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A Smooth and Discontinuous OscillatorPiecewise Linear Approach

A Smooth and Discontinuous Oscillator: Piecewise Linear Approach [This chapter and the following ones introduce a sequence of methodologies for analytical and semi-analytical computations of the SD oscillator whose nonlinearity is of irrational type. We introduce a triple linear procedure in this chapter depending on the property of the nonlinearity of the SD oscillator of the irrational type. And then in the following chapters we will introduce an extended averaging method in Chap. 8, irrational elliptic functions method in Chap. 9 and the so-called cell-mapping method in Chap. 10. These methodologies will be valid for both smooth and discontinuous cases. All the methods which will be presented in the subsequent chapters aim at overcoming or removing the barriers of the conventional methodologies. For example an analytical analysis based upon the polynomial theory is invalid for this irrational nonlinearity and the Runge–Kutta direct integration method is no more applicable for the discontinuous case.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Smooth and Discontinuous OscillatorPiecewise Linear Approach

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Publisher
Springer Berlin Heidelberg
Copyright
© Springer-Verlag Berlin Heidelberg 2017
ISBN
978-3-662-53092-4
Pages
89 –102
DOI
10.1007/978-3-662-53094-8_7
Publisher site
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Abstract

[This chapter and the following ones introduce a sequence of methodologies for analytical and semi-analytical computations of the SD oscillator whose nonlinearity is of irrational type. We introduce a triple linear procedure in this chapter depending on the property of the nonlinearity of the SD oscillator of the irrational type. And then in the following chapters we will introduce an extended averaging method in Chap. 8, irrational elliptic functions method in Chap. 9 and the so-called cell-mapping method in Chap. 10. These methodologies will be valid for both smooth and discontinuous cases. All the methods which will be presented in the subsequent chapters aim at overcoming or removing the barriers of the conventional methodologies. For example an analytical analysis based upon the polynomial theory is invalid for this irrational nonlinearity and the Runge–Kutta direct integration method is no more applicable for the discontinuous case.]

Published: Sep 28, 2016

Keywords: Lyapunov Exponent; Chaotic Attractor; Homoclinic Orbit; Duffing Oscillator; Melnikov Function

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