Abstract
MATCONT is a graphical MATLAB software package for the interactive numerical study of dynamical systems. It allows one to compute curves of equilibria, limit points, Hopf points, limit cycles, period doubling bifurcation points of limit cycles, and fold bifurcation points of limit cycles. All curves are computed by the same function that implements a prediction-correction continuation algorithm based on the Moore-Penrose matrix pseudo-inverse. The continuation of bifurcation points of equilibria and limit cycles is based on bordering methods and minimally extended systems. Hence no additional unknowns such as singular vectors and eigenvectors are used and no artificial sparsity in the systems is created. The sparsity of the discretized systems for the computation of limit cycles and their bifurcation points is exploited by using the standard Matlab sparse matrix methods. The MATLAB environment makes the standard MATLAB Ordinary Differential Equations (ODE) Suite interactively available and provides computational and visualization tools; it also eliminates the compilation stage and so makes installation straightforward. Compared to other packages such as AUTO and CONTENT, adding a new type of curves is easy in the MATLAB environment. We illustrate this by a detailed description of the limit point curve type.
- Allgower, E. L. and Georg, K. 1996. Numerical path following. In Handbook of Numerical Analysis 5, P. G. Ciarlet and J. L. Lions, Eds. North-Holland, Amsterdam, The Neatherlands.Google Scholar
- Arnold, D. and Polking, J. C. 1999. Ordinary Differential Equations using MATLAB, 2nd ed. Prentice-Hall, Englewood Cliffs, NJ. Google Scholar
- Ascher, U. M., Christiansen, J., and Russell, R. D. 1979. A collocation solver for mixed order systems of boundary value problems. Math. Comp. 33, 146, 659--679.Google Scholar
- Back, A., Guckenheimer, J., Myers, M. R., Wicklin, F. J., and Worfolk, P. A. 1992. Dstool: Computer assisted exploration of dynamical systems. Notices Amer. Math. Soc. 39, April, 303--309.Google Scholar
- Beyn, W.-J., Champneys, A., Doedel, E., Govaerts, W., Kuznetsov, Yu. A., and Sandstede, B. 2002. Numerical continuation, and computation of normal forms. In Handbook of Dynamical Systems, Vol. II, B. Fiedler, ed. Elsevier, Amsterdam, The Netherlands, 149--219.Google Scholar
- Choe, W. G. and Guckenheimer, J. 2000. Using dynamical system tools in MATLAB. In Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, IMA Vol. 119, E. J. Doedel and L. S. Tuckerman, Eds. Springer, New York, NY. 85--113.Google Scholar
- De Boor, C. and Swartz, B. 1973. Collocation at Gaussian points. SIAM J. Numer. Anal. 10, 4, 582--606.Google Scholar
- De Feo, O. 2000. MPLAUT: A MATLAB visualization software for AUTO97. EPFL, Lausanne, Switzerland. Available at http://www.math.uu.nl/people/kuznet/cm.Google Scholar
- Dhooge, A., Govaerts, W., Kuznetsov, Yu. A., Mestrom, W., and Riet, A. 2000--2002. CL_MATCONT: A Continuation Toolbox in MATLAB. In Proceedings of the 2003 ACM Symposium on Applied Computing (Melbourne, FL), 161--166. Software available at: http://www.math.uu.nl/people/kuznet/cm. Google Scholar
- Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Yu. A., Sandstede, B., and Wang, X. J. 1997. AUTO97: Continuation and bifurcation software for ordinary differential equations (with HomCont), user's guide. Concordia University, Montreal, P.Q., Canada. Available at http://indy.cs.concordia.ca.Google Scholar
- Doedel, E., Govaerts, W., and Kuznetsov, Yu. A. 2003. Computation of periodic solution bifurcations in ODEs using bordered systems. SIAM J. Numer. Anal. To appear. Google Scholar
- Doedel, E. J., Keller, H. B., and Kernévez, J. P. 1991. Numerical analysis and control of bifurcation problems I : Bifurcation in finite dimensions. Int. J. Bifurc. Chaos 1, 3, 493--520.Google Scholar
- Engelborghs, K., Luzyanina, T., and Roose, D. 2002. Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. 28, 1, 1--21. Google Scholar
- Ermentrout, B. 2002. Simulating, Analyzing and Animating Dynamical Systems, a Guide to XPPAUT for Researchers and Students. SIAM Publications, Philadelphia, PA. Google Scholar
- Govaerts, W., Kuznetsov, Yu. A., and Sijnave, B. 1998. Implementation of Hopf and double Hopf continuation using bordering methods. ACM Trans. Math. Softw. 24, 4, 418--436. Google Scholar
- Govaerts, W. 2000. Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM Publications, Philadelphia, PA. Google Scholar
- Guckenheimer, J. and Holmes, P. 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences 42. Springer-Verlag, New York, NY.Google Scholar
- Henderson, M. 2002. Multiple parameter continuation: Computing implicitly defined k-manifolds. Int. J. Bifurcation Chaos 12, 3 451--476.Google Scholar
- Keller, H. B. 1977. Numerical solution of bifurcation and nonlinear eigenvalue problems. Applications of Bifurcation Theory. Academic Press, New York, NY., 359--384.Google Scholar
- Khibnik, A. I., Kuznetsov, Yu. A., Levitin, V. V., and Nikolaev, E. V. 1993. Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps. Physica D 62, 1--4, 360--371. Google Scholar
- Kuznetsov, Yu. A. 1995/1998. Elements of Applied Bifurcation Theory, Applied Mathematical Sciences 112. Springer-Verlag, New York, NY. Google Scholar
- Kuznetsov, Yu. A. and Levitin, V. V. 1995--1997. content: A multiplatform environment for analyzing dynamical systems. Dynamical Systems Laboratory, CWI, Amsterdam, The Netherlands. Available at ftp.cwi.nl/pub/CONTENT.Google Scholar
- Mestrom, W. 2002. Continuation of limit cycles in MATLAB. Master's thesis. Mathematical Institute, Utrecht University, Utrecht, The Netherlands.Google Scholar
- Polking, J. C. 1997--2003. dfield and pplane software. Available at http://math.rice.edu/∼dfield.Google Scholar
- Riet, A. 2000. A continuation toolbox in MATLAB. Master's thesis. Mathematical Institute, Utrecht University, Utrecht, The Netherlands.Google Scholar
- Russell, R. D. and Christiansen, J. 1978. Adaptive mesh selection strategies for solving boundary value problems. SIAM J. Numer. Anal. 15, 1, 59--80.Google Scholar
- Shampine, L. F. and Reichelt, M. W. 1997. The MATLAB ODE suite. SIAM J. Sci. Compt. 18, 1, 1--22. Google Scholar
Index Terms
- MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs
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