I came across the following alternate definition for Lyapunov stability of continuous linear time varying (CLTV) systems in a textbook:
A CLTV system is sait to be stable in the sense of Lyapunov (isL) if for every initial condition $x(t_0)=x_0\in\mathbb{R}^n$, the homogeneous state response $$ x(t)=\Phi(t,t_0)x_0, \forall t \geq 0$$ is uniformly bounded.
But what about the CLTV (time invariant even) system $$\dot x (t) = 0,$$ which is stable isL according to the original defn. If $M$ is it's upper bound then choose $x_0=2M$ then $x(t)=2M, \forall t>0$, which is a contradiction, so these can't be equivalent definitions. What am I missing?
Usually these definitions include
M(x(0)), and thus your example is fulfilled, since you cannot choose the bound before the initial condition. The same applies to the original definition of Lyapunov stable for LTI systems. Think about local stability, where you have to restrict your set of initial conditions. Then extend the notion to a global definition.