Behaviour of solutions to ODE $\dot y(t)= Bh(y(t))$ from behaviour of solutions to ODE $\dot x(t) = h(x(t))$, if $B$ is positive definite

Question

Suppose $B$ is a positive definite $d \times d$ matrix. Then given the asymptotic behaviour of an o.d.e $\dot{x}(t) = h(x(t))$ where $h:\Bbb R^d \to \Bbb R^d$, will the asymptotic behaviour of the o.d.e $\dot y(t)= Bh(y(t))$ be same as the former o.d.e ?

Answer 1

In general, the answer is no. Consider the unstable system $$\tag{1} \left\{\begin{array}{lll} \dot x_1&=&x_2\\ \dot x_2&=&0 \end{array}\right. $$ or $$ \dot x=Ax,\qquad A=\left( \begin{array}{ll} 0&1\\0&0\end{array}\right). $$ Let $$ B=\left( \begin{array}{rr} 2&-1\\-1&2\end{array}\right). $$ One can see that the system $$ \dot y= BAy,\qquad BA=\left( \begin{array}{rr} 0&2\\0&-1\end{array}\right) $$ is stable.