Determination of periodic orbits of nonlinear oscillators by means of piecewise-linearization methods

Abstract

An approximate method based on piecewise linearization is developed for the determination of periodic orbits of nonlinear oscillators. The method is based on Taylor series expansions, provides piecewise analytical solutions in three-point intervals which are continuous everywhere and explicit three-point difference equations which are P-stable and have an infinite interval of periodicity. It is shown that the method presented here reduces to the well-known Stor- mer technique, is second-order accurate, and yields, upon applying Taylor series expansion and a Padeapproximation, another P-stable technique whenever the Jacobian is different from zero. The method is generalized for single degree- of-freedom problems that contain the velocity, and (approximate) analytical solutions are presented. Finally, by intro- ducing the inverse of a vector and the vector product and quotient, and using Taylor series expansions and a Pade ´ approximation, the method has been generalized to multiple degree-of-freedom problems and results in explicit three-point finite difference equations which only involve vector multiplications. � 2005 Elsevier Ltd. All rights reserved.