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Oscillations in a scalar differential equation coupled to a diffusive field

Merlin Pelz, Arnd Scheel

Abstract

We study the emergence of periodic oscillations through a Hopf bifurcation in a scalar diffusion equation on the half line coupled to a dynamic boundary condition. Our results quantify the effect of delay through the buffering in the diffusive field on boundary kinetics, drawing a parallel to the emergence of oscillations in delay equations. Technically, the Hopf bifurcation occurs in the presence of essential spectrum induced by the diffusive field, preventing a simple approach via center-manifold reduction. The results are motivated by observations in biological systems where dynamic boundary conditions arise when modeling surface dynamics coupled to bulk diffusion.

Oscillations in a scalar differential equation coupled to a diffusive field

Abstract

We study the emergence of periodic oscillations through a Hopf bifurcation in a scalar diffusion equation on the half line coupled to a dynamic boundary condition. Our results quantify the effect of delay through the buffering in the diffusive field on boundary kinetics, drawing a parallel to the emergence of oscillations in delay equations. Technically, the Hopf bifurcation occurs in the presence of essential spectrum induced by the diffusive field, preventing a simple approach via center-manifold reduction. The results are motivated by observations in biological systems where dynamic boundary conditions arise when modeling surface dynamics coupled to bulk diffusion.

Paper Structure

This paper contains 28 sections, 7 theorems, 79 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main result --- assumptions.
  3. Main result --- statement.
  4. Dynamic or Wentzell boundary conditions.
  5. Gradient sensing and polarity preservation in planarian regeneration.
  6. Buffering and delay equations.
  7. Bifurcation in the presence of essential spectrum.
  8. Outline.
  9. Proof of Theorem \ref{['theorem:hopf']}
  10. Reduction to boundary-integral equation.
  11. Properties of the linearization.
  12. Lyapunov-Schmidt splitting and reduction.
  13. Reduced equation and proof of Theorem \ref{['theorem:hopf']}.
  14. Leading-order expansions in an example
  15. Stability and instability in an example
  16. ...and 13 more sections

Key Result

Theorem 1.4

Let Assumption assu:equi, assu:degen, and assu:bif be satisfied. Then for sufficiently small amplitudes $|r|<r_0$ and sufficiently small $|\sigma - \sigma_\ast|$, there exist smooth functions $\mu_\mathrm{nl}$ and $\omega_\mathrm{nl}$ of $r$ and $\sigma$, even in $r$, with and, with time-reparameterization $s = \omega_{\mathrm{nl}} t$, periodic functions such that $u(t,x; r, \sigma)=\tilde{u}(\o

Figures (5)

  • Figure 1: The critical curve $\gamma = (\frac{4\sqrt{2}}{9}-\frac{2}{3}) \beta^2$ from \ref{['e:gamcrit']} for $\alpha=1$, separating supercritical (below) and subcritical branching is shown together with some sample values of $\mu_2 = 0$ from Corollary \ref{['cor:m_w_cubicorder']}. Bifurcation diagrams at the sample points are shown in Figures \ref{['fig:hopfcurve_abovecrit_negor0_g']}, \ref{['fig:hopfcurve_g1']}, and \ref{['fig:hopfcurve_g-1']}, below.
  • Figure 2: Bifurcation diagram for \ref{['eq:sys_cubicorder_degrad']} with $\sigma=0$ as discussed in Corollary \ref{['cor:m_w_cubicorder']}, with parameters in the weakly subcritical regime, close to the transition to supercriticality. Top: $\alpha=1$, $\beta=1$, and $\gamma=0$ (purple 4-star in Figure \ref{['fig:betagammacrit']}). Bottom: $\alpha=1$, $\beta=5$, and $\gamma=-0.1$ (blue $+$ in Figure \ref{['fig:betagammacrit']}). Both diagrams exhibit hysteresis with a slightly subcritical branch turning towards supercritical parameter values. Insets show sample plots of solutions and enlarged diagrams near onset. Asymptotics are shown in magenta for comparison. Note the limit on a homoclinic profile with slowly decaying tails far from the bifurcation point.
  • Figure 3: Bifurcation diagram for \ref{['eq:sys_cubicorder_degrad']} with $\sigma=0$ as discussed in Corollary \ref{['cor:m_w_cubicorder']}, with parameters in the subcritical regime. Top: $\alpha=1$, $\beta=0$, and $\gamma=1$ (light blue 5-star in Figure \ref{['fig:betagammacrit']}). Bottom: $\alpha=1$, $\beta=1$, and $\gamma=1$ (dark blue 5-star in Figure \ref{['fig:betagammacrit']}). Insets show sample plots of solutions and enlarged diagrams near onset. Asymptotics are shown in magenta for comparison. Note the limit on a heteroclinic loop profile (top) and a homoclinic profile (bottom), both with long tails far from the bifurcation point.
  • Figure 4: Bifurcation diagram for \ref{['eq:sys_cubicorder_degrad']} with $\sigma=0$ as discussed in Corollary \ref{['cor:m_w_cubicorder']}, with parameters in the subcritical regime. Top: for $\alpha=1$, $\beta=0$, and $\gamma=-1$ (yellow 8-star in Figure \ref{['fig:betagammacrit']}). Bottom: for $\alpha=1$, $\beta=1$, and $\gamma=-1$ (red 8-star in Figure \ref{['fig:betagammacrit']}). Insets show sample plots of solutions and enlarged diagrams near onset. Asymptotics are shown in magenta for comparison. Note that the amplitude of solutions diverges and solutions approach a double-homoclinic loop in both cases, respecting the symmetry $u^-(\cdot)=-u^-(\cdot+\pi)$ of the odd nonlinearity (where quadratic terms are negligible at large amplitudes).
  • Figure 5: Comparison of frequencies with asymptotics (magenta) and homoclinic or heteroclinic loop (both zero frequency) limits shown in plots of $\omega$ versus bifurcation parameter $\mu$. See captions in subfigures for parameters.

Theorems & Definitions (18)

  • Theorem 1.4
  • Remark 1.5: Far-field asymptotics
  • Lemma 2.1
  • proof
  • Lemma 2.2: Fredholm properties of $\mathcal{L}$
  • proof
  • Remark 2.3: Non-resonance
  • Lemma 3.1: Linear Hopf bifurcation
  • proof
  • Theorem 3.2: Expansion of bifurcating branch
  • ...and 8 more