Continuation of periodic solutions of various types of delay differential equations using asymptotic numerical method and harmonic balance method

Abstract

This article presents an extension of the asymptotic numerical method combined with the harmonic balance method to the continuation of periodic orbits of delay differential equations. The equations can be forced or autonomous and possibly of neutral type. The approach developed in this paper requires the system of equations to be written in a quadratic formalism which is detailed. The method is applied to various systems, from Van der Pol and Duffing oscillators to toy models of clarinet and saxophone. The harmonic balance method is ascertained from a comparison to standards time-integration solvers. Bifurcation diagrams are drawn which are sometimes intricate, showing the robustness of this method.

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References

  1. 1.

    Atay, F.M.: Van der Pol’s oscillator under delayed feedback. J. Sound Vib. 218(2), 333–339 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Barton, D.A., Krauskopf, B., Wilson, R.E.: Collocation schemes for periodic solutions of neutral delay differential equations. J. Differ. Equ. Appl. 12(11), 1087–1101 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Charpentier, I., Cochelin, B.: Towards a full higher order AD-based continuation and bifurcation framework. Optim. Methods Softw. 33, 945–962 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Cochelin, B.: A path-following technique via an asymptotic-numerical method. Comput. Struct. 53(5), 1181–1192 (1994)

    Article  MATH  Google Scholar 

  5. 5.

    Cochelin, B., Damil, N., Potier-Ferry, M.: Asymptotic-numerical methods and pade approximants for non-linear elastic structures. Int. J. Numer. Methods Eng. 37(7), 1187–1213 (1994)

    Article  MATH  Google Scholar 

  6. 6.

    Cochelin, B., Damil, N., Potier-Ferry, M.: The asymptotic-numerical method: an efficient perturbation technique for nonlinear structural mechanics. Revue européenne des éléments finis 3(2), 281–297 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Cochelin, B., Medale, M.: Power series analysis as a major breakthrough to improve the efficiency of asymptotic numerical method in the vicinity of bifurcations. J. Comput. Phys. 236, 594–607 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Cochelin, B., Vergez, C.: A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. J. Sound Vib. 324(1), 243–262 (2009)

    Article  Google Scholar 

  9. 9.

    Colinot, T., Kergomard, J.: Formulation analytique de la pression interne d’un modèle idéalisé d’instrument conique à anche. In: CFA 2018/VISHNO (2018)

  10. 10.

    Cottanceau, E., Thomas, O., Véron, P., Alochet, M., Deligny, R.: A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables. Finite Elem. Anal. Des. 139, 14–34 (2018)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dalmont, J.P., Gilbert, J., Kergomard, J.: Reed instruments, from small to large amplitude periodic oscillations and the Helmholtz motion analogy. Acta Acust. United Acust. 86(4), 671–684 (2000)

    Google Scholar 

  12. 12.

    Engelborghs, K., Luzyanina, T., Roose, D.: Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. (TOMS) 28(1), 1–21 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Fourier, J.: Theorie analytique de la chaleur, par M. Fourier. Chez Firmin Didot, père et fils (1822)

  14. 14.

    Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, vol. 105. SIAM, Philadelphia (2008)

    Google Scholar 

  15. 15.

    Guillot, L., Cochelin, B., Vergez, C.: A generic and efficient taylor series based continuation method using a quadratic recast of smooth nonlinear systems. Int. J. Numer. Methods Eng. (2019). https://doi.org/10.1002/nme.6049

    Google Scholar 

  16. 16.

    Guillot, L., Vigué, P., Vergez, C., Cochelin, B.: Continuation of quasi-periodic solutions with two-frequency harmonic balance method. J. Sound Vib. 394, 434–450 (2017)

    Article  Google Scholar 

  17. 17.

    Hu, H., Dowell, E.H., Virgin, L.N.: Resonances of a harmonically forced duffing oscillator with time delay state feedback. Nonlinear Dyn. 15(4), 311–327 (1998)

    Article  MATH  Google Scholar 

  18. 18.

    Karkar, S., Cochelin, B., Vergez, C.: A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: the case of non-polynomial nonlinearities. J. Sound Vib. 332(4), 968–977 (2013)

    Article  Google Scholar 

  19. 19.

    Kergomard, J., Guillemain, P., Silva, F., Karkar, S.: Idealized digital models for conical reed instruments, with focus on the internal pressure waveform. J. Acoust. Soc. Am. 139(2), 927–937 (2016)

    Article  Google Scholar 

  20. 20.

    Krylov, N.M., Bogoliubov, N.N.: Introduction to Non-linear Mechanics. Princeton University Press, Princeton (1949)

    Google Scholar 

  21. 21.

    Liu, L., Kalmár-Nagy, T.: High-dimensional harmonic balance analysis for second-order delay-differential equations. J. Vib. Control 16(7–8), 1189–1208 (2010)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Nakhla, M., Vlach, J.: A piecewise harmonic balance technique for determination of periodic response of nonlinear systems. IEEE Trans. Circuits Syst. 23(2), 85–91 (1976)

    Article  Google Scholar 

  23. 23.

    Seydel, R.: From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis. North-Holland, Amsterdam (1988)

    Google Scholar 

  24. 24.

    Shampine, L.F.: Dissipative approximations to neutral DDEs. Appl. Math. Comput. 203(2), 641–648 (2008)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Shampine, L.F., Thompson, S.: Solving DDEs in Matlab. Appl. Numer. Math. 37(4), 441–458 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Taillard, P.A., Kergomard, J., Laloë, F.: Iterated maps for clarinet-like systems. Nonlinear Dyn. 62(1–2), 253–271 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Vigué, P., Vergez, C., Karkar, S., Cochelin, B.: Regularized friction and continuation: comparison with Coulomb’s law. J. Sound Vib. 389, 350–363 (2017)

    Article  Google Scholar 

  28. 28.

    Vigué, P., Vergez, C., Lombard, B., Cochelin, B.: Continuation of periodic solutions for systems with fractional derivatives. Nonlinear Dyn. 95, 1–15 (2018)

    Google Scholar 

  29. 29.

    Zahrouni, H., Cochelin, B., Potier-Ferry, M.: Computing finite rotations of shells by an asymptotic-numerical method. Comput. Methods Appl. Mech. Eng. 175(1–2), 71–85 (1999)

    Article  MATH  Google Scholar 

  30. 30.

    Zahrouni, H., Potier-Ferry, M., Elasmar, H., Damil, N.: Asymptotic numerical method for nonlinear constitutive laws. Revue européenne des éléments finis 7(7), 841–869 (1998)

    Article  MATH  Google Scholar 

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Acknowledgements

The authors want to thank Tom Colinot for the help with the saxophone model and its associated time-integration method. This work has been carried out in the framework of the Labex MEC (ANR-10-LABX-0092) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the Investissements d’Avenir French Government program managed by the French National Research Agency (ANR). Conflict of Interest: The authors declare that they have no conflict of interest.

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Correspondence to Louis Guillot.

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Guillot, L., Vergez, C. & Cochelin, B. Continuation of periodic solutions of various types of delay differential equations using asymptotic numerical method and harmonic balance method. Nonlinear Dyn 97, 123–134 (2019). https://doi.org/10.1007/s11071-019-04958-y

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Keywords

  • Nonlinear dynamics
  • Delay equations
  • Harmonic balance method
  • Numerical continuation
  • Periodic solutions
  • Quadratic recast
  • Asymptotic numerical method