361 research outputs found
Tensor Products, Positive Linear Operators, and Delay-Differential Equations
We develop the theory of compound functional differential equations, which
are tensor and exterior products of linear functional differential equations.
Of particular interest is the equation with a single delay, where the delay
coefficient is of one sign, say with .
Positivity properties are studied, with the result that if then
the -fold exterior product of the above system generates a linear process
which is positive with respect to a certain cone in the phase space.
Additionally, if the coefficients and are periodic of
the same period, and satisfies a uniform sign condition, then there
is an infinite set of Floquet multipliers which are complete with respect to an
associated lap number. Finally, the concept of -positivity of the exterior
product is investigated when satisfies a uniform sign condition.Comment: 84 page
A condition on delay for differential equations with discrete state-dependent delay
Parabolic differential equations with discrete state-dependent delay are
studied. The approach, based on an additional condition on the delay function
introduced in [A.V. Rezounenko, Differential equations with discrete
state-dependent delay: uniqueness and well-posedness in the space of continuous
functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (11)
(2009), 3978-3986] is developed. We propose and study a state-dependent
analogue of the condition which is sufficient for the well-posedness of the
corresponding initial value problem on the whole space of continuous functions
. The dynamical system is constructed in and the existence of a compact
global attractor is proved
Dichotomy results for delay differential equations with negative Schwarzian
We gain further insight into the use of the Schwarzian derivative to obtain
new results for a family of functional differential equations including the
famous Wright's equation and the Mackey-Glass type delay differential
equations. We present some dichotomy results, which allow us to get easily
computable bounds of the global attractor. We also discuss related conjectures,
and formulate new open problems.Comment: 16 pages, submitted to Chaos,Solitons,Fractal
Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation
AbstractPoincarĂ© observed that for a differential equation xâČ = Æ(x, α) depending on a parameter α, each periodic orbit generally lies in a connected family of orbits in (x, α)-space. In order to investigate certain large connected sets (denoted Q) of orbits containing a given orbit, we introduce two indices: an orbit index Ï and a âcenterâ index defined at certain stationary points. We show that genetically there are two types of Hopf bifurcation, those we call âsourcesâ ( = 1) and âsinksâ ( = â1). Generically if the set Q is bounded in (x, α)-space, and if there is an upper bound for periods of the orbits in Q, then Q must have as many source Hopf bifurcations as sink Hopf bifurcations and each source is connected to a sink by an oriented one-parameter âsnakeâ of orbits. A âsnakeâ is a maximal path of orbits that contains no orbits whose orbit index is 0. See Fig. 1.1
Travelling waves for a non-monotone bistable equation with delay: existence and oscillations
We consider a bistable (0\textless{}\theta\textless{}1 being the three
constant steady states) delayed reaction diffusion equation, which serves as a
model in population dynamics. The problem does not admit any comparison
principle. This prevents the use of classical technics and, as a consequence,
it is far from obvious to understand the behaviour of a possible travelling
wave in . Combining refined {\it a priori} estimates and a Leray
Schauder topological degree argument, we construct a travelling wave connecting
0 in to \lq\lq something" which is strictly above the unstable
equilibrium in . Furthemore, we present situations
(additional bound on the nonlinearity or small delay) where the wave converges
to 1 in , whereas the wave is shown to oscillate around 1 in
when, typically, the delay is large
The numerical solution of forwardâbackward differential equations: Decomposition and related issues
NOTICE: this is the authorâs version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 234,(2010), doi: 10.1016/j.cam.2010.01.039This journal article discusses the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of âforwardâ solutions and âbackwardâ solutions
Traveling Wave Solutions for Systems of ODEs on a Two-Dimensional Spatial Lattice
This is the published version, also available here: http://dx.doi.org/10.1137/S0036139996312703.We consider infinite systems of ODEs on the two-dimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction , and we explore the relation between the wave speed c, the angle , and the detuning parameter a of the nonlinearity. Of particular interest is the phenomenon of "propagation failure," and we study how the critical value depends on , where is defined as the value of the parameter a at which propagation failure (that is, wave speed c=0) occurs. We show that \theta\tan\theta\tan\theta$ is rational or infinite
On the geometry of wave solutions of a delayed reaction-diffusion equation
The aim of this paper is to study the existence and the geometry of positive
bounded wave solutions to a non-local delayed reaction-diffusion equation of
the monostable type.Comment: 25 pages, several important modifications are made. Some references
added to the previous versio
- âŠ