Hamiltonian systems bifurcation

Vol. 1806 H. Broer, I. Hoveijn. G. Lunther, G. Vegter, Bifurcations in Hamiltonian Systems. Computing Singularities by Grobner Bases (2003)... 

The stochastic aspect of a complex bifurcation arising in a two variables chemical system is studied. The dynamics reduces, in a suitable region of the phase space, to a normal form for which both roots of the characteristic equation vanish simultaneously. In conditions close to this degenerate situation, the normal form can be viewed as a perturbation of an exactly soluble hamiltonian system, of hamiltonian h, which exhibits a homoclinic trajectory, h = 0. BAESENS and IMICOLIS [l ] have shown that the phase portrait of the dissipative sytem displays two steady states that coalesce, a focus F and a saddle S. [ Moreover, as one moves in the parameter space, a limit cycle surrounding F, bifurcates from a homoclinic trajectory and then disappears by Hopf bifurcation. ... 

Bifurcation theory studies the changes in the phase space as we vary the parameters of the system. In essence, this is the authentic notion of bifurcation theory proposed originally by Henry Poincare when he studied Hamiltonian systems with one degree of freedom. We must, however, note that this intuitively evident definition is not always sufficient at the contemporary stage of the development of the theory. One needs, in fact, to have an appropriate mathematical foundation to define the notions of the structure of the phase space and the changes in the structure. 

Chenciner, A. 1985 Hamiltonian-like phenomena in saddle-node bifurcations of invariant curves for plane diffeomorphisms. In Singularities and dynamical systems (ed. S. N. Puevmat-ikos). Amsterdam Elsevier Science Publishers/North Holland. 

Fig. 2. Global bifurcation for equations (21)—(23). (a) Hamiltonian reference system, (b) Stable separ-atrix loop arising from the effect of the dissipative" perturbation I2. (c) Asymptotically stable...

It turns out that the language of normal and local modes that emerged from the bifurcation analysis of the Darling-Dennison Hamiltonian is not sufficient to describe the general Fermi resonance case, because the bifurcations are qualitatively different from the normal-to-local bifurcation in figure Al.2.10. For example, in 2 1 Fermi systems, one type of bifurcation is that in which resonant collective modes are bom [54]. The resonant collective modes are illustrated in figure A12.11 their difference from the local modes of the Darling-Dennison system is evident. Other types of bifurcations are also possible in Fermi resonance systems a detailed treatment of the 2 1 resonance can be found in [44]. 

Vol. 1893 H. HanOmann, Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems, Results and Examples (2007)... 

Although the De Leon-Berne Hamiltonian is apparently the only system for which these fairly complex surface plots have been made, the reactive islands Poincare map structure (which is a unique signature of the cylindrical geometry) has been observed in models of 3-phospholene as well as in a symmetric triple-well prototype. 24 ij has also been observed in the bi-molecular reactions H + H2 - H2 + H52,i24 anj fn (he unimolecular decomposition of the (He I2) cluster - and of HNSi. 24 These studies have shown that even in a strong-coupling limit, where the repulsive PODS wanders away from the barrier to some extent or even bifurcates into multiple P0DS, > - 124 cylindrical separatrix manifolds mediate the pre- and postreaction dynamics. 

A problem that sometimes occurs in reaction-path Hamiltonians, especially for bend potentia1s, is the bifurcation of the reaction path. This occurs when a harmonic frequency becomes imaginary, and for the Raff surface this occurs for bends on both sides of the saddle point. initio calculations can be helpful in determining if the bifurcation is an artifact of the form of the analytic potential function or if it is present in the actual system. When the MEP bifurcates it is probably best to base the RPH on a reference path centered on the ridge between two equivalent MEP s. l This requires extra effort when computing vibrational energy levels since the vibrational potential becomes a double-minimum one, but it probably reduces mode-mode coupling, which (see Sect. 2) is hard to treat accurately. 

Pismen, L. M. Dynamics of lumped chemically reacting systems near singular bifurcation points—II. Almost Hamiltonian dynamics. Chem. Eng. Sci., 40, 905-16. 

Remark An arbitrary Hamiltonian of normal series is represented in the form H = fci/i -f A 2/2 + kzfzy where ki,k2ykz do not depend on X j. Therefore, the bifurcational diagrams for a system with an aribtrary Hamiltonian of our form are obtained from the bifurcational diagrams for a system with a Hamiltonian H (see Fig. 94) by means if a non-degenrate linear transformation(Oshemkov). 

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