When dealing with (nonlinear) dynamical systems, one often deals with state space representation, i.e. systems of the form

$$\dot{x}=f(x),\quad x(t)\in\mathbb{R}^n.$$ Let $x^*$ be a solution of this system, then the variational equation reads

$$\dot{x}= \underbrace{\frac{\partial f}{\partial x}(x^*)}_{=:A}\cdot x$$

or with $D:=\frac{d}{dt}$ in operator form

$$\underbrace{(DI_n-A)}_{=:P(D)}x=0$$ where $I_n$ is the $n$-th unit matrix. The Elements of $P(D)$ in general are meromorphic functions in $D$, so we are dealing with Ore polynomial matrices with the shifting rule

$$Da=\dot{a}+aD$$ for all meromorphic functions $a$. Here is what I am very confused about: How can be assumed $\dot{x}=Dx$ and not $\dot{x}=Dx-xD$ (according to the shifting rule)? And why can't i factor out $D$ to the right, i.e. $\dot{x}=xD$ (which by the shifting rule would result in something different)?


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