Practicalities of State-Dependent and Threshold Delay Differential Equations

Abstract

Delays are ubiquitous in applied problems, but often do not arise as the simple constant discrete delays that analysts and numerical analysts like to treat. In this chapter we show how state-dependent delays arise naturally when modeling and the consequences that follow. We treat discrete state-dependent delays, and delays implicitly defined by threshold conditions. We will consider modeling, formulation as dynamical systems, linearization, and numerical techniques. For discrete state-dependent delays we show how breaking points can be tracked efficiently to preserve the order of numerical methods for simulating solutions. For threshold conditions we will discuss how a velocity ratio term arises in models, and present a heuristic linearization method that avoids Banach spaces and sun-star calculus, making the method accessible to a wider audience. We will also discuss numerical implementations of threshold and distributed delay problems which allows them to be treated numerically with standard software.

Figures (21)

• Figure 1: Illustration of \ref{['eq:rfdeic']}, the initial condition for an RFDE, and a function segment $u_t$ for $t=t_0+\tau/2$ illustrating that $u_t$ is not necessarily continuously differentiable, even when $\varphi$ is smooth.
• Figure 2: Schematic representation of the classical $G_{0}$ model for HSCs. The proliferating phase of the cell cycle is divided between 4 subphases: gap one $G_{1}$, synthesis $S$, gap two $G_{2}$, and mitosis $M$. Cells in the resting phase (gap zero $G_{0}$), may differentiate with rate $\kappa$ or entry the cell cycle with rate $\beta(Q)$. Cells in the proliferation phase may be lost by apoptosis with rate $\gamma$, otherwise they re-enter the resting phase after mitosis, $\tau$ time units after they left the resting phase. (© Society for Industrial and Applied Mathematics (SIAM); Reproduced from SIADS19 with permission.)
• Figure 3: Schematic of WF45WF49 electrodynamics. A proton $p(t)$ and electron $e(t)$ only feel forces (propagating at the speed of light $c$) from the other charge in the past and future at the points where the world lines and light cones intersect. Thus delays and advances are implicitly defined by $|p(t)-e(t\pm\tau_e^\pm)|=c\tau_e^\pm$ and $|e(t)-p(t\pm\tau_p^\pm)|=c\tau_p^\pm.$
• Figure 4: Solutions of the IVP for \ref{['eq:twostatedepmod']} with constant initial functions $\varphi(t)=u_0$ for several different values of $u_0$ with other parameters $1=a_1=\gamma<\kappa_2=\kappa_1=a_2=2$, $c_1=0.5$, $c_2=0.4$. The dashed line at $u=-2$ indicates where $u(t)=-a_1/c_1$ and $\tau_1=0$. (© American Institute of Mathematical Sciences (AIMS); Reproduced from HDMM12 with permission.)
• Figure 5: Solutions of \ref{['eq:twostatedep']} projected onto $(u(t),u(t-a_1),u(t-a_2))$-space. Parameter values are $\gamma=4.75>\kappa_2=3.0$, $a_1=1.3$, $a_2=6.0$ and $c_1 = c_2 =1.0$. (a) With $\kappa_1=4.44$ a quasi-periodic torus is observed. (b) For $\kappa_1=6.93$, a 1:4 phase-locked torus-like object (grey), with embedded stable (blue) and saddle (red) periodic orbits. (© SIAM; Reproduced from CHK17 with permission.)

Theorems & Definitions (8)

• Theorem 1: Existence, Boundedness and Uniqueness HDMM12
• Theorem 2: Extended Existence Theorem Driver63
• Theorem 3: Uniqueness Theorem Driver63
• Definition 4: BZ03
• Theorem 5
• proof
• Example 6: FN84
• Example 7