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In Control & System theory, does a simple integrator consider a static system? i.e.: $$\dot{x}=u , y=x$$ While, a general nonlinear dynamical system can be described by :

$$\dot{x}=f(x,u) , y= h(x)$$ Where $$x,u,y$$ is the state, input and output , respectively.

If my understanding is correct, can I have a formal definition for a static system.

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    Dont know a formal defintion but take $x_{n+1}=f(x_n)$ ist static, because it depends only on $x_n$. In contrast $x_{n+1}=f(x_n,x_{n-1})$ is dynamic. – 2017-01-23
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    In short, if the state depends on time, the system is dynamic. In your equation, we have: $$x=\int_{t_0}^{t}u(\tau)d\tau$$ which is clearly a function of $t$ if $u\ne 0$ – 2017-01-23
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    @user160069 But you can also define your state as $z_n=[x_n^T\quad x_{n-1}^T]^T$ in this case you have $z_{n+1}=f(z_n)$ again. – 2017-01-23

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If a system is dynamic, this means that "the future states depend on the current one", so it evolves with time. For example, if there is a derivative term in the system model, then it is a (continuous) dynamical system.

A static system solution does not depend on time.

Also see http://www.scholarpedia.org/article/Dynamical_systems.

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    The point of confusion, for the simple integrator system, the output at time (t),which is the state at time (t), depends only on the current input at time (t). Will this be a Static or Dynamic System? – 2017-01-24
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    It does depend on previous states. The actual solution is $x(t)=x_0 + \int_{t_0}^t u(\tau) d \tau$, where $x_0 := x(t_0)$. If this is not convincing for you note that at every time instance, the next state depends on current state and the input at that time instance, per the definition of derivative. – 2017-01-24