Erratum to: Large-Amplitude Periodic Solutions for Differential Equations with Delayed Monotone Positive Feedback

1 Erratum to: J Dyn Diff Equat (2011) 23:727–790 DOI 10.1007/s10884-011-9225-2

The purpose of this note is to correct a mistake left in our previous paper [2]. The paper concerns the scalar equation

$$\begin{aligned} \dot{x}\left( t\right) =-\mu x\left( t\right) +f\left( x\left( t-1\right) \right) \end{aligned}$$

(1)

with $\mu =1$ and a special strictly increasing, continuously differentiable $f$. The natural phase space for Eq. (1) is $C=C\left( \left[ -1,0\right] ,\mathbb {R}\right)$ equipped with the supremum norm. For any $\varphi \in C$, there is a unique solution $x^{\varphi }:\left[ -1,\infty \right) \rightarrow \mathbb {R}$ of (1). For each $t\ge 0$, the segment $x_{t}^{\varphi }\in C$ is defined by $x_{t}^{\varphi }\left( s\right) =x^{\varphi }\left( t+s\right)$, $-1\le s\le 0$. Let $\Phi$ denote the semiflow induced by Eq. (1):

$$\begin{aligned} \Phi :\left[ -1,\infty \right) \times C\ni \left( t,\varphi \right) \mapsto x_{t}^{\varphi }\in C. \end{aligned}$$

Theorem 1.1 of paper [2] gives a periodic solution $p:\mathbb {R}\rightarrow \mathbb {R}$ of Eq. (1) with $p\left( -1\right) =0$ and $\dot{p}\left( -1\right) \ne 0$. The proof of Theorem 1.2 in Section 8 then applies a Poincaré return map defined on a neighborhood of $p_{0}$ in $H$, where $H=\left\{ \varphi :\,\varphi \left( -1\right) =0\right\}$ is a hyperplane transversal to the periodic orbit $\mathcal {O}_{p}=\left\{ p_{t}:\, t\in \mathbb {R}\right\}$. As we shall see, this hyperplane was not selected appropriately.

We evoke results from Floquet theory before pointing at the error and showing its correction.

2 Floquet Theory

Let $\omega \in \left( 1,2\right)$ denote the minimal period of $p$. Consider the period map $Q=\Phi \left( \omega ,\cdot \right)$ with fixed point $p_{0}$. Consider its derivative $M=D_{2}\Phi \left( \omega ,p_{0}\right)$ at $p_{0}$. Then $M\varphi =u_{\omega }^{\varphi }$ for all $\varphi \in C$, where $u^{\varphi }:\left[ -1,\infty \right) \rightarrow \mathbb {R}$ is the solution of the variational equation

$$\begin{aligned} \dot{u}(t)&=-u(t)+f'\left( p\left( t-1\right) \right) u\left( t-1\right) \end{aligned}$$

(2)

with $u_{0}^{\varphi }=\varphi$. $M$ is called the monodromy operator. $M$ is a compact operator, $0$ belongs to its spectrum $\sigma =\sigma \left( M\right)$, and eigenvalues of finite multiplicity—the so called Floquet multipliers—form $\sigma \left( M\right) \backslash \left\{ 0\right\}$.

As $\dot{p}$ is a nonzero solution of the variational equation, $1$ is a Floquet multiplier with eigenfunction $\dot{p}_{0}$. The paper [2] proves that $\mathcal {O}_{p}$ is hyperbolic, which means that the generalized eigenspace of $M$ corresponding to the eigenvalue $1$ is one-dimensional, furthermore there are no Floquet multipliers on the unit circle besides $1$.

If $\varphi$ is a nonzero element of the phase space $C=C\left( \left[ -1,0\right] ,\mathbb {R}\right)$, let $V\left( \varphi \right)$ denote the number of sign changes of $\varphi$ if it is even or $\infty$, otherwise let $V\left( \varphi \right)$ be the number of sign changes plus one. This is the so-called discrete Lyapunov functional of Mallet-Paret and Sell [4].

