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Coordinate-independent criteria for Hopf bifurcations. (English) Zbl 1443.34038

The Hopf (or Poincaré-Andronov-Hopf) bifurcation is one of the basic building block in the study of dynamical systems; the determination of parameters’ values for which it takes place is rather simple if one deals with a system in normal form, but can be quite complicate if the system is not normalized, in particular, when we deal – as frequent in concrete applications – with multi-parameter systems.
In fact, the usual statement of the Hopf bifurcation theorem assumes that the system is given in a standardized form; this includes a distinguished real bifurcation parameter and a convenient choice of coordinates.
For multi-parameter systems, prior to determining a single parameter effectively driving the bifurcation, one has to address the problem of finding the critical parameter values from which Hopf bifurcations emanate (for some choice of a curve through this point in parameter space). This part of the problem is concerned only with the linearization of the vector field, and has been resolved by [W.-m. Liu, J. Math. Anal. Appl. 182, No. 1, 250–256 (1994; Zbl 0794.34033)].
For polynomial systems, H. Errami et al. [J. Comput. Phys. 291, 279–302 (2015; Zbl 1349.92168)] devised an algorithm to find all possible stationary points and critical parameter values for Hopf bifurcations, but did not proceed to determine the nature of the bifurcations. This part of the problem involves nonlinear terms in the Taylor expansion, and this is addressed (and solved) in this paper.
Here, the Poincaré-Dulac normal form is the fundamental tool, but it is considered in a coordinate-independent formulation following previous work by S. Mayer et al. [ZAMM, Z. Angew. Math. Mech. 84, No. 7, 472–482 (2004; Zbl 1046.37012)].
Some applications from mathematical biology are also considered, in particular, a FitzHugh-Nagumo system and a three-species predator-prey system introduced by N. Kruff et al. in recent work [J. Math. Biol. 78, No. 1–2, 413–439 (2019; Zbl 1410.92101)].

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C23 Bifurcation theory for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
92C20 Neural biology
92D25 Population dynamics (general)

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References:

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