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A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces
Coordinate-independent criteria for Hopf bifurcations
Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany |
We discuss the occurrence of Poincaré-Andronov-Hopf bifurcations in parameter dependent ordinary differential equations, with no a priori assumptions on special coordinates. The first problem is to determine critical parameter values from which such bifurcations may emanate; a solution for this problem was given by W.-M. Liu. We add a few observations from a different perspective. Then we turn to the second problem, viz., to compute the relevant coefficients which determine the nature of the Hopf bifurcation. As shown by J. Scheurle and co-authors, this can be reduced to the computation of Poincaré-Dulac normal forms (in arbitrary coordinates) and subsequent reduction, but feasibility problems quickly arise. In the present paper we present a streamlined and less computationally involved approach to the computations. The efficiency and usefulness of the method is illustrated by examples.
References:
[1] |
H. Amann, Ordinary Differential Equations, W. de Gruyter, Berlin, 1990.
doi: 10.1515/9783110853698. |
[2] |
B. Aulbach, Gewöhnliche Differentialgleichungen, Springer Spektrum, Heidelberg, 2004. |
[3] |
Yu. N. Bibikov, Local Theory of Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702, Springer, New York, 1979. |
[4] |
C. Chicone, Ordinary Differential Equations with Applications, Second Edition. Springer, New York, 2006. |
[5] |
D. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms, Third Edition. Springer, New York, 2007.
doi: 10.1007/978-0-387-35651-8. |
[6] |
W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-1-0 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de (2016). |
[7] |
F. Dumortier, J. Llibre and J. C. Artes, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006. |
[8] |
H. Errami, M. Eiswirth, D. Grigoriev, W. M. Seiler, T. Sturm and A. Weber,
Detection of Hopf bifurcations in chemical reaction networks using convex coordinates, J. Comp. Phys., 291 (2015), 279-302.
doi: 10.1016/j.jcp.2015.02.050. |
[9] |
R. FitzHugh,
Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[10] |
F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers Ltd., London, 1959. |
[11] |
K. Gatermann, Computer Algebra Methods for Equivariant Dynamical Systems, Lecture Notes in Mathematics, 1728, Springer-Verlag, New York - Berlin, 2000.
doi: 10.1007/BFb0104059. |
[12] |
K. Gatermann, M. Eiswirth and A. Sensse,
Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems, J. Symb. Comput., 40 (2005), 1361-1382.
doi: 10.1016/j.jsc.2005.07.002. |
[13] |
A. Goeke, S. Walcher and E. Zerz,
Determining "small parameters" for quasi-steady state, J. Diff. Equations, 259 (2015), 1149-1180.
doi: 10.1016/j.jde.2015.02.038. |
[14] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[15] |
B. Hassard and Y. H. Wan,
Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl., 63 (1978), 297-312.
doi: 10.1016/0022-247X(78)90120-8. |
[16] |
N. Kruff, Local Invariant Sets of Analytic Vector Fields, Doctoral dissertation, RWTH Aachen, 2018. |
[17] |
N. Kruff, C. Lax, V. Liebscher and S. Walcher, The Rosenzweig-MacArthur system via reduction of an individual based model, Journal of Mathematical Biology, (2018), 1–27.
doi: 10.1007/s00285-018-1278-y. |
[18] |
S. Lang, Algebra, Third Ed. Springer, New York, 2002.
doi: 10.1007/978-1-4613-0041-0. |
[19] |
W.-M. Liu,
Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256.
doi: 10.1006/jmaa.1994.1079. |
[20] |
J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer, New York, 1976. |
[21] |
S. Mayer, J. Scheurle and S. Walcher,
Practical normal form computations for vector fields, Z. Angew. Math. Mech., 84 (2004), 472-482.
doi: 10.1002/zamm.200310115. |
[22] |
J. D. Murray, Mathematical Biology. I. An Introduction, 3rd Ed. Springer, New York, 2002. |
[23] |
J. S. Nagumo, S. Arimoto and S. Yoshizawa,
An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2071.
