What does a fixed point mean in a autonomous dynamical system, I mean I know the definition of it, but I keep hearing that if a dynamical system starts at a fixed point then it will remain there, why is this true.
Say $$ \frac{dx}{dt} = f(x,y) \\ \frac{dy}{dt} = g(x,y)$$
And suppose $(a,b)$ is a fixed point. Can someone show me why taking initial condition to be this fixed point then the system will remain at this fixed point, I see $\frac{dx}{dt}(0)= 0$ and $\frac{dy}{dt}(0) = 0$ but why must $x(t)=a$ and $y(t)=b$ at all times. I will be very grateful is someone were to clear this up for me.
As Did wrote, the theorem of Chauchy-Lipschitz/Picard-Lindelöf (see below for a note) states for us that, if the functions $f$ and $g$ are uniformly Lipschitz-continuous, and we are given an inital condition $x(0)=x_0$, $y(0)=y_0$ then we have a unique solution $(x,y)$ satisfying the differential equations.
For a stable fixed point, the constant functions $(x,y) \equiv (a,b)$ do satisfy the initial conditions $x(0)=a$, $y(0)=b$ and the differential equation.
So if we do have any solution of the differential equation which satisfies $x(0)=a, y(0)=b$, then by uniqueness, it must coincide with the constant solution.
Note that the theorem in question is called Cauchy-Lipschitz in French and Picard–Lindelöf in English and German and refer to the desired uniqueness result for an initial value problem $v'(t) = f(t,v(t))$, $v(t_0)=v_0$, where we need continuity of $f$ in the real variable $t$, and global Lipschitz continuity w.r.t. the second argument. Now in the most basic setting, the second argument is a real number as well, which is shown in the english wikipedia. However, the way of proving it for the second argument being an element of $\mathbb R^n$ or even of some Banach space $E$ is exactly the same, which is shown in the German and French wikipedia.