I have the following non linear difference equation:
$$ \rho_{{t}}=- \left( -4+ \left( 2+\rho_{{t-2}} \right) \rho_{{t-1}} \right) ^{-1} $$
I found the three equilibrium points, which are: $$ \hat \rho^1 = 1 $$ $$ \hat \rho^2 = -3/2 + 1/2\,\sqrt {13} $$ $$ \hat \rho^3 = -3/2 - 1/2\,\sqrt {13} $$
I "know" that only the second one is stable (globally as well): I found out via simulation. However, I would like to study the stability of this equation through more "theoretic" methods such a Lyapunov's methods.
I tried taking a look at various papers but they either mention the continuous time case or they concern first order equations, and I'm not sure if such methods apply in this particular case.
So, my actual question is the following: Is there a method to study the stability of the fixed points of this equation and if yes, could someone illustrate how to apply it in this particular case? Even if it's Lyapunov's method that I really did not understand that well.
Thanks in advance for all the answers.
Update
Here is my attempt to apply Lyapunov's direct method to study the stability of the first equilibrium point: (through simulation I found this to be unstable) $$ \hat \rho^1 = 1 $$
I use this as a Lyapunov Function
$$ V \left( \rho_{{t-1}},\rho_{{t-2}} \right) = \left( \rho_{{t-1}}-1 \right) ^{2}+ \left( \rho_{{t-2}}-1 \right) ^{2} $$
$$ \Delta V \left( \rho \right) = \left( \rho_{{t}}-1 \right) ^{2}- \left( \rho_{{t-2}}-1 \right) ^{2} $$
$$ \Delta V \left( \rho \right) = \left( - \left( -4+ \left( 2+\rho_{{t-2 }} \right) \rho_{{t-1}} \right) ^{-1}-1 \right) ^{2}- \left( \rho_{{t- 2}}-1 \right) ^{2} $$
Since the difference is neither semidefinite positive nor semidefinite negative I cannot say anything about the stability of this point of equilibrium, right? Or did I do something wrong?