TITLE:
The Extended Non-Elementary Amplitude Functions as Solutions to the Forced and Damped Pendulum Equation, Ueda’s Chaotic Nonlinear Oscillator, the Shimizu-Morioka System, Lorenz System, Rössler System, Sprott-Linz F Chaotic Attractor
AUTHORS:
Magne Stensland
KEYWORDS:
The Forced Pendulum Equation, Ueda’s Chaotic Oscillator, Shimizu-Morioka System, Lorenz System, Rössler System, Sprott-Linz F Chaotic Attractor
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.4,
April
24,
2025
ABSTRACT: In this paper, we define some non-elementary amplitude functions that are giving solutions to some second-order nonlinear ODEs with forcing term and systems of ODEs with chaotic behavior, such as the chaotic cases of the Lorenz system. For this purpose, we will introduce a special function, that is a function of the dependent variable
φ
and the independent variable t, and place it into the solution-function. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. The first derivative to these amplitude functions contains one or two integrals that disappear at the second derivation or at the third derivation. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. Using these integral amplitude functions, we can define solutions to some well-known second-order ODEs and systems of ODEs exhibiting chaotic behavior.