Global Bifurcation of Periodic Solutions in Delay Equations with Symmetric Monotone Feedback

Abstract

We study the periodic solutions of the delay equation $\dot{x}(t)=f(x(t),x(t-1))$, where $f$ scalar is strictly monotone in the delayed component and has even-odd symmetry. We completely describe the global bifurcation structure of periodic solutions via a period map originating from planar ordinary differential equations. Moreover, we prove that the first derivative of the period map determines the local stability of the periodic orbits. This article builds on the pioneering work of Kaplan and Yorke, who found some symmetric periodic solutions for $f$ with even-odd symmetry. We enhance their results by proving that all periodic solutions are symmetric if $f$ is in addition monotone.

Figures (6)

• Figure 1: (Left) Schematic graph of ${p}_f$ for a nonlinearity $f\in\mathfrak{X^+}$, by Theorem \ref{['Theorem1.2']}, the DDE \ref{['eq:1.2']} possesses solely three periodic solutions with amplitudes $a_i$, marked by circles, the number inside each circle indicates the unstable dimension given by Theorem \ref{['Theorem1.3']}. (Right) Schematic graph of ${p}_f$ for a nonlinearity $f\in\mathfrak{X^-}$, by Theorem \ref{['Theorem1.2']}, the DDE \ref{['eq:1.2']} possesses a total of five periodic solutions, marked by circles, the number inside each circle indicates the unstable dimension given by Theorem \ref{['Theorem1.3']}.
• Figure 2: First branches of periodic orbits in the DDE \ref{['eq:1.15b']} with $f\in\mathfrak{X}^+$. The unstable dimension is indicated in circles. The dashed curves indicate the higher unstable dimension out of the two possible values on each branch. The changes from dashed to solid correspond to saddle-node bifurcations of periodic orbits and the points of intersection with the horizontal axis are Hopf bifurcations.
• Figure 3: (Left) A supercritical Hopf bifurcation in the DDE \ref{['eq:1.15']} where $f\in\mathfrak{X}^+$ and ${p}_f"(0)>0$. At parameter value ${r}=1$, a periodic orbit $\gamma^\ast$ is emitted by the origin and increases its amplitude as ${r}$ grows. The unstable dimension of $\gamma^\ast$ is $2n-1$. (Right) A saddle-node bifurcation of periodic orbits in the DDE \ref{['eq:1.15']} where $f\in\mathfrak{X}^+$ and ${p}_f"(a^\ast)>0$. At ${r}=1$, the family $\gamma_1$ of periodic orbits with unstable dimension $2n$ collides with the family $\gamma_2$ with unstable dimension $2n-1$ forming a nonhyperbolic periodic solution $\gamma^\ast$ which has unstable dimension $2n-1$. For ${r}<1$, no periodic solutions exist near the amplitude $a^\ast$ of $\gamma^\ast$.
• Figure 4: (Left) Schematic picture for a sinusoidal periodic solution of the DDE \ref{['eq:1.2']}. (Right) Nested planar projections of two different periodic solutions.
• Figure 5: (Left) Branches of periodic solutions $\mathcal{B}^+_n$ of the DDE \ref{['eq:3.15']} (solid). At parameter value $r = 1 + m^\ast p$, there are periodic solutions on the branch $\mathcal{B}^+_{n^\ast}$ whose projection intersects $P\gamma^\ast$ in the $(x(t), x(t - 1))$-plane. (Right) Planar projection $P\gamma^\ast$ of the orbits $\gamma(x_0^{(m)})$ for all $m\in\mathbb{Z}$ (solid) and orbits of the ODEs \ref{['eq:1.12']} (dashed). All the periodic orbits of \ref{['eq:1.12']} possessing amplitudes within the gray region intersect the planar projection $P\gamma^\ast$.

Theorems & Definitions (24)

• Remark 1.1
• Lemma 1.2: period map
• Theorem 1.3: characterization of periodic solutions
• Remark 1.4
• Theorem 1.5: local stability of periodic solutions
• Remark 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof