Stability of Synchronous Slowly Oscillating Periodic Solutions for Systems of Delay Differential Equations with Coupled Nonlinearity

Abstract

We study stability of so-called synchronous slowly oscillating periodic solutions (SOPSs) for a system of identical delay differential equations (DDEs) with linear decay and nonlinear delayed negative feedback that are coupled through their nonlinear term. Under a row sum condition on the coupling matrix, existence of a unique SOPS for the corresponding scalar DDE implies existence of a unique synchronous SOPS for the coupled DDEs. However, stability of the SOPS for the scalar DDE does not generally imply stability of the synchronous SOPS for the coupled DDEs. We obtain an explicit formula, depending only on the spectrum of the coupling matrix, the strength of the linear decay and the values of the nonlinear negative feedback function near plus/minus infinity, that determines the stability of the synchronous SOPS in the asymptotic regime where the nonlinear term is heavily weighted. We also treat the special cases of so-called weakly coupled systems, near uniformly coupled systems, and doubly nonnegative coupled systems, in the aforementioned asymptotic regime. Our approach is to estimate the characteristic (Floquet) multipliers for the synchronous SOPS. We first reduce the analysis of the multidimensional variational equation to the analysis of a family of scalar variational-type equations, and then establish limits for an associated family of monodromy-type operators. We illustrate our results with examples of systems of DDEs with mean-field coupling and systems of DDEs arranged in a ring.

Appendix A: Functional Analysis Results

The proofs of Lemma 10 and Corollary 4 rely on the following theorem, which is a slight variant of [46, Theorem 10] that immediately follows from the linearity of the operators. Suppose $X$ is a complex Banach space and $V\in B_0(X)$. For $\lambda \in \sigma (V)$ recall that $m_V(\lambda )$ is equal to the dimension of the generalized eigenspace $E_\lambda =\cup _{j=1}^\infty {{\,\mathrm{Null}\,}}(V-\lambda I_X)^j$.

Theorem 11

Let $X$ be a complex Banach space and $V\in B_0(X)$ satisfy $V\chi _0=\mu \chi _0$ for some $\mu \in {\mathbb {C}}{\setminus }\{0\}$ and $\chi _0\in X{\setminus }\{0\}$. If $L\in B(X,{\mathbb {C}})$ is a continuous linear functional such that $L\chi _0=1$ and $W:X\rightarrow X$ is defined by $W\chi =V\chi -L(V\chi )\chi _0$ for $\chi \in X$, then $W\in B_0(X)$ and

$$\begin{aligned} m_W(\lambda )={\left\{ \begin{array}{ll} m_V(\lambda ) &{} \text {if } \lambda \ne \mu ,\\ m_V(\lambda )-1 &{} \text {if } \lambda = \mu . \end{array}\right. } \end{aligned}$$

In particular, if $\mu =1$ then $\sigma _{-1}(V)=\sigma (W)$.

The following proof of Lemma 10 closely parallels the proof of [46, Theorem 9].

Proof of Lemma 10

Define $L\in B_0({\mathcal {C}}({\mathbb {R}}),{\mathbb {R}})$ by $L\phi =\phi (-1)$ for all $\phi \in {\mathcal {C}}({\mathbb {R}})$ so that $T(\beta ,\phi ,t)=LS(\beta ,\phi ,t)$ for all $\beta >0$, $\phi \in {\mathcal {C}}({\mathbb {R}})$ and $t\ge 0$. Recall that for each $\beta >\beta _0$, by the definition of the monodromy operator, $D_\phi S(\beta ,p_0^\beta ,\omega ^\beta )=U_1^\beta (0)$ (see, e.g., [21, Chapter 2, Theorem 4.1]). It follows that $D_\phi T(\beta ,\phi ,\omega ^{\beta })=LU_1^{\beta }(0)$ for each $\beta >\beta _0$. Let $\beta >\beta _0$. By Lemma 9, $T({\tilde{\beta }},\phi ,q^\beta ({\tilde{\beta }},\phi ))=0$ for all $({\tilde{\beta }},\phi )\in W^\beta$ and so

$$\begin{aligned} 0&=D_\phi \left[ T({\tilde{\beta }},\phi ,q^\beta ({\tilde{\beta }},\phi ))\right] \big |_{({\tilde{\beta }},\phi )=(\beta ,p_0^\beta )}\\&=D_\phi T(\beta ,p_0^\beta ,\omega ^\beta -1)+D_tT(\beta ,p_0^\beta ,\omega ^\beta -1)D_\phi q^\beta (\beta ,p_0^\beta )\\&=LU_1^\beta (0)+\dot{p}^\beta (-1)D_\phi q^\beta (\beta ,p_0^\beta ). \end{aligned}$$

Hence,

$$\begin{aligned} D_\phi q^\beta (\beta ,p_0^\beta )=-\frac{LU_1^\beta (0)}{\dot{p}^\beta (-1)}. \end{aligned}$$