doi: 10.1109/JRPROC.1962.288235. |
[24] |
J. Scheurle and S. Walcher, On normal form computations, In: P.K. Newton, P. Holmes, A. Weinstein (eds.): Geometry, Dynamics and Mechanics, Springer, New York, (2002), 309–325.
doi: 10.1007/0-387-21791-6_10. |
[25] |
S. Walcher,
On transformations into normal form, J. Math. Anal. Appl., 180 (1993), 617-632.
doi: 10.1006/jmaa.1993.1420. |
show all references
References:
[1] |
H. Amann, Ordinary Differential Equations, W. de Gruyter, Berlin, 1990.
doi: 10.1515/9783110853698. |
[2] |
B. Aulbach, Gewöhnliche Differentialgleichungen, Springer Spektrum, Heidelberg, 2004. |
[3] |
Yu. N. Bibikov, Local Theory of Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702, Springer, New York, 1979. |
[4] |
C. Chicone, Ordinary Differential Equations with Applications, Second Edition. Springer, New York, 2006. |
[5] |
D. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms, Third Edition. Springer, New York, 2007.
doi: 10.1007/978-0-387-35651-8. |
[6] |
W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-1-0 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de (2016). |
[7] |
F. Dumortier, J. Llibre and J. C. Artes, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006. |
[8] |
H. Errami, M. Eiswirth, D. Grigoriev, W. M. Seiler, T. Sturm and A. Weber,
Detection of Hopf bifurcations in chemical reaction networks using convex coordinates, J. Comp. Phys., 291 (2015), 279-302.
doi: 10.1016/j.jcp.2015.02.050. |
[9] |
R. FitzHugh,
Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[10] |
F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers Ltd., London, 1959. |
[11] |
K. Gatermann, Computer Algebra Methods for Equivariant Dynamical Systems, Lecture Notes in Mathematics, 1728, Springer-Verlag, New York - Berlin, 2000.
doi: 10.1007/BFb0104059. |
[12] |
K. Gatermann, M. Eiswirth and A. Sensse,
Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems, J. Symb. Comput., 40 (2005), 1361-1382.
doi: 10.1016/j.jsc.2005.07.002. |
[13] |
A. Goeke, S. Walcher and E. Zerz,
Determining "small parameters" for quasi-steady state, J. Diff. Equations, 259 (2015), 1149-1180.
doi: 10.1016/j.jde.2015.02.038. |
[14] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[15] |
B. Hassard and Y. H. Wan,
Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl., 63 (1978), 297-312.
doi: 10.1016/0022-247X(78)90120-8. |
[16] |
N. Kruff, Local Invariant Sets of Analytic Vector Fields, Doctoral dissertation, RWTH Aachen, 2018. |
[17] |
N. Kruff, C. Lax, V. Liebscher and S. Walcher, The Rosenzweig-MacArthur system via reduction of an individual based model, Journal of Mathematical Biology, (2018), 1–27.
doi: 10.1007/s00285-018-1278-y. |
[18] |
S. Lang, Algebra, Third Ed. Springer, New York, 2002.
doi: 10.1007/978-1-4613-0041-0. |
[19] |
W.-M. Liu,
Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256.
doi: 10.1006/jmaa.1994.1079. |
[20] |
J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer, New York, 1976. |
[21] |
S. Mayer, J. Scheurle and S. Walcher,
Practical normal form computations for vector fields, Z. Angew. Math. Mech., 84 (2004), 472-482.
doi: 10.1002/zamm.200310115. |
[22] |
J. D. Murray, Mathematical Biology. I. An Introduction, 3rd Ed. Springer, New York, 2002. |
[23] |
J. S. Nagumo, S. Arimoto and S. Yoshizawa,
An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2071.
doi: 10.1109/JRPROC.1962.288235. |
[24] |
J. Scheurle and S. Walcher, On normal form computations, In: P.K. Newton, P. Holmes, A. Weinstein (eds.): Geometry, Dynamics and Mechanics, Springer, New York, (2002), 309–325.
doi: 10.1007/0-387-21791-6_10. |
[25] |
S. Walcher,
On transformations into normal form, J. Math. Anal. Appl., 180 (1993), 617-632.
doi: 10.1006/jmaa.1993.1420. |
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