A new class of point-interaction problem characterizing the time evolution of spatially localized spots for reaction-diffusion (RD) systems on the surface of the sphere is introduced and studied. This problem consists of a differential algebraic system (DAE) of ODE’s for the locations of a collection of spots on the sphere, and is derived from an asymptotic analysis in the large diffusivity rat...

We demonstrate the emergence of spontaneous temperature regulation by the combined action of two sets of dissipative structures. Our model system comprised an incompressible, non-isothermal fluid in which two sets of Gray-Scott reaction diffusion systems were embedded. We show that with a temperature dependent rate constant, self-reproducing spot patterns are extremely sensitive to temperature ...

There exists a countable structure M of Scott rank ωCK 1 where ωM 1 = ω CK 1 and where the LωCK 1 ,ω-theory of M is not ω-categorical. The Scott rank of a model is the least ordinal β where the model is prime in its Lωβ,ω-theory. Most well-known models with unbounded atoms below ωCK 1 also realize a non-principal LωCK 1 ,ω-type; such a model that preserves the Σ1-admissibility of ωCK 1 will hav...

We study a reaction-diffusion system with four morphogens which has been suggested in [23]. This system is a generalization of the Gray-Scott model [10, 11] and allows for multiple activators and multiple substrates. We construct single-spike solutions on the real line and establish their stability properties in terms of conditions of connection matrices which describe the interaction of the co...

In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high-order parameterization of the invariant manifolds at the steady states. Using this paramet...

Exponential Runge–Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction stiffly accurate exponential methods, however, relies on a convergence result that requires weakening many order conditions, resulting in schemes whose stages must implemented sequential way. In this work, after showing stronger result, we are able d...

The Scott process of a relational structure M is the sequence of sets of formulas given by the Scott analysis of M . We present axioms for the class of Scott processes of structures in a relational vocabulary τ , and use them to give a proof of an unpublished theorem of Leo Harrington from the 1970’s, showing that a counterexample to Vaught’s Conjecture has models of cofinally many Scott ranks ...