CHAPTER 8. The Schwarzian Derivative

8.1 The Schwarzian Derivative.

The Schwarzian derivative is important in determining the nature of non-hyperbolic fixed points. It is defined by
						  
S f(x)=f'''(x)/f'(x)-((3[f''(x)/f'(x)])^2)/2, 
						  
and we are particularly interested where S f(x)<0. Let us consider functions of the form  f(x)=(x^3)-a x, for different values of the parameter  a.

Clear[f]
f[x_,a_]:=x^3-a*x

Doing a ManipulatePlot  allows us to consider these maps as the parameter varies:

Manipulate[Plot[f[x,a],{x,-2,2}],{a,-10,10}]

We also do a ManipulatePlot on the Schwarzian derivative, using f'(x)=3(x^2)-a, f''(x)=6x, f'''(x)=6,

S[x_,a_]:=6/(3x^2-a)-3/2(6x/(3x^2-a))^2

Manipulate[Plot[S[x,a], {x,-2,2}],{a,-10,10}]

From the ManipulatePlot of the Schwarzian derivative, we estimate that the it becomes negative when a>0.

Exercises 8.1.

1.     Use Mathematica to find the fixed points of  f(x)=-2(x^3)+2(x^2)+x,  and show that it has both types of non-hyperbolic fixed points. Find the Schwarzian derivative of this map and determine the nature of the fixed points.

2.     Estimate the values of a  for which the functions 
(i)   f(x)=(x^4)-a (x^2),  (ii)   f(x) = (x^4) - a (x^2)+x, (iii)    f(x)=arctan(x)-a x,
have a negative Schwarzian derivative. By simplifying the Schwarzian derivative using Mathematica, verify your finding in (i) and (iii).

3.     Find the fixed points of  (Subscript[f, p])(x)=x (e^(x-p)),  x>0, p>0, and use a ManipulatePlot to examine their stability. Find the Schwarzian derivative of this map for this range of values.

4.     Show that if a polynomial  p(x ), of degree n, has derivative  p'(x )  with n-1 roots (some of them possibly repeating), then  p(x )  has a negative Schwarzian derivative. Consider examples to conjecture whether the polynomial  p(x ) has negative Schwarzian derivative when it has n roots (some of then possibly repeating).