Dynamics of a state-dependent delay-differential equation

TL;DR

This work advances the theory of scalar state-dependent delay differential equations with threshold delays by providing a rigorous semiflow formulation, positivity/global attractor results, and a comprehensive bifurcation analysis across all combinations of Hill-function nonlinearities for delay and feedback. By treating the threshold delay directly (without time-transforming to a constant-delay distributed DDE), the authors reveal a far richer dynamical tapestry than in prior constant-delay models, including multiple Hopf, fold, cusp, Bogdanov–Takens, and Bautin bifurcations, as well as homoclinic phenomena and sausage-type oscillations. The study systematically compares single- and two-Hill-function scenarios, identifies limiting switch-like behaviors, and documents intricate two-parameter bifurcation structures, including BT and fold-Hopf points, across various threshold configurations. The results illuminate how state-dependence and thresholding fundamentally alter stability and oscillatory dynamics in gene-regulatory-inspired DDE models, with implications for understanding multi-stability, periodic forcing, and the design of synthetic networks.

Abstract

We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov-Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable.

Figures (34)

• Figure 5.1: Steady states of \ref{['eq:basic']}, given by \ref{['eq:h']}, occur at the intersections of $\xi\mapsto\beta e^{-\mu\tau}g(\xi)$ and $\xi\mapsto\gamma\xi$. These are illustrated for various $\gamma$ in the limiting case of \ref{['eq:gpwconst']} and \ref{['eq:vpwconst']} with $v^\pm= v^-=v^+$, so the delay $\tau=a/v^\pm$ is constant, and $g^->g^+$, so $g$ is monotonically decreasing: $(g \downarrow, v \leftrightarrow)$.
• Figure 5.2: Bifurcations of \ref{['eq:basic']}-\ref{['eq:thres']} for $(g\downarrow,v\leftrightarrow)$ with parameters $\beta=1.4$, $\mu=0.2$, $g^-=1$, $g^+=1/2$, $\theta_g=1$, $\gamma=a=1$, and $v=v^-=v^+=2$. (a) The limiting case with $g$ defined by \ref{['eq:gpwconst']} showing the stable (green solid line) and singular (black dashed line) steady states. (b) With smooth nonlinearity $g$ defined by \ref{['eq:vghill']} and $n=50$. Solid lines denote stable objects including the stable steady state (green) and a stable limit cycle (represented by maximum and minimum of $x(t)$ on the periodic solution). Dashed lines represent unstable steady states which have two eigenvalues with positive real part (in black). (c) As in (b) but with $n=23$. (d) Two-parameter continuations in $n$ and $\gamma$ of the Hopf bifurcations defined by \ref{['eq:gam']}-\ref{['eq:gdash']} with the other parameters as above. Solid curves indicate the parts of the branch where there are no characteristic values with positive real part (and hence a stability change at the bifurcation), and dashed lines indicate the parts of the branch where there are already unstable characteristic values. The outermost curve of Hopf bifurcations is associated with the stability change seen in (b). The dash-dotted vertical black lines denote $\gamma=\gamma_1$ and $\gamma=\gamma_2$, the locations of the Hopf bifurcations in the limiting case as $n\to\infty$. (e) Profiles of the stable periodic orbits from the outermost curve of Hopf bifurcations in (d) at $\gamma=1$ for different values of the continuation parameter $n$. (f) The same periodic orbits as in (e), but now shown as a projection onto the plane $(x(t), x(t-\tau))$ where $\tau=0.5$. The arrow indicates the direction of the flow. The square denotes the singular steady state in the limiting case.
• Figure 5.3: Illustration of how the number of steady states of \ref{['eq:basic']} given by \ref{['eq:h']} changes with the intersections of $\xi\mapsto\beta e^{-\mu\tau}g(\xi)$ and $\xi\mapsto\gamma\xi$. These are shown in the limiting case with $v^\pm =v^-=v^+$ so $\tau=a/v^\pm$ is constant, and $g^-<g^+$ in \ref{['eq:gpwconst']} so $g$ is piecewise constant and monotonically increasing: $(g\uparrow,v\leftrightarrow)$.
• Figure 5.4: Bifurcations of \ref{['eq:basic']}-\ref{['eq:thres']} with $(g \uparrow, v \leftrightarrow)$ and parameters $\beta=2$, $\mu=0.02$, $g^-=0.1$, $g^+=1$, $\theta_g=1$, $\gamma=1$, $a=2$ and $v=v^-=v^+=2$. (a) The limiting case with $g$ defined by \ref{['eq:gpwconst']}. Stable steady states are shown as green solid lines, and the singular steady state as a black dashed line. (b) With a smooth nonlinearity $g$ defined by \ref{['eq:vghill']} with $n=30$. (c) Two-parameter continuations in $n$ and $\gamma$ of the fold (blue) and the Hopf (black) bifurcations with the other parameters as above. The dashed vertical lines denote $\gamma=\gamma_1$ and $\gamma=\gamma_2$, the location of the fold bifurcations in the limiting case as $n\to\infty$. The red dash-dotted curve denotes the bound on the fold bifurcations given by \ref{['eq:foldbound']}.
• Figure 5.5: Steady states of \ref{['eq:basic']} are given by \ref{['eq:h']}, and hence occur at the intersections of $\xi\mapsto\beta e^{-\mu\tau(\xi)}g$ and $\xi\mapsto\gamma\xi$. These are illustrated for various $\gamma>0$ in the limiting case of \ref{['eq:gpwconst']} and \ref{['eq:vpwconst']} with $g^\pm= g^-=g^+$, so $g(\xi)=g^\pm$ is a constant function, and $v^->v^+$, so $v$ is piecewise constant and monotonically decreasing : $(g\leftrightarrow,v\downarrow)$. Then $\tau(\xi)=a/v(\xi)$ is state-dependent; $\tau(\xi)=\tau^-=a/v^-$ for $\xi<\theta_v$, $\tau(\xi)=\tau^+=a/v^+$ for $\xi>\theta_v$ and $\tau(\xi)$ is set-valued when $\xi=\theta_v$.

Theorems & Definitions (31)

• Proposition 2.1
• Proof 1: Proof of Proposition \ref{['prop:1']}
• Proposition 2.2
• Proposition 3.1
• Proof 2
• Proposition 3.2
• Proof 3
• Corollary 3.3
• Proof 4
• Theorem 3.4