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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 and 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. [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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##### 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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