Imagine you have two dynamical systems, given by the statespace equations: $\frac{dx}{dt}=F_1(x)$ and $\frac{dx}{dt}=F_2(x)$, and you are concerned with trajectories form a point in phase space $x_0$. Where $x_{0},x\in \mathbb{R}^n$, and $F_1(x)$ and $F_2(x)$ are mappings from $\mathbb{R}^n$ to $\mathbb{R}^n$.
Its clear that all trajectories from $x_0$ in either system are identical if $F_1(x)=F_2(x) \forall x$ or if $F_1(x)=cF_2(x) \forall x$ where c is a positive constant real number (the second system will just move faster or slower along the same trajectory). But what if they aren't?
What is required for the trajectories to be equal in the following two cases?
- What if $F_{1,i}(x)=v_i(t)F_{2,i}(x) \forall x$ where $F_{1,i}(x)$ and $F_{2,i}(x)$ are components of the statespace equation vectors, and $v_i(t)$ is a function of time?
- What if $F_{1,i}(x)=G_i(x)F_{2,i}(x) \forall x$ where $F_{1,i}(x)$ and $F_{2,i}(x)$ are components of the statespace equation vectors, and $G_i(t)$ is a function of the statespace variables?
If you are not able to answer in full i'd still like to get your thoughts, because although I think can partially answer these questions, I'd love to get a fresh perspective on them.