Approximating Lyapunov exponents and Sacker–Sell spectrum for retarded functional differential equations

Abstract

We consider Lyapunov exponents and Sacker–Sell spectrum for linear, nonautonomous retarded functional differential equations posed on an appropriate Hilbert space. A numerical method is proposed to approximate such quantities, based on the reduction to finite dimension of the evolution family associated to the system, to which a classic discrete QR method is then applied. The discretization of the evolution family is accomplished by a combination of collocation and generalized Fourier projection. A rigorous error analysis is developed to bound the difference between the computed stability spectra and the exact stability spectra. The efficacy of the results is illustrated with some numerical examples.

Appendix A

Besides the notation introduced in Sect. 3, we recall that $G_{s}$ is defined in (11), $V$ in (21) and ${\fancyscript{L}}_{N}^{+}$ is specified right before Proposition 1. We first recall in Theorem 4 basic approximation results from [15], namely (9.4.6) and (9.4.24), and then prove some technical lemmas which are used in Sect. 5.1.

Theorem 4

Let $a<b\in \mathbb R$, $L:=L^{2}(a,b;\mathbb R ^{d})$ and $H^{k}:=H^{k}(a,b;\mathbb R ^{d})$, $k\ge 1$ a positive integer. Given positive integers $M$ and $N$, if $\varphi \in H^{k}$, then there exists a constant $c_{F}$ independent of $M$ s.t.

$$\begin{aligned} \Vert \varphi -F_{M}\varphi \Vert _{L}\le c_{F}M^{-k}\Vert \varphi \Vert _{H^{k}} \end{aligned}$$

(49)

where $F_{M}$ is the Fourier projection operator w.r.t. the Legendre polynomials of $L$ and a constant $c_{{\fancyscript{L}}}$ independent of $N$ s.t.

$$\begin{aligned} \Vert \varphi -{\fancyscript{L}}_{N}\varphi \Vert _{L}\le c_{{\fancyscript{L}}}N^{\frac{1}{2}}N^{-k}\Vert \varphi \Vert _{H^{k}} \end{aligned}$$

(50)

where ${\fancyscript{L}}_{N}$ is the interpolation operator w.r.t. the Legendre-Gauss zeros of $[a,b]$.

Lemma 4

$\Vert V\Vert _{L^{\pm }\leftarrow L^{+}}\le r$.

Proof

Let $y\in L^{+}$. Then

$$\begin{aligned} \Vert Vy\Vert _{L^{\pm }}\!=\!\int \limits _{-\tau }^{r}\left| (Vy)(t)\right| ^{2}dt \!=\!\!\!\int \limits _{0}^{r}\left| \int \limits _{0}^{t}y(\sigma )d\sigma \right| ^{2}dt \!\le \!\int \limits _{0}^{r}\left( \int \limits _{0}^{r}\left| y(\sigma )\right| ^{2}d \sigma \right) dt\!=\!r\Vert y\Vert _{L^{+}}. \end{aligned}$$

$\square$

Lemma 5

$(I_{L^{+}}-G_{s}V)^{-1}\in {\fancyscript{B}}(L^{+})$.

Proof

The assertion corresponds to proving that for any given $h\in L^{+}$ there exists a unique $f\in L^{+}$ solution of $(I_{L^{+}}-G_{s}V)f=h$. This in turn corresponds to the existence and uniqueness of the solution $y$ to the IVP

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle y^{\prime }(t)=(G_{s}y)(t)+h(t), &{}\quad \text{ a.e. } \text{ for } t\in [0,r],\\ \displaystyle y(0)=0, &{}\\ \displaystyle y(\theta )=0,&{}\quad \text{ a.e. } \text{ for } \theta \in [-\tau ,0). \end{array} \right. \end{aligned}$$

The latter follows from standard results on RFDEs, see e.g. [25].$\square$

Lemma 6

Assume $G_{s}:H^{1,\pm }\rightarrow H^{1,+}$. Then $\left\| (I_{L^{+}}-{\fancyscript{L}}_{N}^{+})G_{s}V\right\| _{L^{+}}\rightarrow 0$ as $N\rightarrow \infty$.

Proof

Let $y\in L^{+}$. Then (21) implies $Vy\in H^{1,\pm }$ and, by the hypothesis, $G_{s}Vy\in H^{1,+}$. Now apply (50) in Theorem 4.$\square$

Lemma 7

Assume $G_{s}:H^{1,\pm }\rightarrow H^{1,+}$. Then, for sufficiently large $N$, $(I_{L^{+}}-{\fancyscript{L}}_{N}^{+}G_{s}V)^{-1}\in {\fancyscript{B}}(L^{+})$ and

$$\begin{aligned} \left\| (I_{L^{+}}-{\fancyscript{L}}_{N}^{+}G_{s}V)^{-1}\right\| _{L^{+}} \le 2\left\| (I_{L^{+}}-G_{s}V)^{-1}\right\| _{L^{+}}. \end{aligned}$$

Proof

The thesis follows by applying the Banach’s Perturbation Lemma, Lemmas 5 and 6 since $I_{L^{+}}-{\fancyscript{L}}_{N}^{+}G_{s}V=(I_{L^{+}}-G_{s}V)+(I_{L^{+}} -{\fancyscript{L}}_{N}^{+})G_{s}V$.$\square$