I want to find a diffeomorphism $G$ defined on some neighborhood of the origin with $G(0)=0$ such that $G$ transforms the linear system $\dot{x}=x$ to the nonlinear system $\dot{y}=y-y^2$. (These are one-dimensional systems).
Any comments or responses are greatly appreciated!
The diffeomorphism is $$ y=\frac{x}{x-1}. $$ Indeed, $$ \dot y=\frac{\dot x(x-1)-x\dot x}{(x-1)^2}= \frac{x(x-1)-x^2}{(x-1)^2}= \frac{x}{x-1}-\frac{x^2}{(x-1)^2}=y-y^2. $$ This result can be observed by a comparison between the general solutions $ x= Ce^t $ and $ y=\frac{Ce^t}{Ce^t-1} $.