Interior Transition Layers Research Articles

Interior transition layers in flight-path optimization

Interior transition layers occur in flight path optimization by singular perturbation methods whenever the reduced solution exhibits multiple branches. Although such phenomena occur in many problems, it has been little studied in the literature. In this paper, we consider interior transition layers in vertical-plan e climb path optimization. Our approach is to treat the interior layer associated with the transonic energy state discontinuity as two boundary layers, one in forward time and the other in backward time. The initial states of the two boundary layers are matched to give continuous composite solutions at the point of reduced solution discontinuity.

Following homology in singularly perturbed systems

The fast systems associated to a system of first-order equations where a small parameter multiplies the derivatives (hereinafter, an SPMD system) are obtained by stretching time by a factor of one over the small parameter and then reducing the small parameter to zero. Curves of isolated invariant sets associated to the fast system serve as approximate solutions to the SPMD system over finite intervals in the following sense: A manifold of initial points representing non-zero homology α in the Morse-Conley index of the invariant set at the left endpoint has a submanifold which is carried by the flow of the SPMD system to a manifold representing that non-zero homology class in the index of the invariant set at the right endpint which is the continuation of α along the curve of invariants sets. This is a type of linearization of the approximation problem since the continuation of homology classes along the curve of invariant sets is given by a linear transformation which varies functorially with the curve of invariant sets. Examples are given for systems in R 2 and R 3. The results are used elsewhere to prove existence theorems for two point boundary value problems to obtain solutions with boundary and interior transition layers.

Global Structure of Bifurcating Solutions of Some Reaction-Diffusion Systems

We study the global structure of the set of bifurcating solutions of a class of coupled nonlinear reaction-diffusion systems. Our main result is that when one of the diffusion coefficients is sufficiently large, the bifurcating branch emanating from a uniform state continues to exist until it is connected to the singularly perturbed solutions which contain interior transition layers (Theorem 6.1). We also present a global branching theorem for the bifurcating branch which shows that in the general situation it does not fall entirely on the trivial branch (Theorem 2.2).

Boundary and interior transition layer phenomena for pairs of second-order differential equations

Boundary and interior transition layer phenomena for pairs of second-order differential equations