Neural field equations are integro-differential systems describing the macroscopic activity of spatially extended pieces of cortex. In such cortical assemblies, the propagation of information and the transmission machinery induce communication delays, due to the transport of information (propagation delays) and to the synaptic machinery (constant delays). We investigate the role of these delays...

Rapid and dramatic changes in climate and glacial conditions have taken place during the last 2.5 million years of the earth’s history. Huge ice sheets expanded and contracted periodically, at times covering large areas of North America and Europe. Global sea levels dropped and rose 100 m to 150 m in response to the growth and melting of glaciers, causing continental coast lines to move far int...

My parents, my sister and all my friends who helped me and prayed for me throughout the work on my thesis My God and heavenly father for never leaving me alone, for the love and patience He granted me The Rector of my congregation in Rabenau-Kesselbach and all my brothers and sisters for their prayers The Konrad-Adenauer-Foundation which granted a Ph.D.-scholarship for three years and supported...

We study the global Hopf bifurcation of periodic solutions for one-parameter systems of state-dependent delay differential equations, and specifically we obtain a priori estimates of the periods in terms of certain values of the state-dependent delay along continua of periodic solutions in the Fuller space C(R;RN+1) × R2. We present an example of three-dimensional state-dependent delay differen...

We consider G-equivariant semilinear parabolic equations where G is a finite-dimensional possibly non-compact symmetry group. We treat periodic forcing of relative equilibria and resonant periodic forcing of relative periodic orbits as well as Hopf bifurcation from relative equilibria to relative periodic orbits using LyapunovSchmidt reduction. Our main interest are drift phenomena caused by re...

Normal form theory is very important and useful in simplifying the forms of equations restricted on the center manifolds in studying nonlinear dynamical problems. In this paper, using the center manifold theorem associated with the integrated semigroup theory, we develop a normal form theory for semilinear Cauchy problems in which the linear operator is not densely defined and is not a Hille–Yo...

In this paper, a five-neuron bidirectional associative memory (BAM) neural network with two time delays is studied. Since the study of Hopf bifurcation is very important for the design and application of BAM neural networks, we investigate that Hopf bifurcation occurs and a family of periodic solutions appear when the sum of two delays passes through a critical value.

Manifolds with non-negative sectional curvature have been of interest since the beginning of global Riemannian geometry, as illustrated by the theorems of Bonnet-Myers, Synge, and the sphere theorem. Some of the oldest conjectures in global Riemannian geometry, as for example the Hopf conjecture on S × S, also fit into this subject. For non-negatively curved manifolds, there are a number of obs...

This book is a thin gem having two main parts: Part mDifferential Dynamical Systems and Part 2--Bifurcations. The approach to these subjects is a thorough grounding in dynamics on manifolds in finite-dimensional and Banach spaces, invariant sets and attractors, especially stable, unstable, and center manifolds for maps and semiflows; however, there is no mention of inertial manifolds, which are...

The diffusive Lotka–Volterra predator–prey system with two delays is reconsidered here. The stability of the coexistence equilibrium and associated Hopf bifurcation are investigated by analyzing the characteristic equations, and our results complement earlier ones. We also obtain that in a special case, a Hopf bifurcation of spatial inhomogeneous periodic solutions occurs in the system. © 2011 ...