Gumovski-Mira Equation

Gumovski-Mira Equation has the form:

xn+1 = 2axn  /( 1 +  xn2 ) -  xn-1 ,

where a > 1.

This equation has an equilibrium point p = (a - 1)1/2 .

This equation possesses an invariant
 
 

I(xn , xn-1 ) = xn2  xn-1 2 - a2( xn2 + xn-1 2 ) + 2 xn  xn-1 .


THEOREM (Discrete version of  the Dirichlet's theorem).  Consider the difference equation

xn+1 = f( xn)
where xn is in Rk and f: D® D is continuous, where D Ì Rk. Suppose that I:Rk® R is a continuous invariant that is I(f(x)) = I(x) for every x Î D. If I attains an isolated local minimum or maximum value at the equilibrium (fixed) point p of this system, then there exists a Liapunov function equal to ±(I(x) - I(p)) and so the equilibrium p is stable.
 

By using the above theorem, the Liapunov function, V(x,y) for this equation is found to be given by:
 


V(x, y) = I(x , y ) + (a - 1)1/2


which shows, that "p" is a  stable equilibrium.

Shown below is the surface of the invariant for the specific value a = 2.
 
 

Here is the animated graph of invariant for the specific value a = 2.
 
 

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Click here to simulate the solutions of this equation for different choices of parameters and initial points