By Section 4 of [2], $\mathcal {O}_{p}$ has two real and simple Floquet multipliers $\lambda _{1}$ and $\lambda _{2}$ outside the unit circle with $\lambda _{1}>\lambda _{2}>1$. Regarding the associated eigenspaces, we have the following information from [4] and from Appendix VII of [3]. The eigenvector $u_{1}$ of $M$ corresponding to $\lambda _{1}$ is strictly positive. The realified generalized eigenspace $C_{<\lambda _{1}}$ associated with the spectral set $\left\{ z\in \sigma :\,\left| z\right| <\lambda _{1}\right\}$ satisfies

$$\begin{aligned} C_{<\lambda _{1}}\cap V^{-1}\left( 0\right) =\emptyset . \end{aligned}$$

(3)

Let $C_{\le \rho },\; \rho >0$, denote the realified generalized eigenspace of $M$ associated with the spectral set $\left\{ z\in \sigma :\,\left| z\right| \le \rho \right\}$. The set

$$\begin{aligned} \left\{ \rho \in \left( 0,\infty \right) :\,\sigma \left( M\right) \cap \rho S_{\mathbb {C}}^{1}\ne \emptyset ,\, C_{\le \rho }\cap V^{-1}\left( \left\{ 0,2\right\} \right) =\emptyset \right\} \end{aligned}$$

is nonempty and has a maximum $r_{M}$. Then

$$\begin{aligned} C_{\le r_{M}}\cap V^{-1}\left( \left\{ 0,2\right\} \right) =\emptyset ,\quad C_{r_{M}<}\setminus \left\{ \hat{0}\right\} \subset V^{-1}\left( \left\{ 0,2\right\} \right) \hbox {and dim}C_{r_{M}<}\le 3, \end{aligned}$$

(4)

where $C_{r_{M}<}$ is the realified generalized eigenspace of $M$ associated with the nonempty spectral set $\left\{ z\in \sigma :\,\left| z\right| >r_{M}\right\}$. It follows from the construction of $p$ in [2] that $V\left( \dot{p}_{0}\right) =2$. Hence $r_{M}<1$ in our case, and $V\left( u_{2}\right) =2$ for the eigenvector $u_{2}$ of $M$ corresponding to $\lambda _{2}$.

3 Poincaré Return Maps

Choose $X$ to be a hyperplane with codimension $1$ so that $\dot{p}_{0}\notin X$. An application of the implicit function theorem yields a convex bounded open neighborhood $N$ of $p_{0}$ in $C$, $\varepsilon \in \left( 0,\omega \right)$ and a $C^{1}$-map $\gamma :N\rightarrow \left( \omega -\varepsilon ,\omega +\varepsilon \right)$ with $\gamma \left( p_{0}\right) =\omega$ so that for each $\left( t,\varphi \right) \in \left( \omega -\varepsilon ,\omega +\varepsilon \right) \times N$, the segment $x_{t}^{\varphi }$ belongs to $p_{0}+X$ if and only if $t=\gamma (\varphi )$ (see [1], Appendix I in [3]). The Poincaré return map $P_{X}$ is defined by

$$\begin{aligned} P_{X}:N\cap \left( p_{0}+X\right) \ni \varphi \mapsto \Phi \left( \gamma (\varphi ),\varphi \right) \in p_{0}+X. \end{aligned}$$

Then $P_{X}$ is continuously differentiable with fixed point $p_{0}$.

Let $\sigma \left( P_{X}\right)$ and $\sigma \left( M\right)$ denote the spectra of $DP_{X}\left( p_{0}\right) :X\rightarrow X$ and the monodromy operator $M$, respectively. We obtain the following result from Theorem XIV.4.5 in [1].

