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  1. Could someone explain me what is the fundamental difference between the dynamical system of the kind $\dot x = f(x)$ and $E \dot x= f(x)$ where $E$ is an matrix with real entries. For the first system, a state is $x(t)$ at time $t$; is it the same for the second system? I wish to visualise trajectories of such type of dynamical system to be more precise.

  2. Consider a continuous time dynamical system \begin{align} \dot x= f\left(x\right) \end{align} on a state-space $\mathcal{X}$, where $x$ is a coordinate vector of the state, $f$ is a non-linear vector valued smooth function of the same dimension as its argument $x$.

    Could anyone tell me why assume $f$ to be smooth and what it means by ''function of the same dimension as its argument $x$'', what is a dimension of function, by the way, and why its dimension needs to be equal as $x$?

Thank you for your clarification.

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