Systems of ODEs that fulfill a matrix relationship at steady state [closed]

Question

It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$ with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, z_n)$, all entries of $x(t_1)$ can be written down as a linear combination of $x(t_0) = (y_1, \ldots, y_n)$ plus a constant: $$ z_i = c_i +\sum_{j=1}^{n} b_{ij} y_j,$$ for all $i$ where the $b_i, c_{ij}$ are constants.

My question is now: What are other common systems of ODEs, where we can describe the entries of the solution vector at a different time point (or of a steady state to which the initial condition onverged to) by such a linear combination from above from the initial condition $x(t_0)$?

Answer 1

What you're saying, I think, is that (for a particular $t_1$) the map $\Psi$ from $x(0)$ to $x(t_1)$ is affine: $\Psi(x) = \Phi x + c$ where $\Phi$ is linear. If this was true for all $t_1$, the differential equation would have to be linear, because $x'(t) = \lim_{h \to 0} \dfrac{x(t+h) - x(t)}{h}$ and a limit of affine maps is affine. But for a single $t_1$, there are many other possibilities. For any invertible $\Phi$ and any $c$, let $\psi(x,t)$ be any smooth function on $\mathbb R^{n+1}$ such that $\psi(x,0) = x$, $\psi(x,t_1) = \Phi x + c$, and $\psi(\cdot,t)$ is a diffeomorphism of $\mathbb R^n$ for each $t \in [0,t_1]$. Let $\phi(\cdot,t)$ be the inverse function of $\psi(\cdot,t)$. Then $X(t) = \psi(x,t)$ satisfies the differential equation $$ \dfrac{dX}{dt} = \psi_t(x,t) = \psi_t(\phi(X,t),t)$$ (the subscript indicating a partial derivative with respect to $t$).

Answer 2

For a system $\dot x = f(t, x(t))$ with $f\in C^0(I\times \mathbb{R}^n,\mathbb{R}^n)$, the answer is: $f$ is necessarily of the form $f(t,x)=A(t)x+b(t)$, if we assume that the affine dependence of $x(t_1)$ from $x(t_0)$ holds for all pairs $t_0, t_1$ in $I$ (or, for all $t_1$ and some fixed $t_0$, hence for all). Of course, together with the implicit assumption that any IVP for this ODE has unique solution in the whole interval $I$.

Premise. Let $I$ be an open interval and let $W:I\times I\rightarrow M_n (\mathbb{R})$ a $2$-parameter family of matrices verifying the chain law: $$W(r,s)=W(r,t)W(t,s),$$ $$W(t,t)=\mathbb{I},$$ for all $r,s,t$ in $I$. Assume that $W(t,s)$ is continuous in the pair $(t,s)\in I\times I$ and partially differentiable in the first variable.

Then it follows that $\partial_1W(t,s)W(s,t)$ is a matrix $A(t)$ independent from $s$, and that $W$ is continuously differentiable in $I\times I$, with $\partial_1W(t,s)=A(t)W(t,s)$, and $\partial_2W(t,s)=- W(t,s)A(s)$, for all $t,s$ in $I$. In other words, $W(t,s)$ is the transition operator associated with the linear system $\dot u(t)=A(t)u(t).$ (Note that this corresponds to your situation in the special case where $c_j$ are identically zero for all $t_0$ and $t_1$).

Indeed, by the chain law we have that all $W(t,s)$ are invertible, and $W^{-1}(t,s)=W(s,t)$ for all $t,s$, so $W(t,s)$ is differentiable in its second variable too. If we derive w.r.to $t$ the identity $W(r,s)=W(r,t)W(t,s)$ we get $0=\partial_2W(r,t)W(t,s)+W(r,t)\partial_1W(t,s) $, whence multiplying conveniently on the left and on the right we see by separation of variables that both sides of $ \partial_1W(t,s) W(s,t)=- W(t,r) \partial_2W(r,t)$ only depend on $t$. If we denote the common value $A(t):=\partial_1W(t,s) W(s,t)=- W(t,r) \partial_2W(r,t)$, a continuous path of matrices, we have that $W$ solves $\partial_1W(t,s)=A(t)W(t,s)$ (and also verifies $\partial_2W(t,s)=-W(t,s)A(s)$ ), with initial condition $W(t,t)=\mathbb{1}$, that is, it is the transition operator associated with the linear system $\dot u(t)=A(t)u(t).$ (Incidentally, $W$ is actually $C^1$ by the Total Differential Theorem). $$*$$ Now, as in your assumptions, let be given a system of ODE's, all of whose IVP have unique solutions $x:I\to\mathbb{R}^n$, and assume that for all $s,t$ in $I$ the value of the solution $x $ at time $t$ depends affinely from the initial data at time $s$. In other words, assume that for any solution $x:I\to\mathbb{R}^n$ there holds $$x(t)=W(t,s)x(s)+c(t,s),$$ for some $2$-parameter family of matrices $W(s,t)$ and some $2$-parameter family of vectors $c(t,s)$. Since by assumption for any $s\in I$ the values $x(s)$ can be chosen in the whole space, the above form implies that $W(t,t)=\mathbb{I}$ and $c(t,t)=0$, for any $t\in I$. Since any solution $x(t)$ is differentiable, we have that both $W(t,s)$ and $c(t,s)$ are partially differentiable in their first variable; moreover $W$ and $c$ are continuous in the pair $(t,s)$, as the continuous dependence from the initial data is in fact a general consequence of the uniqueness property of solutions of the IVP.

The above form also implies, for any $r,s,t$ in $I$ and for any solution $x$: $$x(r )=W(r,s)x(s)+c(r,s)=W(r,t)x(t)+c(r,t)=W(r,t)\left[ W(t,s)x(s)+c(t,s) \right]+c(r,t) $$ $$ =W(r,t)W(t,s)x(s)+ W(r,t) c(t,s) +c(r,t) . $$ Again this implies two separate identities, because for any fixed $s$, $x(s)$ varies in the whole space $\mathbb{R}^n$: $$W(r,s)=W(r,t)W(t,s)$$ and
$$c(r,s)= W(r,t) c(t,s) +c(r,t),$$ together with $W(t,t)=\mathbb{I}$ and $c(t,t)=0$, for all $r,s,t$.

By the initial remark, we have that $W$ is the transition operator of a linear system $\dot u(t)=A(t)u(t)$ corresponding to a continuous path $A(t)$ of matrices. Finally, differentiating $c(r,s)= W(r,t) c(t,s) +c(r,t)$ w.r.to $r$ we have

$$\partial_1c(r,s)= A(r)W(r,t) c(t,s) +\partial_1c(r,t)$$ $$=A(r )\left[ W(r,t) c(t,s) + c(r,t) \right] -A(r )c(r,t) + \partial_1c(r,t)$$ $$=A(r )c(r,s) -A(r )c(r,t) + \partial_1c(r,t).$$ Separating variables, this tells that $\partial_1c(r,s)-A(r )c(r,s)= \partial_1c(r,t)-A(r )c(r,t) $ is a continuous path $b(t)$ only depending on $t$. We therefore have for all $s,t$ in $I$ and for all solution $x$ $$\dot x(t)=A(t)W(t,s)+\partial_1c(t,s)=A(t)\left[W(t,s)+c(t,s)\right]+\partial_1c(t,s)-A(t)c(t,s)=A(t)x(t)+b(t),$$ proving that the ODE is actually affine.