Bifurcation of space periodic solutions in symmetric reversible FDEs. (English) Zbl 1413.37039

Balanov, Zalman; Wu, Hao-Pin

Differ. Integral Equ. 30, No. 3-4, 289-328 (2017).

The reversible codimension-one Hopf bifurcation in parameterized ODEs may occur as a result of a collision of eigenvalues of the linearization on the imaginary axis (the so-called \(1:1\) resonance; see [C. A. Buzzi and J. S. W. Lamb, Arch. Ration. Mech. Anal. 175, No. 1, 39–84 (2005; Zbl 1073.34038); A. Vanderbauwhede, Proc. R. Soc. Edinb., Sect. A 116, No. 1–2, 103–128 (1990; Zbl 0719.34072)]). Higher \(1:k\) resonances, related to the case \(i\beta_1=k i\beta_2\; (k\in\mathbb{Z}\) and \(i\beta_1\), \(i\beta_2\) are the eigenvalues at the moment of the bifurcation) were studied in [V. I. Arnol’d, “Reversible systems”, in: R. Z. Sagdeev (ed.), Nonlinear and Turbulent Processes in Physics 3, 1161–1174 (1984); V. I. Arnold and M. B. Sevryuk, “Oscillations and bifurcations in reversible systems”, in: R. Z. Sagdeev (ed.), Nonlinear Phenomena in Plasma Physics and Hydrodynamics, 31–64 (1986)].
The eigenvalues moving along the imaginary axis may give rise to a bifurcation of periodic solutions of constant period even without any resonance. This bifurcation, considered in systems of functional differential equations (FDEs) respecting a finite group of symmetries, is the main subject of the present paper. Results on existence, multiplicity and symmetric properties of bifurcation branches of periodic solutions in symmetric networks of reversible FDEs are obtained.
The authors give an illustrative example of such symmetries, the octahedral group \(S_4\), for which the equivariant bifurcation invariant is fully estimated.

Reviewer: Irina V. Konopleva (Ul’yanovsk)

MSC:

37G40	Dynamical aspects of symmetries, equivariant bifurcation theory
34K18	Bifurcation theory of functional-differential equations
70K28	Parametric resonances for nonlinear problems in mechanics
70K30	Nonlinear resonances for nonlinear problems in mechanics

Keywords:

reversible functional differential equations; Hopf bifurcation

Citations:

Zbl 1073.34038; Zbl 0719.34072; Zbl 0546.34033