American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020075

Coordinate-independent criteria for Hopf bifurcations

Niclas Kruff , and  Sebastian Walcher

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.

Keywords: Normal form, stability, characteristic polynomial, FitzHugh-Nagumo equation, predator-prey system.
Mathematics Subject Classification: Primary: 34C20, 34C23, 37G15.
Citation: Niclas Kruff, Sebastian Walcher. Coordinate-independent criteria for Hopf bifurcations. Discrete & Continuous Dynamical Systems - S, 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

show all references

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

[1]

Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205
[2]

Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639
[3]

Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150
[4]

John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851
[5]

Leonid Braverman, Elena Braverman. Stability analysis and bifurcations in a diffusive predator-prey system. Conference Publications, 2009, 2009 (Special) : 92-100. doi: 10.3934/proc.2009.2009.92
[6]

Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173
[7]

Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216
[8]

Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203
[9]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979
[10]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
[11]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[12]

Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441
[13]

Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3453-3474. doi: 10.3934/dcdsb.2018251
[14]

Jing-An Cui, Xinyu Song. Permanence of predator-prey system with stage structure. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 547-554. doi: 10.3934/dcdsb.2004.4.547
[15]

Dongmei Xiao, Kate Fang Zhang. Multiple bifurcations of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 417-433. doi: 10.3934/dcdsb.2007.8.417
[16]

Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236
[17]

S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Mathematical Biosciences & Engineering, 2006, 3 (1) : 173-187. doi: 10.3934/mbe.2006.3.173
[18]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002
[19]

Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877
[20]

Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (25)
  • HTML views (349)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences