Forward and backward continuation for neutral functional differential equations

Basic theory of existence, uniqueness, and continuation for neutral functional differential equation

Attracting Manifolds for Evolutionary Equations

Attracting Manifolds for Evolutionary Equation

Coupled Oscillators on a Circle

We consider a continuum of diffusively coupled oscillators on a circle. When each oscillator is of Lienard type, very little is known about the corresponding hyperbolic POE. When each oscillator is represented by a lossless transmission line, we obtain a partial neutral delay differential equation and give the beginnings of a qualitative theory for the dynamics. In particular, we discuss the properties of the solution map, the existence of the global attractor, behavior near an equilibrium point including the Hopf bifurcation, and some elementary properties near a periodic orbit

Limits of Semigroups Depending on Parameters

nuloIt is reasonable to compare dissipative semigroups with a global attractor by restricting the flows to the attractor. However, if the rate of approach to the attractor is not uniform with respect to parameters, then the transient behavior near the attractor will give more information. We introduce a concept which takes into account this transient behavior. The concept also is useful when the limit system is conservative. We give the general theory with applications to parabolic and hyperbolic PDE on thin domains as well as situations where the limit problem is conservative

Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool

Time delay has been incorporated in models to reflect certain physical or biological meaning. The theory of delay differential equations (DDEs), which has seen extensive growth in the last seventy years or so, can be used to examine the effects of time delay in the dynamical behavior of systems being considered. Numerical tools to study DDEs have played a significant role not only in illustrating theoretical results but also in discovering interesting dynamics of the model. DDE-Biftool, which is a Matlab package for numerical continuation and numerical bifurcation analysis of DDEs, is one of the most utilized and popular numerical tools for DDEs. In this paper, we present a guide to using the latest version of DDE-Biftool targeted to researchers who are new to the study of time delay systems. A short discussion of an example application, which is a harvested predator-prey model with a single discrete time delay, will be presented first. We then implement this example model in DDE-Biftool, pointing out features where beginners need to be cautious. We end with a comparison of our theoretical and numerical results

Polynomial approximation of quasipolynomials based on digital filter design principles

This contribution is aimed at a possible procedure approximating quasipolynomials by polynomials. Quasipolynomials appear in linear time-delay systems description as a natural consequence of the use of the Laplace transform. Due to their infinite root spectra, control system analysis and synthesis based on such quasipolynomial models are usually mathematically heavy. In the light of this fact, there is a natural research endeavor to design a sufficiently accurate yet simple engineeringly acceptable method that approximates them by polynomials preserving basic spectral information. In this paper, such a procedure is presented based on some ideas of discrete-time (digital) filters designing without excessive math. Namely, the particular quasipolynomial is subjected to iterative discretization by means of the bilinear transformation first; consequently, linear and quadratic interpolations are applied to obtain integer powers of the approximating polynomial. Since dominant roots play a decisive role in the spectrum, interpolations are made in their very neighborhood. A simulation example proofs the algorithm efficiency. © Springer International Publishing Switzerland 2016