Abstract
In this paper, we propose a path-following method for computing a curve of equilibria of a dynamical system, based upon the smooth Singular Value Decomposition (SVD) of the Jacobian matrix. Our method is capable of detecting fold points, and continuing past folds. It is also able to detect branch points and to switch branches at such points. Algorithmic details and examples are given.
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Allgower, E., Georg, K.: Numerical Continuation Methods. Springer, New York (1990)
Beyn, W.J., Kleß, W., Thümmler, V.: Continuation of low-dimensional invariant subspaces in dynamical systems of large dimension. In: Fiedler, B. (ed.) Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems, pp. 47–72. Springer, Heidelberg (2001)
Bindel, D., Demmel, J., Friedman, M.: Continuation of Invariant Subspaces for Large Bifurcation Problems. In: Proceedings of the SIAM Conference on Applied Linear Algebra. The College of William and Mary, Williamsburg, VA (2003), http://www.siam.org/meetings/la03/proceedings
Bunse-Gerstner, A., Byers, R., Mehrmann, V., Nichols, N.K.: Numerical Computation of an Analytic Singular Value Decomposition by a Matrix Valued Function. Numer. Math. 60, 1–40 (1991)
Chow, S., Shen, Y.: Bifurcations via Singular Value Decompositions. Appl. Math. Comput. 28, 231–245 (1988)
Dieci, L., Eirola, T.: On Smooth Decomposition of Matrices. SIAM J. Matrix. Anal. Appl. 20, 800–819 (1999)
Dieci, L., Gasparo, M.G., Papini, A.: Continuation of Singular Value Decompositions. Mediterr. j. math. 2, 179–203 (2005)
Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: A MATLAB Package for Numerical Bifurcation Analysis of ODE. ACM Trans. on Math. Software 29, 141–164 (2003)
Doedel, E.J., et al.: AUTO 1997: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), User’s Guide. Concordia University, Montreal, P.Q, Canada (1997), http://indy.cs.concordia.ca
Keller, H.B.: Numerical Methods in Bifurcation Problems, Tata Institute of Fundamental Research, Bombay. Springer, Heidelberg (1987)
Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, Heidelberg (1995)
Mehrmann, V., Rath, W.: Numerical Methods for the Computation of Analytic Singular Value Decompositions. Electron. Trans. Numer. Anal. 1, 72–88 (1993)
Rheinboldt, W.C.: Numerical Analysis of Parametrized Nonlinear Equations. Wiley, New York (1986)
Rheinboldt, W.C., Burkardt, J.V.: A locally parametrizied continuation process. ACM Trans. on Math. Software 9, 215–235 (1983)
Watson, L.T., Sosonkina, M., Melville, R.C., Morgan, A.P., Walker, H.F.: Algorithm 777: HOMPACK 90: a suite of Fortran 90 codes for globally convergent homotopy algorithms. ACM Trans. Math. Software 23, 514–549 (1997)
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Dieci, L., Gasparo, M.G., Papini, A. (2006). Path Following by SVD. In: Alexandrov, V.N., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds) Computational Science – ICCS 2006. ICCS 2006. Lecture Notes in Computer Science, vol 3994. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11758549_92
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DOI: https://doi.org/10.1007/11758549_92
Publisher Name: Springer, Berlin, Heidelberg
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