Article Contents
Article Contents

# Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method

• * Corresponding author: danko@illinois.edu
• This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the COCO software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example COCO code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation. 200 words.

Mathematics Subject Classification: Primary: 37C55, 37M21; Secondary: 49K40, 37E10, 37E45.

 Citation:

• Figure 1.  Illustration of sensitivity analysis along an invariant curve: (top left) Bifurcation diagram, (bottom left) spectrum of $\Gamma_\rho$ and $\hat{\Gamma}_\rho$, (middle column) phase plots of invariant curves at points labeled A and B, and tangents to stable fibers at selected points, (right column) differences between trajectories of initial conditions near a point on the invariant curve (black dot) and forward iterates of the corresponding stable fiber projection

Figure 2.  At each stage of construction of the two-segment periodic orbit problem in (112)-(114), one obtains an extended continuation problem (left column) in terms of a subset of the complete vector of continuation variables and an associated set of adjoint conditions (right column) in terms of a subset of the complete vector of adjoint variables

Figure 3.  A logical matrix representation of the variable dependencies of zero and monitor functions introduced at different stages of problem construction. At the stage of construction illustrated in the figure, two zero/monitor functions are added that depend on three newly introduced continuation variables. Each function also depends on previously introduced continuation variables identified by ones in the corresponding row

Figure 4.  Fields of a function data structure may be used to store context-independent integer indices for subsets of continuation variables associated with a newly constructed function for later reference by subsequent constructor calls. In the figure, the fields $\mathtt{{fcndata.x1_idx}}$ and $\mathtt{{fcndata.x2_idx}}$ reference constructor-dependent details that are unknown to COCO and independent of the order of construction

Figure 5.  A logical matrix representation of the dependencies of the adjoint conditions on adjoint variables introduced at different stages of problem construction. At the stage of construction illustrated in the figure, three adjoint conditions are added that include terms linear in two newly introduced adjoint variables. Each variable also appears in contributions to previously introduced adjoint conditions identified by ones in the corresponding rowspan="$[0[]0]" •  [1] Z. Ahsan, H. Dankowicz, M. Li and J. Sieber, Methods of continuation and their implementation in the coco software platform with application to delay differential equations, Nonlinear Dynamics, 107 (2022), 3181-3243. doi: 10.1007/s11071-021-06841-1. [2] Z. Ahsan, H. Dankowicz and J. Sieber, Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay, Nonlinear Dynamics, 99 (2020), 837-854. doi: 10.1007/s11071-019-05304-y. [3] M. Bernardo, C. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, 163. Springer, London, 2008. [4] X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds Ⅰ : Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328. doi: 10.1512/iumj.2003.52.2245. [5] X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds Ⅱ : Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), 329-360. doi: 10.1512/iumj.2003.52.2407. [6] X. Cabré, E. Fontich and R. De La Llave, The parameterization method for invariant manifolds Ⅲ : overview and applications, J. Differential Equations, 218 (2005), 444-515. doi: 10.1016/j.jde.2004.12.003. [7] S. Corner, A. Sandu and C. Sandu, Adjoint sensitivity analysis of hybrid multibody dynamical systems, Multibody Syst. Dyn., 49 (2020), 395-420. doi: 10.1007/s11044-020-09726-0. [8] H. Dankowicz and F. Schilder, Recipes for Continuation, SIAM, 2013. doi: 10.1137/1.9781611972573. [9] R. de la Llave, A tutorial on KAM theory, In Smooth Ergodic Theory and Its Applications, Proceedings of Symposia in Pure Mathematics, 69 (2001), 175–292. doi: 10.1090/pspum/069/1858536. [10] A. Demir, C. Gu and J. Roychowdhury, Phase equations for quasi-periodic oscillators, In 2010 IEEE/ACM International Conference on Computer-Aided Design (ICCAD), (2010), 292–297. [11] W. Govaerts and B. Sautois, Computation of the phase response curve: A direct numerical approach, Neural Comput., 18 (2006), 817-847. doi: 10.1162/neco.2006.18.4.817. [12] Á. Haro, M. Canadell, J.-L. Figueras, A. Luque and J.-M. Mondelo, The Parameterization Method for Invariant Manifolds, Applied Mathematical Sciences, 195. Springer, Cham, 2016. doi: 10.1007/978-3-319-29662-3. [13] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583. Springer-Verlag, Berlin, 1977. [14] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}\$ edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [15] M. Li and H. Dankowicz, Staged construction of adjoints for constrained optimization of integro-differential boundary-value problems, SIAM J. Appl. Dyn. Syst., 17 (2018), 1117-1151.  doi: 10.1137/17M1143563. [16] M. Li and H. Dankowicz, Optimization with equality and inequality constraints using parameter continuation, Appl. Math. Comput., 375 (2020), 125058, 20 pp. doi: 10.1016/j.amc.2020.125058. [17] J. Moser, On the theory of quasiperiodic motions, SIAM Rev., 8 (1966), 145-172. [18] V. Novičenko and K. Pyragas, Phase reduction of weakly perturbed limit cycle oscillations in time-delay systems, Phys. D: Nonlinear Phenomena, 241 (2012), 1090-1098.  doi: 10.1016/j.physd.2012.03.001. [19] Y. Park, K. M. Shaw, H. J. Chiel and P. J. Thomas, The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems, European J. Appl. Math., 29 (2018), 905-940.  doi: 10.1017/S0956792518000128. [20] A. Rubino, M. Pini, P. Colonna, T. Albring, S. Nimmagadda, T. Economon and J. Alonso, Adjoint-based fluid dynamic design optimization in quasi-periodic unsteady flow problems using a harmonic balance method, J. Compu. Phys., 372 (2018), 220-235.  doi: 10.1016/j.jcp.2018.06.023. [21] F. Schilder, H. Dankowicz and M. Li, Continuation Core and Toolboxes (COCO), https://sourceforge.net/projects/cocotools, Accessed: 2022-02-26. [22] S. Shirasaka, W. Kurebayashi and H. Nakao, Phase reduction theory for hybrid nonlinear oscillators, Physical Review E, 95 (2017), 012212.  doi: 10.1103/PhysRevE.95.012212. [23] R. Szalai and H. M. Osinga, Arnol'd tongues arising from a grazing-sliding bifurcation, SIAM J. Appl. Dyn. Syst., 8 (2009), 1434-1461.  doi: 10.1137/09076235X. [24] D. A. Tortorelli and P. Michaleris, Design sensitivity analysis: Overview and review, Inverse Problems in Engineering, 1 (1994), 71-105.  doi: 10.1080/174159794088027573. [25] T. Traverso and L. Magri, Data assimilation in a nonlinear time-delayed dynamical system with Lagrangian optimization, Computational science—ICCS 2019, Lecture Notes in Comput. Sci., 11539 (2019), 156–168. doi: 10.1007/978-3-030-22747-0.

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