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

TL;DR

This paper introduces a higher order Schwarzian derivative and establishes a new invariant condition involving Hankel determinants that guarantees chaotic behavior in higher degree polynomial dynamical systems.

Contribution

It defines the n-th Schwarzian derivative for polynomials and derives a novel invariant condition for chaos based on Hankel determinants of polynomial coefficients.

Findings

Negative Hankel determinant of order 2 indicates chaos in polynomial systems.

The condition is invariant under polynomial degree.

Applicable to solutions of nonlinear differential equations as partial sums.

Abstract

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.