Distance between solutions of two first-order ODEs

Question

Let $f:\mathbb{R}^n\to \mathbb{R}$ be a twice-continuously differentiable function, and $\varphi:\mathbb{R}\to\mathbb{R}$ be a smooth concave function. For a given initial condition $x_0\in\mathbb{R}^n$, consider the gradient system applied to $f$, $$ \begin{cases} \dot x(t) = -\nabla f(x(t)) \\x(0)=x_0 \end{cases},\qquad t\geq 0, $$ and the gradient system applied to $\varphi\circ f$, $$ \begin{cases} \dot z(t) = -\nabla (\varphi \circ f)(z(t)) = -\varphi'(f(z(t))\nabla f(z(t)) \\z(0)=x_0 \end{cases},\qquad t\geq 0. $$ My question is, can we relate the two systems, that is, does there exist an homeomorphism $H$ such that $\forall t\geq 0$, $x(t)=H(z(t))$?

Answer 1

With a parameter change $t=\alpha(s)$ the derivative of $x$ becomes $$ \frac{d}{ds} x(α(s))=\dot x(α(s))α'(s)=-α'(s)∇f(x(α(s))) $$ So if you set $$ α'(s)=φ'(f(x(α(s)))), $$ then $x(t) = x(α(s))=z(s)$ follows, at least as long as $α$ remains monotonously increasing.