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April  2020, 13(4): 1319-1340. doi: 10.3934/dcdss.2020075

Coordinate-independent criteria for Hopf bifurcations

 Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany

* Corresponding author: Niclas Kruff

Dedicated to Jürgen Scheurle on the occasion of his retirement from non-mathematical duties

Received  August 2017 Revised  January 2018 Published  April 2019

Fund Project: The frst author acknowledges support by the DFG Research Training Group GRK 1632 "Experimental and constructive algebra". Both authors thank an anonymous reviewer for helpful comments

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.

Citation: Niclas Kruff, Sebastian Walcher. Coordinate-independent criteria for Hopf bifurcations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1319-1340. doi: 10.3934/dcdss.2020075
References:
 [1] H. Amann, Ordinary Differential Equations, W. de Gruyter, Berlin, 1990. doi: 10.1515/9783110853698.  Google Scholar [2] B. Aulbach, Gewöhnliche Differentialgleichungen, Springer Spektrum, Heidelberg, 2004. Google Scholar [3] Yu. N. Bibikov, Local Theory of Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702, Springer, New York, 1979.  Google Scholar [4] C. Chicone, Ordinary Differential Equations with Applications, Second Edition. Springer, New York, 2006.  Google Scholar [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.  Google Scholar [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). Google Scholar [7] F. Dumortier, J. Llibre and J. C. Artes, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers Ltd., London, 1959.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] N. Kruff, Local Invariant Sets of Analytic Vector Fields, Doctoral dissertation, RWTH Aachen, 2018. Google Scholar [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.  Google Scholar [18] S. Lang, Algebra, Third Ed. Springer, New York, 2002. doi: 10.1007/978-1-4613-0041-0.  Google Scholar [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.  Google Scholar [20] J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer, New York, 1976.  Google Scholar [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.  Google Scholar [22] J. D. Murray, Mathematical Biology. I. An Introduction, 3rd Ed. Springer, New York, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] S. Walcher, On transformations into normal form, J. Math. Anal. Appl., 180 (1993), 617-632.  doi: 10.1006/jmaa.1993.1420.  Google Scholar

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References:
 [1] H. Amann, Ordinary Differential Equations, W. de Gruyter, Berlin, 1990. doi: 10.1515/9783110853698.  Google Scholar [2] B. Aulbach, Gewöhnliche Differentialgleichungen, Springer Spektrum, Heidelberg, 2004. Google Scholar [3] Yu. N. Bibikov, Local Theory of Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702, Springer, New York, 1979.  Google Scholar [4] C. Chicone, Ordinary Differential Equations with Applications, Second Edition. Springer, New York, 2006.  Google Scholar [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.  Google Scholar [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). Google Scholar [7] F. Dumortier, J. Llibre and J. C. Artes, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers Ltd., London, 1959.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] N. Kruff, Local Invariant Sets of Analytic Vector Fields, Doctoral dissertation, RWTH Aachen, 2018. Google Scholar [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.  Google Scholar [18] S. Lang, Algebra, Third Ed. Springer, New York, 2002. doi: 10.1007/978-1-4613-0041-0.  Google Scholar [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.  Google Scholar [20] J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer, New York, 1976.  Google Scholar [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.  Google Scholar [22] J. D. Murray, Mathematical Biology. I. An Introduction, 3rd Ed. Springer, New York, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] S. Walcher, On transformations into normal form, J. Math. Anal. Appl., 180 (1993), 617-632.  doi: 10.1006/jmaa.1993.1420.  Google Scholar
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