Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Models in Ecology

We establish an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe the spatial-temporal pattern, the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry. We apply these general results to obtain the coexistence of multiple large-amplitude wave solutions for the delayed Hopfield-Cohen-Grossberg model of neural networks with a symmetric circulant connection matrix.

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Symmetric functional differential equations and neural networks with memory

The paper is concerned with the local and global bifurcation structure of positive solutions ${u,v\in H^1_0(\Omega)}$ of the system

$$\left\{\begin{array}{lll}-\Delta u + u = \mu_1u^3+\beta v^2u &\quad\text{in $\Omega$}\\-\Delta v + v = \mu_2v^3+\beta u^2v &\quad\text{in $\Omega$}\nonumber\end{array}\right.$$

of nonlinear Schrodinger (or Gross-Pitaevskii) type equations in ${\Omega\subset\mathbb {R}^N}$, N ≤ 3. The system arises in nonlinear optics and in the Hartree–Fock theory for a double condensate. Local and global bifurcations in terms of the nonlinear coupling parameter β of the system are investigated by using spectral analysis and by establishing a new Liouville type theorem for nonlinear elliptic systems which provides a-priori bounds of solution branches. If the domain is radial, possibly unbounded, then we also control the nodal structure of a certain weighted difference of the components of the solutions along the bifurcating branches.

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A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system

In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30

Transformation of Groups

Elements of Differential Topology. Degree in Finite--Dimensional Spaces. Leray--Schauder Degree for Compact Fields. Nussbaum--Sadovskii Degree for Condensing Fields. Applications to Bifurcation Theory. S 1 --Equivariant Degree. Global Hopf Bifurcation Theory. Equivariant Degree of Dold--Ulrich. References. Index.

Wieslaw Krawcewicz

Papers

Theory of Degrees with Applications to Bifurcations and Differential Equations

Applied Equivariant Degree

Theory and applications of Hopf bifurcations in symmetric functional differential equations

S1-degree and global Hopf bifurcation theory of functional differential equations

An Equivariant Degree with Applications to Symmetric Bifurcation Problems. Part 1: Construction of the Degree