By the definition of $\varPhi ^\beta$ in (113), the derivative of $\varPhi ^\beta$ with respect to $\phi$ satisfies

$$\begin{aligned} D_\phi \varPhi ^\beta (\beta ,p_0^\beta )&=D_\phi S(\beta ,p_0^\beta ,\omega ^\beta )+D_tS(\beta ,p_0^\beta ,\omega ^\beta )D_\phi q^\beta (\beta ,p_0^\beta )\\&=U_1^\beta (0)-\frac{{\dot{p}}_0^\beta }{\dot{p}^\beta (-1)}LU_1^\beta (0). \end{aligned}$$

Since $U_1^\beta (0)\in B_0({\mathcal {C}}({\mathbb {R}}))$, we see that $D_\phi \varPhi ^\beta (\beta ,p_0^\beta )\in B_0({\mathcal {C}}({\mathbb {R}}))$. Now an application of Theorem 11 with $X={\mathcal {C}}({\mathbb {R}})$, $V=U_1^\beta (0)$, $\chi _0\in {\mathcal {C}}({\mathbb {R}})$ defined by $\chi _0(\theta )=\frac{{\dot{p}}_0^\beta (\theta )}{\dot{p}^\beta (-1)}$ for $\theta \in [-1,0]$, $\mu =1$, and $W=D_\phi \varPhi ^\beta (\beta ,p_0^\beta )$ implies that $\sigma (D_\phi \varPhi ^\beta (\beta ,p_0^\beta ))=\sigma _{-1}(U_1^\beta (0))$. By [45, Theorem 1], 1 is a simple eigenvalue of $U_1^\beta (0)$, thus completing the proof of the lemma.

Given a Banach space $X$, $\chi \in X$ and $r>0$, recall that ${\mathbb {B}}_X(\chi ,r)$ denotes the ball of radius r centered at $\chi$.

Proof of Corollary 4

Let $\delta \in (0,1)$, $\mu$ and $\psi$ be as in Theorem 10. Let $Y^*=B(Y,{\mathbb {C}})$ denote the dual of $Y$. We claim, and prove below, there is a $\delta _0\in (0,\delta ]$ such that for each $\chi \in {\mathbb {B}}_X(\chi _0,\delta _0)$ there is a linear functional $L(\chi )\in Y^*$ such that $L(\chi )\psi (\chi )=1$, and L is continuous in $\chi$. Assuming the claim holds, define, for each $\chi \in {\mathbb {B}}_X(\chi _0,\delta _0)$, the operator $W(\chi )\in B_0(Y)$ by

$$\begin{aligned} W(\chi )\xi =V(\chi )\xi -(L(\chi )\xi )\psi (\chi ),\qquad \xi \in Y. \end{aligned}$$

Then $W(\chi )$ is continuous in $\chi$, and by Theorem 11 and the fact that $\mu (\chi )$ is a simple eigenvalue of $V(\chi )$ for each $\chi \in {\mathbb {B}}_X(\chi _0,\delta _0)$, it follows that $W(\chi )\in B_0(Y)$ and $\mu (\chi )\not \in \sigma (W(\chi ))$ for each $\chi \in {\mathbb {B}}_X(\chi _0,\delta _0)$. Since $W(\chi _0)$ is compact, it has an isolated spectrum, which along with the continuity of the function $\chi \mapsto \sigma (W(\chi ))$, implies that by choosing $\delta _0>0$ possibly smaller, we can ensure that for each $\chi \in {\mathbb {B}}_X(\chi _0,\delta _0)$, the spectrum of $W(\chi )$ does not contain any elements in ${\mathbb {B}}(\mu _0,\delta _0)$.

We are left to prove the claim. For each $\chi \in O$ let $V^*(\chi )\in B_0(Y^*)$ denote the adjoint of $V(\chi )$. Then $V^*:O\rightarrow B_0(Y^*)$ is continuous and for each $\chi \in {\mathbb {B}}(\chi _0,\delta )$, $\mu (\chi )$ is the unique simple eigenvalue of $V(\chi )$. Moreover, $\psi _0^*\in Y^*$ is a unit eigenfunction associated with $\mu _0$. Applying Theorem 10 again, this time with $Y^*$, $V^*$ and $\psi _0^*$ in place of $Y$, V and $\psi _0$, respectively, there is a $\delta _0>0$ and a continuous function $\psi ^*:{\mathbb {B}}(\chi _0,\delta _0)\rightarrow Y^*$ such that for each $\chi \in {\mathbb {B}}(\chi _0,\delta _0)$, $\psi ^*(\chi )$ is a unit eigenfunction of $V^*(\chi )$ associated with $\mu (\chi )$. Since $\psi _0^*(\psi _0)\ne 0$, by choosing $\delta _0>0$ possibly smaller, we can ensure that $\psi ^*(\chi )(\psi (\chi ))\ne 0$ for all $\psi \in {\mathbb {B}}_X(\chi _0,\delta _0)$. For each $\chi \in {\mathbb {B}}_X(\chi _0,\delta _0)$, define $L(\chi )\in Y^*$ by

$$\begin{aligned} L(\chi )\xi =\frac{\psi ^*(\chi )(\xi )}{\psi ^*(\chi )(\psi (\chi ))},\qquad \xi \in Y. \end{aligned}$$

Then $L(\chi )\psi (\chi )=1$ for all $\chi \in {\mathbb {B}}_X(\chi _0,\delta _0)$, and $L(\chi )$ is a continuous function of $\chi$. This completes the proof of the claim.