The Higher Order Schwarzian Derivative: Its Applications for Chaotic Behavior and New Invariant Sufficient Condition of Chaos

摘要

The Schwarzian derivative of a function f(x) which is defined in the interval(a, b) having higher order derivatives is given bySf(x)=(f''(x)/f'(x))'-1/2(f''(x)/f'(x))^2 . A sufficient condition for afunction to behave chaotically is that its Schwarzian derivative is negative.In this paper, we try to find a sufficient condition for a non-linear dynamicalsystem to behave chaotically. The solution function of this system is a higherdegree polynomial. We define n-th Schwarzian derivative to examine its generalproperties. Our analysis shows that the sufficient condition for chaoticbehavior of higher order polynomial is provided if its highest order threeterms satisfy an inequality which is invariant under the degree of thepolynomial 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 realvariable analytic function. Let this analytic function be the solution ofnon-linear differential equation, then the sufficient condition for thechaotical behavior of this function is the Hankel determinant of order 2negative, where the elements of this determinant are the coefficient of theterms of n, n-1, n-2 in Taylor expansion.