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

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  • Ramos, J.I.

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 Störmer technique, is second-order accurate, and yields, upon applying Taylor series expansion and a Padé approximation, 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 introducing the inverse of a vector and the vector product and quotient, and using Taylor series expansions and a Padé 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.

Suggested Citation

  • Ramos, J.I., 2006. "Determination of periodic orbits of nonlinear oscillators by means of piecewise-linearization methods," Chaos, Solitons & Fractals, Elsevier, vol. 28(5), pages 1306-1313.
    • Handle: RePEc:eee:chsofr:v:28:y:2006:i:5:p:1306-1313
    DOI: 10.1016/j.chaos.2005.08.203
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    References listed on IDEAS

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    1. Ramos, J.I., 2006. "Piecewise-linearized methods for oscillators with limit cycles," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1229-1238.
    2. Starkov, Konstantin E., 2005. "Localization of periodic orbits of polynomial vector fields of even degree by linear functions," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 621-627.
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    Cited by:

    1. Barrio, Roberto & Blesa, Fernando, 2009. "Systematic search of symmetric periodic orbits in 2DOF Hamiltonian systems," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 560-582.
    2. Amaral, Gleison F.V. & Nepomuceno, Erivelton G., 2018. "A smooth-piecewise model to the Cord Attractor," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 31-35.

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