The higher order Schwarzian derivative: Its applications for chaotic behavior and new invariant sufficient condition of chaos

The Schwarzian derivative of a function f ( x ) which is defined in the interval ( a , b ) having higher order derivatives is given by S f ( 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 the 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 the 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 a non-linear differential equation, then the sufficient condition for the chaotical behavior of this function is that the Hankel determinant of order 2 should be negative, where the elements of this determinant are the coefficients of the terms of n , n − 1 , n − 2 in the Taylor expansion.