  The Schwarzian derivative of a function f(x) which is defined in the interval
(a, b) having higher order derivatives is given by
Sf(x)=(f''(x)/f'(x))'-1/2(f''(x)/f'(x))^2 . A sufficient condition for a
function to behave chaotically is that its Schwarzian derivative is negative.
In this paper, we try to find a sufficient condition for a non-linear dynamical
system to behave chaotically. The solution function of this system is a higher
degree polynomial. We define n-th Schwarzian derivative to examine its general
properties. Our analysis shows that the sufficient condition for chaotic
behavior of higher order polynomial is provided if its highest order three
terms satisfy an inequality which is invariant under the degree of the
polynomial and the condition is represented by Hankel determinant of order 2.
Also the n-th order polynomial can be considered to be the partial sum of real
variable analytic function. Let this analytic function be the solution of
non-linear differential equation, then the sufficient condition for the
chaotical behavior of this function is the Hankel determinant of order 2
negative, where the elements of this determinant are the coefficient of the
terms of n, n-1, n-2 in Taylor expansion.
