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I'm new to dynamical systems. I'm trying to prove some equivalence property of the following ordinary equations. The uniqueness and exisistence of the solution is assumed.

$\frac{d\boldsymbol{x}(t)}{dt}=\boldsymbol{f}(t,\boldsymbol{x})$ with initial values $\boldsymbol{x}(t_0)=\boldsymbol{x}_0$,

the solution is denoted as $\boldsymbol{F}(t,t_0,\boldsymbol{x}_0)$, where $t,t_0\in\mathbb{R},\boldsymbol{x},\boldsymbol{x}_0\in\mathbb{R}^2$.

For some specific linear transforms, $t\to\tau(t):\mathbb{R}\to\mathbb{R}$ and $\boldsymbol{x}\to\boldsymbol{y}(\boldsymbol{x}): \mathbb{R}^2\to\mathbb{R}^2$,

the transformed equations $\frac{d\boldsymbol{y}(\tau)}{d\tau}=\boldsymbol{g}(\tau,\boldsymbol{y})$ satisfy $\boldsymbol{g}(\tau,\boldsymbol{y})=\boldsymbol{f}(t,\boldsymbol{x})$.

How should I prove the solution to the transformed equations satisfies $\boldsymbol{G}(\tau,\tau(t_0),\boldsymbol{y}(\boldsymbol{x}_0))=\boldsymbol{y}(\boldsymbol{F}(t,t_0,\boldsymbol{x}_0))$?

Frank Han
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1 Answers1

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This is bad notation, at one place $y$ is a function of $x$, at another it is a function of $τ$. Better denote the first dependence as $y=\phi(x)$ ($x,y$ points) so that then for the solution functions $$y(τ(t))=\phi(x(t)).$$ Now apply the chain rule $$ y'(τ(t))\,τ'(t)=ϕ'(x(t))\,x'(t)\\ g(τ(t),y(τ(t)))\,τ'(t)=ϕ'(x(t))\,f(t,x(t)) $$ Now the assumed identity requires that $ϕ'(x)$ be a constant multiple of the identity, that is, $ϕ(x)=cx+d$ for some constant $c$ and vector $d$. On the other side this makes $τ(t)$ a linear function with linear coefficient $c$.

Lutz Lehmann
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  • Thank you! So I think that following you steps, checking the derivatives of $G-\phi(F)$ and then checking identity of the initial values would finish the proof. – Frank Han May 11 '22 at 05:41
  • I practically started from the claim, as $y(τ)=G(τ;τ_0,y_0)$ and $x(t)=F(t;t_0,x_0)$. The question is if my conclusion is as necessary as I wrote it. One really only has $$g(τ(t),ϕ(x))τ′(t)=ϕ'(x)f(t,x).$$ With $g(τ(t),ϕ(x))=f(t,x)$ this only says that $f(t,x)$ is an eigenvector of $ϕ'(x)$ with eigenvalue $τ′(t)$. The question is if this is rigid enough when including the local picture to determine $ϕ'(x)$ in the way I did. – Lutz Lehmann May 11 '22 at 07:29