Difference equations

I have a question:

suppose I have the difference equation

x(k+1)=0.5[x^3(k) +x(k) ] where ^ is to the power. k=0,1,2,3.....

then I can show easily that the equation has 3 equilibrium points at
x*=0,-1 and +1

Only the one at x=0 is stable. What does equilibrium point mean in
this context? I simulated this and found it is stable if -1<x0<+1.
where x0 is the initial condition of x ie x(0). Should x(0) be less
than the equilibrium points for stability in general?

K.
.