Making sense of a derivative on direct sum

Question

Consider $\mathbb{R}^n=E(t)\bigoplus F(t) $ be a continuous splitting. Is that possible to define a derivative on the direct sum such that $$ (x(t)\bigoplus y(t))'=x'(t)\bigoplus y'(t) $$ The reason being I was looking at the Lyapunov exponent where you can have such nice splitting. However most studies are done on the assumption that there is a direct product $\mathbb{R}^n=E(t)\times F(t)$. It makes some applications much more difficult. I understand that in finite dimensional space they are equivalent and my feeling of why people preferring direct product over direct sum is because it makes it hard to make sense of the dynamical system on the direct sum. i.e. what do you mean by $$ x'(t)\bigoplus y'(t)=(x(t)\bigoplus y(t))'=A(t)(x(t)\bigoplus y(t))=A(t)x(t)\bigoplus A(t)y(t) $$ I think the problem with the first equation is that $$ x'(t)=\lim_{\epsilon\to0}\frac{x(t+\epsilon)-x(t)}{\epsilon}. $$ and $x(t+\epsilon)\in E(t+\epsilon)$ but not $E(t)$. Thus we do not know that $x'(t)\in E(t)$ and therefore it might not make sense to write $x'(t)\bigoplus y'(t)$. Is the continuity of $E(t)$ makes the first equation true?

Answer 1

Thanks for @John B, the answer is 'NO' and talking about differential equation on the notion of direct sum is a bad idea (No wonder people are avoiding it). $x′(t)$ is most likely to be perpendicular to $E(t)$ instead of in $E(t)$.

However another alternative: we can consider the transition operator $\Phi(t,s)=\Phi_e(t,s)+\Phi_u(t,s)$ where $\Phi_e(t,s):E(s)\to E(t)$, $\Phi_u(t,s):F(s)\to F(t)$ and maps everything else to 0. Now we just need to observe that $$\frac{d}{dt}(x(t)\bigoplus y(t))=\frac{d}{dt}(\Phi(t,0)x_0\bigoplus \Phi(t,0)y_0)=A(t)\Phi(t,0)x_0+A(t)\Phi(t,0)y_0=A(t)(\Phi(t,0)x_0\bigoplus \Phi(t,0)y_0)=A(t)(x(t)\bigoplus y(t))$$ Now the dynamical system make sense on the notion of direct sum.

Good thing about this is that we know that there exists $B(t)$ and $C(t)$ such that $$\frac{d}{dt}\Phi_e(t,s)=B(t)\Phi_e(t,s)$$ $$\frac{d}{dt}\Phi_u(t,s)=C(t)\Phi_u(t,s)$$ We can further more require that $B(t)y(t)=0$ for all $y(t)\in F(t)$, $C(t)x(t)=0$ for all $x(t)\in E(t)$. We thus have $A(t)=B(t)+C(t)$.

After all those painful set up, we finally obtain an expression similar to $A(t)=\begin{bmatrix} B(t) & 0\\0 & C(t)\end{bmatrix}$ that allows us to work in stable/unsatble component separately, but under the notion of direct sum instead of direct product.

Now good new is that we can work with Lyapunov normal basis (whatever it call) directly instead of needing to worry about the effect of changing of coordinate.