Abstract
The behavior of center-indices, as introduced by J. Mallet-Paret & J. Yorke, is analyzed for two-parameter flows. The integer sum of center-indices along a one-dimensional curve in parameter space is called the H-index. A nonzero H-index implies global Hopf bifurcation. The index H is not a homotopy invariant. This fact is due to the occurrence of stationary points with an algebraically double eigenvalue zero, which we call B-points. To each B-point we assign an integer B-index, such that the H-index relates to the B-indices by a formula such as occurs in the calculus of residues.
This formula is easily applied to study global bifurcation of periodic solutions in diffusively coupled two-cells of chemical oscillators and to treat spatially heterogeneous time-periodic oscillations in porous catalysts.
Similar content being viewed by others
References
J. C. Alexander: Spontaneous oscillations in two 2-component cells coupled by diffusion. Preprint.
J. C. Alexander & J. A. Yorke: Global bifurcations of periodic orbits, Amer. J. Math. 100 (1978), 263–292.
K. T. Alligood, J. Mallet-Paret, & J. A. Yorke: Families of periodic orbits: local continuability does not imply global continuability, J. Diff. Geom. 16 (1981), 483–492.
K. T. Alligood, J. Mallet-Paret, & J. A. Yorke: An index for the continuation of relatively isolated sets of periodic orbits, in [41], 1–21.
K. T. Alligood & J. A. Yorke: Families of periodic orbits: virtual periods and global continuability, J. Diff. Eq. 55 (1984), 59–71.
W. Alt: Some periodicity criteria for functional differential equations, Man. Math. 23 (1978), 295–318.
R. Aris: The mathematical theory of diffusion and reaction in permeable catalysts I, II, Clarendon Press, Oxford 1975.
V. I. Arnold: Lectures on bifurcations and versal systems, Russ. Math. Surveys 27, 5 (1972), 54–123.
V. I. Arnold: Geometrical methods in the theory of ordinary differential equations (engl. transl.), Springer, Berlin, Heidelberg, New York 1983.
M. Ashkenazi & H. G. Othmer: Spatial patterns in coupled biochemical oscillators, J. Math. Biol. 5 (1978), 305–350.
R. I. Bogdanov: Bifurcations of a limit cycle of a family of vector fields in the plane (russ.), Trudy Sem. I. G. Petrovskogo 2 (1976), 23–36.
A versal deformation of a singular point of a vector field in the plane in the case of zero eigenvalues (russ.), ibid. Trudy Sem. I. G. Petrovskogo 2 (1976) 37–65.
R. I. Bogdanov: Versal deformations of a singular point on the plane in the case of zero eigenvalues, Funct. Anal. Appl. 9 (1975), 144–145.
S.-N. Chow & J. K. Hale: Methods of bifurcation theory, Springer, Berlin, Heidelberg, New York, 1982.
S.-N. Chow, J. Mallet-Paret & J. A. Yorke: A periodic orbit index which is a bifurcation invariant, in [41], 109–131.
M. G. Crandall & P. H. Rabinowitz: The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal. 67 (1977), 53–72.
B. Fiedler: Global Hopf bifurcation in porous catalysts, in Equadiff 82, H. W. Knobloch & K. Schmitt eds., Lect. Notes Math. 1017, 177–183, Springer, Berlin, Heidelberg, New York, Tokyo 1983.
B. Fiedler: An index for global Hopf bifurcation in parabolic systems, J. Reine u. Angew. Math., 359 (1985), 1–36.
B. Fiedler: Global Hopf bifurcation for Volterra integral equations, to appear in SIAM J. Math. Anal.,
M. Golubitsky & V. Guillemin: Stable mappings and their singularities, Springer, Berlin, Heidelberg, New York 1973.
J. Guckenheimer: Multiple bifurcation problems of codimension two, SIAM J. Math. Anal. 15 (1984), 1–49.
J. Guckenheimer & P. J. Holmes: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer, Berlin, Heidelberg, New York, Tokyo 1983.
J. K. Hale: Generic bifurcations with applications, in Nonlinear analysis and mechanics: Heriot-Watt Symposium I, R. J. Knops ed., 59–157, Pitman, London, San Francisco, Melbourne 1977.
J. Harrison & J. A. Yorke: Flows on S 3and ℝ3 without periodic orbits, in [41], 401–407.
M. W. Hirsch: Differential Topology, Springer, Berlin, Heidelberg, New York 1976.
L. N. Howard: Nonlinear oscillations, in Nonlinear oscillations in biology, F. Hoppensteadt ed., AMS Lect. Appl. Math. 17 (1979), 1–69.
J. Ize: Obstruction theory and multiparameter Hopf bifurcation. Preprint.
K. F. Jensen & W. H. Ray: The bifurcation behavior of tubular reactors, Chem. Eng. Sci. 37 (1982), 199–222.
J. P. Keener: Infinite period bifurcation in simple chemical reactors, in Modelling of chemical reaction systems, K. H. Ebert, P. Deuflhard, W. Jäger eds., Springer, Berlin, Heidelberg, New York 1981.
J. P. Keener: Infinite period bifurcation and global bifurcation branches, SIAM J. Appl. Math. 41 (1981), 127–144.
R. Lefever & I. Prigogine: Symmetry-breaking instabilities in dissipative systems II, J. Chem. Phys. 48 (1968), 1695–1700.
J. Mallet-Paret & J. A. Yorke: Snakes: oriented families of periodic orbits, their sources, sinks and continuation, J. Diff. Eq. 43 (1982), 419–450.
M. Medveď: Generic properties of parametrized vector fields I, II, Czech. Math. J. 25 (1975), 376–388 and 26 (1976), 71–83.
M. Medveď: On two-parametric systems of matrices and diffeomorphisms, Czech. Math. J. 33 (1983), 176–192.
I. Schreiber & M. Marek: Strange attractors in coupled reaction-diffusion cells, Physica 5D (1982), 258–272.
S. Smale: An infinite dimensional version of Sard's theorem. Amer. J. Math. 87 (1965), 861–866.
S. Smale: A mathematical model of two cells via Turing's equation, in Some mathematical questions in biology V, J. D. Cowen ed., AMS Lect. Math. in the Life Sciences 6 (1974), 15–26.
J. Sotomayor: Generic bifurcations of dynamical systems, in: Salvador Symposium on Dynamical Systems 1971, M. M. Peixoto ed., 561–582, Academic Press, New York, San Francisco, London 1973.
F. Takens: Singularities of vector fields, Publ. IHES 43 (1974), 47–100.
A. J. Tromba: Fredholm vector fields and a transversality theorem, J. Funct. Anal. 23 (1976), 362–368.
J. A. Yorke & K. T. Alligood: Cascades of period-doubling bifurcations: a prerequisite for horseshoes, Bull. American Math. Soc. 9 (1983), 319–322.
Geometric Dynamics, Proceedings 1981, J. Palis Jr. ed., Lect. Notes Math. 1007, Springer, Berlin, Heidelberg, New York, Tokyo 1983.
Author information
Authors and Affiliations
Additional information
Communicated by M. Golubitsky
Dedicated to the memory of Charlie Conley
Rights and permissions
About this article
Cite this article
Fiedler, B. Global Hopf bifurcation of two-parameter flows. Arch. Rational Mech. Anal. 94, 59–81 (1986). https://doi.org/10.1007/BF00278243
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00278243