Exercise :
Find competent and necessary conditions for $a,b \in \mathbb R$, such that $0 \in \mathbb R^2$ is an exponentially stable stationary point for the the linear dynamical system of discrete time : $$x_1^+=x_2$$ $$x_2^+=ax_1 + bx_2$$
Discussion :
We have just been introduced to such matters on our Dynamical Systems course and I do not have a lot of experience regarding such a problem, which means I didn't know how to proceed. I did though some research regarding exponential stability, which is listed below.
Going over exponential stability in dynamical system matters, I found the following definition and theorem on internet :
Definition (Exponential Stability, rate of convergence) :
The equilibrium point $x^* = 0$ is an exponentially stable equilibrium point of the dynamical system : $$\dot{x} = f(x,t)$$ $$x(t_0)=x_0$$ $$x \in \mathbb R^n$$ if there exist constants $a,m >0$ and $\epsilon >0$ such that : $$\| x(t)\| \leq me^{-a(t-t_0)}\| x(t_0)\|$$ for all $\|x(t_0)\|\leq\epsilon$ and $t\geq t_0$. The largest constant $a$ is called rate of convergence.
Exponential Stability Theorem : The point $x^* =0$ is an exponentially stable equilibrium point of $\dot{x} = f(x,t)$ if and only if there exist an $\epsilon > 0$ and a function $V(x,t)$ which satisfies : $$a_1\|x\|^2 \leq V(x,t) \leq a_2 \|x\|^2$$ $$\dot{V}(x,t) \leq -a_3\| x \|^2$$ $$\bigg\|\frac{\partial V(x,t)}{\partial x}\bigg\| \leq a_4\|x\|$$ for some positive constants $a_1,a_2,a_3,a_4$ and $\|x\| \leq \epsilon$.
Now, I can say that the theorem looks a bit like the Lyapunov Stability ones (correct me if I'm wrong) but I can't seem on how I should proceed on the particular exercise listed up above. I would appreciate some thorough help regarding the particular problem and its solution, to get some comfort with such problems over exponential stability.