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Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay

  • Zaid AhsanEmail author
  • Harry Dankowicz
  • Jan Sieber
Original paper
  • 46 Downloads

Abstract

This paper generalizes a previously conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and quasiperiodic solutions of delay-differential equations. A Lagrange formalism is used to construct adjoint conditions that are linear and homogenous in the unknown Lagrange multipliers. As a consequence, it is shown how critical points on the constraint manifold can be found through several stages of continuation along a sequence of connected one-dimensional manifolds of solutions to increasing subsets of the necessary optimality conditions. Due to the presence of delayed and advanced arguments in the original and adjoint differential equations, care must be taken to determine the degree of smoothness of the Lagrange multipliers with respect to time. Such considerations naturally lead to a formulation in terms of multi-segment boundary-value problems (BVPs), including the possibility that the number of segments may change, or that their order may permute, during continuation. The methodology is illustrated using the software package coco on periodic orbits of both linear and nonlinear delay-differential equations, keeping in mind that closed-form solutions are not typically available even in the linear case. Finally, we demonstrate optimization on a family of quasiperiodic invariant tori in an example unfolding of a Hopf bifurcation with delay and parametric forcing. The quasiperiodic case is a further original contribution to the literature on optimization constrained by partial differential BVPs.

Keywords

Delay-differential equations Lagrange multipliers Adjoint equations Successive continuation 

Notes

Acknowledgements

We would like to thank Mingwu Li for insightful discussions during the course of this study.

Funding

J.S. gratefully acknowledges the financial support of the EPSRC via Grants EP/N023544/1 and EP/N014391/1, and from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 643073.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Calver, J., Enright, W.: Numerical methods for computing sensitivities for ODEs and DDEs. Numer. Algorithms 74(4), 1101–1117 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dankowicz, H., Schilder, F.: Recipes for Continuation. SIAM, Philadelphia (2013)CrossRefGoogle Scholar
  3. 3.
    Engelborghs, K., Doedel, E.J.: Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations. Numer. Math. 91(4), 627–648 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Göllmann, L., Kern, D., Maurer, H.: Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Opt. Control Appl. Methods 30(4), 341–365 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Heinkenschloss, M.: PDE Constrained Optimization. https://archive.siam.org/meetings/op08/Heinkenschloss.pdf. Accessed 25 Dec 2018
  6. 6.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Berlin (2008)zbMATHGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    Iglesias, A., Dombovari, Z., Gonzalez, G., Munoa, J., Stepan, G.: Optimum selection of variable pitch for chatter suppression in face milling operations. Materials 12(1), 112 (2018)CrossRefGoogle Scholar
  9. 9.
    Iglesias, A., Munoa, J., Ciurana, J.: Optimisation of face milling operations with structural chatter using a stability model based process planning methodology. Int. J. Adv. Manuf. Technol. 70(1–4), 559–571 (2014)CrossRefGoogle Scholar
  10. 10.
    Insperger, T., Stépán, G.: Semi-discretization method for delayed systems. Int. J. Numer. Methods Eng. 55(5), 503–518 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kernévez, J., Doedel, E.: Optimization in bifurcation problems using a continuation method. In: Küpper, T., Seydel, R., Troger, H. (eds.) Bifurcation: Analysis, Algorithms, Applications, pp. 153–160. Springer, Berlin (1987)CrossRefGoogle Scholar
  12. 12.
    Li, M., Dankowicz, H.: Staged construction of adjoints for constrained optimization of integro-differential boundary-value problems. SIAM J. Appl. Dyn. Syst. 17(2), 1117–1151 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Liao, H.: Nonlinear dynamics of duffing oscillator with time delayed term. Comput. Model. Eng. Sci. 103(3), 155–187 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Olikara, Z.P.: Computation of Quasi-Periodic Tori and Heteroclinic Connections in Astrodynamics Using Collocation Techniques. Ph.D. thesis, University of Colorado at Boulder (2016)Google Scholar
  15. 15.
    Orosz, G., Krauskopf, B., Wilson, R.E.: Bifurcations and multiple traffic jams in a car-following model with reaction-time delay. Physica D Nonlinear Phenom. 211(3–4), 277–293 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rubino, A., Pini, M., Colonna, P., Albring, T., Nimmagadda, S., Economon, T., Alonso, J.: Adjoint-based fluid dynamic design optimization in quasi-periodic unsteady flow problems using a harmonic balance method. J. Comput. Phys. 372, 220–235 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rusinek, R., Weremczuk, A., Kecik, K., Warminski, J.: Dynamics of a time delayed duffing oscillator. Int. J. Non-Linear Mech. 65, 98–106 (2014)CrossRefGoogle Scholar
  18. 18.
    Schilder, F., Dankowicz, H.: Continuation Core and Toolboxes (COCO). https://sourceforge.net/projects/cocotools. Accessed 13 Dec 2018
  19. 19.
    Sieber, J., Krauskopf, B.: Tracking oscillations in the presence of delay-induced essential instability. J. Sound Vib. 315(3), 781–795 (2008)CrossRefGoogle Scholar
  20. 20.
    Tlusty, J., Polacek, M.: The stability of machine tools against self-excited vibrations in machining. Int. Res. Prod. Eng. ASME 1, 465–474 (1963)Google Scholar
  21. 21.
    Wojciechowski, S., Maruda, R., Barrans, S., Nieslony, P., Krolczyk, G.: Optimisation of machining parameters during ball end milling of hardened steel with various surface inclinations. Meas. J. Int. Meas. Confed. 111, 18–28 (2017)CrossRefGoogle Scholar
  22. 22.
    Yusoff, A., Sims, N.: Optimisation of variable helix tool geometry for regenerative chatter mitigation. Int. J. Mach. Tools Manuf. 51(2), 133–141 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.College of Engineering, Mathematics and Physical SciencesUniversity of ExeterExeterUK

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