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

Abstract

The Schwarzian derivative of a function f (x) which is defined in the interval (a, b) having higher order derivatives is given by

Research on the Statistical Characteristics of Crosstalk in Naval Ships Wiring Harness Based on Polynomial Chaos Expansion Method

“…Polynomial chaos expansion method is a powerful and effective mehtod which is widely applied in numerous aspects such as industrial, material research and mathematical field [13][14][15][16][17][18]. The fundamentals of polynomial chaos expansion and uniform distribution will be briefly summarized here only to guide the audients and define units.…”

Complex Partial Fraction Decompositions of Rational Functions

“…These algorithms especially focus for determining the real PFDs for given rational functions. On the other hand, real PFDs may fail in some applications; for example a failure of real PFD is mentioned in [3]. From the point of view, complex PFD can alternatively be used and this yields simplest form for representation of rational function (1).…”

“…Rational functions are widely used in many branches of mathematics such as numerical analysis i.e Pade Approximations, mathematical analysis as well as mathematical modelling and they appear in mathematical representations of many problems in science and engineering [1][2][3][4][5][6][7]. Unfortunately, working with rational functions except polynomials is generally not very easy.…”

Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes

“…They also proved that the higher-order Schwarzian derivatives of Aharonov and Tamanoi cannot be extended to holomorphic functions between projective Riemann surfaces unlike the classical Schwarzian derivatives. The higher order Schwarzian derivative are useful in the study of the properties of non-linear dynamical system and has been studied extensively by several researchers [17]. For many applications of the higher order Schwarzian derivatives related to the real functions, the reader may refer to [17,18] and the references cited therein.…”

“…The higher order Schwarzian derivative are useful in the study of the properties of non-linear dynamical system and has been studied extensively by several researchers [17]. For many applications of the higher order Schwarzian derivatives related to the real functions, the reader may refer to [17,18] and the references cited therein. Kwon and Sim [19], in 2017, using the theory of admissible functions investigated some sufficient conditions for normalized analytic functions to be star-like, associated with Tamanoi's Schwarzian derivative of third order.…”