arXiv subjects

John Mallet-Paret

Publications and source records attributed to John Mallet-Paret.

2 recordsLinked to original sources

arXiv2013

Analyticity and Nonanalyticity of Solutions of Delay-Differential Equations

We consider the equation $$ \dot x(t)=f(t,x(t),x(η(t))) $$ with a variable time-shift $η(t)$. Both the nonlinearity $f$ and the shift function $η$ are given, and are assumed to be analytic (that is, holomorphic) functions of their arguments. Typically the time-shift represents a delay, namely that $η(t)=t-r(t)$ with $r(t)\ge 0$. The main problem considered is to determine when solutions (generally $C^\infty$ and often periodic solutions) of the differential equation are analytic functions of $t$; and more precisely, to determine for a given solution at which values of $t$ it is analytic, and at which values it is not analytic. Both sufficient conditions for analyticity, and also for nonanalyticity, at certain values of $t$ are obtained. It is shown that for some equations there exists a solution which is $C^\infty$ everywhere, and is analytic at certain values of $t$ but is not analytic at other values of $t$. Throughout our analysis, the dynamic properties of the map $t\to η(t)$ play a crucial role.

math.CA↗

arXiv2012

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 $\dot x(t)=-α(t)x(t)-β(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $δβ(t)\ge 0$ with $δ\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=δ$ then the $k$-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 $α(t)$ and $β(t)$ are periodic of the same period, and $β(t)$ 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 $u_0$-positivity of the exterior product is investigated when $β(t)$ satisfies a uniform sign condition.

math.DS↗