Definition And Application Of Elliptic Functions Of An Irrational Nonlinear System

Irrational nonlinear systems often occur in both science and engineering, suchas irrational elastic term, oil-film force and so on. The method of harmonicbalance method (HBM), Hamiltonian approach (HA), parameter-expansion method(PEM) and Taylor’s expansion are applied to obtain the approximate solution ofthe irrational nonlinear system. However, the method of Taylor’s expansionavailable used to transform the original systems into local approximate polynomialsystems is not suitable sometimes. In order to investigate the original irrationalequation avoiding Taylor’s expansion, a series of irrational functions are definedto express the analytical solutions. By the application of the analytical solution,the curve of the chaotic threshold of the irrational nonlinear system can beobtained. And we can predict the border of chaos which may occur or disappear ofsome system parameters. These results can provide a reference for the furtherstudy. The main content of this article is as follows:Firstly, the equilibrium, Hamiltonian, potential energy and the phase structureof system are investigated for the unperturbed system. A parameter k isintroduced to link the energy and orbits together. So the first kind of irrationalelliptic function, the second kind of irrational elliptic function, the homoclinicorbit function and the third kind of irrational elliptic function of the dipteran flightoscillator are defined to express the analytical solution. Furthermore, some of thefundamental properties of the irrational functions are given in this article.The Melnikov function of the homoclinic orbit is defined on the basis of thehomoclinic orbit function which has been defined. A numerical method ispresented to compute the Melnikov function, and to get the curve of thresholdabove which chaos may occur. We can predict the border of chaos which mayoccur or disappear of the amplitude of external excitation, frequency and thedamping ratio.Numerical analysis is carried out to explore the influence of the parameters bythe fourth order Runge-Kutta method. And the bifurcation diagram, time historydiagram, phase diagram and the Poincare sections are given to depict the periodsolutions and chaotic solutions. These results not only verify the prediction butalso help us better understand the dynamical behaviours of the dipteran flight.