Nondegenerate Hamiltonian Hopf Bifurcations in $$\omega:3:6$$ Resonance $$(\omega=1$$ or $$2)$$
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Nondegenerate Hamiltonian Hopf Bifurcations in ω : 3 : 6 Resonance (ω = 1 or 2) Reza Mazrooei-Sebdani1 * and Elham Hakimi1** 1
Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111 Isfahan, Iran Received January 11, 2020; revised July 14, 2020; accepted July 22, 2020
Abstract—This paper deals with the analysis of Hamiltonian Hopf bifurcations in threedegree-of-freedom systems, for which the frequencies of the linearization of the corresponding Hamiltonians are in ω : 3 : 6 resonance (ω = 1 or 2). We obtain the truncated second-order normal form that is not integrable and expressed in terms of the invariants of the reduced phase space. The truncated first-order normal form gives rise to an integrable system that is analyzed using a reduction to a one-degree-of-freedom system. In this paper, some detuning parameters are considered and nondegenerate Hamiltonian Hopf bifurcations are found. To study Hamiltonian Hopf bifurcations, we transform the reduced Hamiltonian into standard form. MSC2010 numbers: 70K30, 37J35, 70H06, 70H33, 70K45 DOI: 10.1134/S1560354720060027 Keywords: Hamiltonian ω : 3 : 6 resonance (ω = 1 or 2), integrability, reduction, normal forms, Hamiltonian Hopf bifurcation.
1. INTRODUCTION Hamiltonian dynamical systems near equilibria obtained by modeling nonlinear oscillators are present in many fields in science. There are many mechanical and physical systems that are modeled by Hamiltonian equations (see, for instance, [18, 23]). Hamiltonian systems can undergo an enormous variety of bifurcations. One can see [17, 24–26] as good references for studying Hamiltonian bifurcations. One of the most interesting bifurcations which may happen only in two or more degrees of freedom is the Hamiltonian Hopf bifurcation. There are many publications on this interesting bifurcation, so we just refer to [36] and references therein for extensive discussions. For Hamiltonian Hopf bifurcations in three or more degrees of freedom, one can see [14, 15, 19]. Hanßmann et al. [19] determined Hamiltonian Hopf bifurcations by an algebraic method for a class of 1 : 1 : 1 Hamiltonian resonance, i. e., 3D H´enon – Heiles family. In [14] the authors studied an integrable class of 1 : 1 : 1 : 1 Hamiltonian resonance and found some bifurcations and especially Hamiltonian Hopf bifurcations. Also, in the master thesis [15], the 1 : 3 : 4 resonance is considered and Hamiltonian Hopf bifurcations with respect to detuning parameters are found by the method of [19]. The detuning parameters are considered due to the presence in physical models. This bifurcation appears in various examples such as the Lagrange top [9, 37], the Kirchhoff top [3], the double spherical pendulum [10, 22], the circular restricted three-body problem at Routh’s critical mass ratio [36] and the 3D H´enon – Heiles family [19]. The Hamiltonian Hopf bifurcation was first discovered at the L4 Lagrange point of the planar restricted three-body problem in a series of papers [5, 11, 26, 28, 30]. Also, recently Palaci´an et al. in [29] considere
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