Lemma

• (i) $\sigma \left( P_{X}\right) \setminus \left\{ 0,1\right\} =\sigma \left( M\right) \setminus \left\{ 0,1\right\}$, and for every $\lambda \in \sigma \left( M\right) \setminus \left\{ 0,1\right\}$, the projection along $\mathbb {R}\dot{p}_{0}$ onto $X$ defines an isomorphism from the realified generalized eigenspace of $\lambda$ and $M$ onto the realified generalized eigenspace of $\lambda$ and $DP_{X}\left( p_{0}\right)$.

• (ii) $1\notin \sigma \left( P_{X}\right)$.

In Section 8 of [2] we selected the hyperplane $H=\left\{ \varphi :\,\varphi \left( -1\right) =0\right\}$ and the associated Poincaré map $P=P_{H}$. It follows from the above proposition that $DP\left( p_{0}\right)$ has exactly two real eigenvalues $\lambda _{1}>\lambda _{2}>1$ outside the unit circle. Let $v_{1}$ and $v_{2}$ denote the eigenvectors of $DP\left( p_{0}\right)$ corresponding to $\lambda _{1}$ and $\lambda _{2}$, respectively. Section 8 of [2] used the statement that $V\left( v_{1}\right) =0$ and $V\left( v_{2}\right) =2$. This is not necessarily true. The mistake can be corrected by selecting a different hyperplane.

Let $C_{s}$ and $C_{u}$ be the closed subspaces of $C$ chosen so that $C=C_{s}\oplus \mathbb {R}\dot{p}_{0}\oplus C_{u},\, C_{s}$ and $C_{u}$ are invariant under $M$, and the spectra $\sigma _{s}\left( M\right)$ and $\sigma _{u}\left( M\right)$ of the induced maps $C_{s}\ni x\mapsto Mx\in C_{s}$ and $C_{u}\ni x\mapsto Mx\in C_{u}$ are contained in $\left\{ \mu \in \mathbb {C}:\,\left| \mu \right| <1\right\}$ and $\left\{ \mu \in \mathbb {C}:\,\left| \mu \right| >1\right\}$, respectively. As $\mathcal {O}_{p}$ has two real and simple Floquet multipliers $\lambda _{1}$ and $\lambda _{2}$ outside the unit circle with eigenvectors $u_{1}$ and $u_{2}$, we have $C_{u}=\left\{ c_{1}u_{1}+c_{2}u_{2}\right\}$.

Set $Y=C_{s}\oplus C_{u}$. Then $Y$ is a hyperplane in $C$, $\dot{p}_{0}\notin Y$ and $C=Y\oplus \mathbb {R}\dot{p}_{0}$.

The special choice of $Y$ and Lemma imply that $\lambda _{i}$ and $u_{i}$ is an eigenvalue-eigenvector pair of $DP_{Y}\left( p_{0}\right)$ for both $i\in \left\{ 1,2\right\}$. In addition, $C_{s}$ and $C_{u}$ are invariant under $DP_{Y}\left( p_{0}\right)$, and the spectra $\sigma _{s}\left( P_{Y}\right)$ and $\sigma _{u}\left( P_{Y}\right)$ of the induced maps $C_{s}\ni x\mapsto DP_{Y}\left( p_{0}\right) x\in C_{s}$ and $C_{u}\ni x\mapsto DP_{Y}\left( p_{0}\right) x\in C_{u}$ are contained in $\left\{ \mu \in \mathbb {C}:\,\left| \mu \right| <1\right\}$ and $\left\{ \mu \in \mathbb {C}:\,\left| \mu \right| >1\right\}$, respectively. Summing up, $DP_{Y}\left( p_{0}\right)$ has exactly two real and simple eigenvalues $\lambda _{1}>\lambda _{2}>1$ outside the unit circle, and for the corresponding eigenvectors $u_{1}$ and $u_{2}$, we have the desired properties $V\left( u_{1}\right) =0$ and $V\left( u_{2}\right) =2$.

In accordance, $H$ and $P=P_{H}$ should be changed to $Y$ and $P_{Y}$ in Section 8 of [2]. Then the proof of Theorem 1.2. (found in Section 8 of [2]) becomes correct without any further change.