Concrete differences between a map and a flow

Question

The fixed points of a Flow (say, autonomous flow) is the set of all points $x=x^{*}$ for $x \in$ state space $\mathbb{R}^{n}$ such that $F\left ( x=x^{*} \right )$ =0.

The fixed points of a Map are the set of all points $x=x^{*}$ such that $G\left ( x=x^{*} \right )=x^{*}$. Really, G is just the identity map.

In my lecture notes, the Maps equations are always denoted with a subscript. Is this a coincidence?

The biggest question I do have for now is the physical differences illustrated between a Flow and a Map.

Both the equation $\dot{x}=F\left ( x \right )=x\left ( N-x \right )$ and $\dot{x_{t+1}}=F\left ( x_{t} \right )=\lambda x_{t}\left ( 1-x_{t} \right )$ are used to model equation-the latter, being a Map, having been normalised. An intuitive explanation as to the "why" and the "when" Flows and Maps are used would be very helpful. What does either one accomplishes that the other doesn't?

Thanks in advance.

Answer 1

Actually a discrete Dynamical System is known as a map and being discrete the system is of the form $x_{n+1} = f(x_{n})$ ( a difference equation ),$x_{n}$ represents the state variable $x$ at the $n$th time instant and $f$ is a mathematical rule governing the evolution of the system.

But Flows are continuous Dynamical systems.The study of these systems requires Calculus.Flow is given by $\frac{dx(t)}{dt} = F(x(t))$ where $F(x(t))$ is a function that governs the evolution of the system and this results in an Oridnary differential equation.

I found these after googling- You can refer this and more here - http://math.arizona.edu/~shankar/efa/efa1.pdf

Hope this helps!

Answer 2

If you were actually writing down the flow $\phi(t;t_0,x_0)$ as solution of the initial value problem $$\frac{d}{dt}\phi(t;t_0,x_0)=F(\phi(t;t_0,x_0)),\quad \phi(t_0;t_0,x_0)=x_0 $$ the fixed point equation takes a rather similar form, $$ x^*=\phi(t;t_0,x^*)\quad\forall t. $$ However, the localization of constant solutions is much easier by examining the condition that the derivative is zero, which together with the differential equation gives the condition $0=F(x^*)$. The uniqueness theorem then guarantees that the constant solution is the only one for these initial conditions.