Implementation of Hopf and double-Hopf continuation using bordering methods

ABSTRACT

We discuss the computational study of curves of Hopf and double-Hopf points in the software package CONTENT developed at CWI, Amsterdam. These are important points in the numerical study of dynamical systems characterized by the occurrence of one or two conjugate pairs of pure imaginary eigenvalues in the spectrum of the Jacobian matrix. The bialternate product of matrices is extensively used in three codes for the numerical continuation of curves of Hopf points and in one for the continuation of curves of double-Hopf points. In the double-Hopf and two of the single-Hopf cases this is combined with a bordered matrix method. We use this software to find special points on a Hopf curve in a model of chemical oscillations and by computing a Hopf and a double-Hopf curve in a realistic model of a neuron.

REFERENCES

• 1: BYKOV, V. I., YABLONSKII, G. S., AND KIM, V. F. 1978. On the simple model of kinetic self-oscillations in catalitic reaction of CO oxidation. Dokl. AN SSSR 242, 3, 637-639. In Russian.
• 2: K.-W. E. Chu , W. Govaerts , A. Spence, Matrices with rank deficiency two in eigenvalue problems and dynamical systems, SIAM Journal on Numerical Analysis, v.31 n.2, p.524-539, April 1994 [doi>10.1137/0731028]
• 3: 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, Canada.
• 4: GOVAERTS, W. AND PRYCE, J.D. 1989. A singular value inequality for block matrices. Lin. Alg. Appl. 125, 141-148.
• 5: W. Govaerts , J. Guckenheimer , A. Khibnik, Defining Functions for Multiple Hopf Bifurcations, SIAM Journal on Numerical Analysis, v.34 n.3, p.1269-1288, June 1997 [doi>10.1137/S0036142995282182]
• 6: John Guckenheimer , Mark Myers , Bernd Sturmfels, Computing Hopf Bifurcations I, SIAM Journal on Numerical Analysis, v.34 n.1, p.1-21, Feb. 1997 [doi>10.1137/S0036142993253461]
• 7: HOLODNIOK, M., AND KUBICEK, M. 1984. New algorithms for the evaluation of complex bifurcation points in ordinary differential equations. A comparative numerical study. Appl. Math. Comput. 15, 261-274.
• 8: KHIBNIK, A. I., BYKOV, V. I., AND YABLONSKII, G.S. 1987. 23 phase portraits of the simplest catalitic oscillator. J. Fiz. Khim. 61, 1388-1390. In Russian.
• 9: KHIBNIK, A. I., KUZNETSOV, YU. A., LEVITIN, V. V., AND NIKOLAEV, E.V. 1993. Interactive LO Cal BIFurcation Analyzer: LOCBIF, v.2. Computer Algebra Nederland Expertise Center, Amsterdam, The Netherlands.
• 10: Yuri A. Kuznetsov, Elements of applied bifurcation theory (2nd ed.), Springer-Verlag New York, Inc., New York, NY, 1998
• 11: KUZNETSOV, YU. A., AND LEVITIN, V.V. 1997. CONTENT: A multiplatform environment for analyzing dynamical systems. Dynamical Systems Laboratory, Centrum voor Wiskunde en Informatica. Software under development.
• 12: KUZNETSOV, YU. A., LEVITIN, V. V., AND SKOVORODA, A.R. 1996. Continuation of stationary solutions to evolution problems in CONTENT. Rep. AM-R9611, Centrum voor Wiskunde en Informatica, Amsterdam.
• 13: LEVITIN, V.V. 1995. Computation of functions and their derivatives in CONTENT. Rep. AM-R9512, Centrum voor Wiskunde en Informatica, Amsterdam.
• 14: ROOSE, D. AND HLAVACEK, V. 1985. A direct method for the computation of Hopf bifurcation points. SIAM J. Appl. Math. 45, 879-894.
• 15: Bodo Werner, Computation of Hopf bifurcation with bordered matrices, SIAM Journal on Numerical Analysis, v.33 n.2, p.435-455, April 1996 [doi>10.1137/0733023]
• 16: STEPHANOS, C. 1990. Sur une extension du calcul des substitutions lin~aires. Journal de Math~matiques pures et Appliqu~es 6, 73-128.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.5 Roots of Nonlinear Equations
          Subjects: Continuation (homotopy) methods

General Terms:
Algorithms, Design

Keywords:
CONTENT, bialternate product, dynamical system