Tensor Products, Positive Linear Operators, and Delay-Differential Equations

Abstract

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 $\dot{x}(t)=-\alpha (t)x(t)-\beta (t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\delta \beta (t)\ge 0$ with $\delta \in -1,1$. Positivity properties are studied, with the result that if $(-1{)}^k=\delta$ 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 $\alpha (t)$ and $\beta (t)$ are periodic of the same period, and $\beta (t)$ 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 $\beta (t)$ satisfies a uniform sign condition.

84 pages

Cited by in corpus (5)

• Coexistence of infinitely many large, stable, rapidly oscillating periodic solutions in time-delayed Duffing oscillators
• Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. III. Parabolic equations and delay systems
• Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez-Schopf delayed oscillator and the irregularity of ENSO
• The unstable set of a periodic orbit for delayed positive feedback
• Variational description of uniform Lyapunov exponents via adapted metrics on exterior products