R}^{k\times k}\) is a symmetric semi-positive definite matrix; \(f:\mathbb{R}^k\times \mathbb{R}^p\rightarrow \mathbb{R}^k\) is a smooth vector-function, arise in chemical and biological systems, population dynamics and nuclear reactor physics. The nonlinearity of this system implies the bifurcational character of its solutions, i.e. their number and stability change with the variation of the control parameters \(\lambda\). The book is devoted to the numerical analysis of bifurcational problems in RDEqs, to a systematic investigation of generic bifurcations and mode interactions by numerical methods.
Chapters 1-4 give a general introduction to RDEqs, summarize the numerical methods for continuation of solutions branches, their parametrizations, detection of bifurcation points, and the branch switching at simple bifurcation points.
Ch. 5 introduces the group-thoretical notions and methods which are useful in the investigation of nonlinear equations under group symmetry conditions.
Ch. 6-8 describe the techniques for reducing of bifurcation problems to equivalent low dimensional systems. These are the Lyapounov-Schmidt method and center manifold theory with their interconnections, the bifurcation function for homoclinic orbits. Here is the introduction in the theory of normal forms. For the third approach its applications to the Kuramoto-Sivashinsky equation are given.
In Chs. 9-13 the generic bifurcations and mode interactions for RDEqs are discussed. Ch. 9 studies systems of 1-dimensional RDEqs, in particular, in the Bruesselator equations. To ensure a correct reflection of the bifurcation scenario in discretizations and to reduce the imperfection of singularities a preservation of multiplicities of the bifurcation points in the discrete problems is considered. The Arnoldi-continuation algorithm is exploited to trace the solution branches and to detect secondary bifurcations.
In Ch. 10 the properties of RDEqs on square domains are considered, specifically symmetries, eigenpairs of the Laplacian, bifurcation points, and occurence of mode interactions. The Lyapounov-Schmidt method is used for obtaining the complete bifurcation scenarios at simple and double bifurcation points.
Ch. 11 is devoted to the calculation of normal forms for generic Hopf bifurcations of RDEqs on a square suitable for reducible and irreducible representations of this symmetry group. An algorithm for finding the number of branches and their symmetries is suggested. As example the normal form analysis is used for the interpretation of the bifurcation diagram for the Brusselator equations with Robin boundary conditions.
The last two chapters of this part are concerned with steady/steady state mode interactions of RDEqs. Symmetries and group theoretic concepts are applied for derivation of normal forms of the reduced bifurcation equations and for the analysis of bifurcation scenarios.
The concluding three chapters deal with the influence of boundary conditions on the bifurcation scenarios. Ch. 15 studies steady state bifurcations for scalar RDEq. The variation of the bifurcation picture along a homotopy from Neumann to Dirichlet boundary conditions is analyzed. The influence of boundary conditions on the changes of symmetries, numbers and directions of bifurcating solutions branches is clearified. Forced symmetry-breaking in the boundary conditions is used to explore how symmetries of a problem are inherited by its solutions manifold. The same questions for steady/steady state bifurcations are studied in Ch. 16.
Literature on bifurcation theory is supplemented by one more excellent book highlighting its numerical aspect. The reviewed book will be very helpful for all specialists applying bifurcation theory mathods in their investigations.

MSC:

65P30 Numerical bifurcation problems
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35K57 Reaction-diffusion equations
65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
58E09 Group-invariant bifurcation theory in infinite-dimensional spaces
37M20 Computational methods for bifurcation problems in dynamical systems
35B32 Bifurcations in context of PDEs
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
92E20 Classical flows, reactions, etc. in chemistry
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
92D25 Population dynamics (general)
82D75 Nuclear reactor theory; neutron transport

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