Towards a full higher order AD-based continuation and bifurcation framework. (English) Zbl 1453.65439

Summary: Some of the theoretical aspects of continuation and bifurcation methods devoted to the solution for nonlinear parametric systems are presented in a higher order automatic differentiation (HOAD) framework. Besides, benefits in terms of generality and ease of use, HOAD is used to assess fold and simple bifurcations points. In particular, the formation of a geometric series in successive Taylor coefficients allows for the implementation of an efficient detection and branch switching method at simple bifurcation points. Some comparisons with the Auto and MatCont continuation software are proposed. Strengths are then exemplified on a classical case study in structural mechanics.


65P30 Numerical bifurcation problems
70E60 Robot dynamics and control of rigid bodies
Full Text: DOI HAL


[1] Allgower, E. L.; Georg, K., Numerical Continuation Methods: An Introduction, (1990), Springer-Verlag, New York · Zbl 0717.65030
[2] Bilasse, M.; Charpentier, I.; Daya, E. M.; Koutsawa, Y., A generic approach for the solution of nonlinear residual equations. part II: homotopy and complex nonlinear eigenvalue method, Comput. Methods Appl. Mech. Engrg., 198, 3999-4004, (2009) · Zbl 1231.74133
[3] Charpentier, I., checkpointing schemes for adjoint codes: application to the meteorological model meso-NH, SIAM J. Sci. Comput., 22, 2135-2151, (2001) · Zbl 0989.86003
[4] Charpentier, I., on higher-order differentiation in nonlinear mechanics, Optim. Methods Softw., 27, 221-232, (2012) · Zbl 1242.41031
[5] Charpentier, I.; Gustedt, J., arbogast, higher-order AD for special functions with modular C, Optim. Methods Softw., (2018)
[6] Charpentier, I.; Lampoh, K., sensitivity computations in higher order continuation methods, Appl. Math. Model., 40, 3365-3380, (2016)
[7] Diamanlab—An interactive Taylor-based continuation tool in MATLAB, 2013, Archives ouvertes HAL. Available at
[8] Clauss, P.-N.; Gustedt, J., iterative computations with ordered Read-write locks, J. Parallel Distrib. Comput., 70, 496-504, (2010) · Zbl 1233.68048
[9] Cochelin, B., A path-following technique via an asymptotic-numerical method, Comput. Struct., 53, 1181-1192, (1994) · Zbl 0918.73337
[10] Cochelin, B.; Médale, 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)
[11] Cochelin, B.; Damil, N.; Potier-Ferry, M., Méthode Asymptotique Numérique, (2007), Hermes Science Publications, Paris
[12] Numerical computation of normal form coefficients of ODEs in Matlab, in Discrete and Continuous Dynamical Systems Supplements, Vol. I, American Institute of Mathematical Sciences (AIMS), Dresden, 2011, pp. 362–372. doi:
[13] Dhooge, A.; Govaerts, W.; Kuznetsov, Yu. A.; Meijer, H. G.E.; Sautois, B., new features of the software matcont for bifurcation analysis of dynamical systems, Math. Comput. Model Dyn. Syst., 14, 147-175, (2008) · Zbl 1158.34302
[14] Lecture notes on numerical analysis of nonlinear equations, in Numerical Continuation Methods for Dynamical Systems, B. Krauskopf, H. Osinga, and J. Galan-Vioque, eds., Springer, New York, 2007, pp. 1–49 · Zbl 1130.65119
[15] AUTO-07P: Continuation and bifurcation software for ordinary differential equations, 2012
[16] Doedel, E.; Keller, H. B.; Kernevez, J. P., numerical analysis and control of bifurcation problems (i) bifurcation in finite dimensions, Int. J. Bifurcat. Chaos, 1, 493-520, (1991) · Zbl 0876.65032
[17] Auto-07p, 2015. Available at
[18] Dyke, M. V., analysis and improvement of perturbation series, Q. J. Mech. Appl. Math., 27, 423-450, (1974) · Zbl 0295.65066
[19] The computation of disconnected bifurcation diagrams, preprint (2016). Available at
[20] Golubitsky, M.; Schaeffer, D., Singularities and Groups in Bifurcation Theory - Volume I, (1985), Springer, New-York · Zbl 0607.35004
[21] Golubitsky, M.; Stewart, M.; Schaeffer, D., Singularities and Groups in Bifurcation Theory - Volume II, (1988), Springer, New-York · Zbl 0691.58003
[22] Matcont: Numerical bifurcation analysis toolbox in matlab, 2013. Available at
[23] Griewank, A.; Walther, A., algorithm 799: revolve: an implementation of checkpoint for the reverse or adjoint mode of computational differentiation, ACM Trans. Math. Softw., 26, 19-45, (2000) · Zbl 1137.65330
[24] Guckenheimer, J.; Meloon, B., computing periodic orbits and their bifurcations with automatic differentiation, SIAM J. Sci. Comput., 22, 951-985, (2000) · Zbl 0976.65111
[25] Hentz, G.; Charpentier, I.; Renaud, P., higher-order continuation for the determination of robot workspace boundaries, C. R. Mecanique, 344, 95-101, (2016)
[26] Keller, H., Lectures on Numerical Methods in Bifurcation Problems, (1987), Springer, Berlin
[27] Kernevez, J.-P., Enzyme mathematics, Studies in mathematics and its applications, 10, (1980), Amsterdam, North-Holland
[28] Tutorial II: One-parameter bifurcation analysis of equilibria with matcont, 2011
[29] Seydel, R., numerical computation of branch points in nonlinear equations, Numer. Math., 33, 339-352, (1979) · Zbl 0396.65023
[30] Seydel, R., Practical Bifurcation and Stability Analysis, 5, (2009), Springer, New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.