Bifurcations in Hamiltonian systems: computing singularities by Henk Broer, Igor Hoveijn, Gerton Lunter, Gert Vegter

By Henk Broer, Igor Hoveijn, Gerton Lunter, Gert Vegter

The authors consider purposes of singularity conception and desktop algebra to bifurcations of Hamiltonian dynamical structures. They limit themselves to the case have been the subsequent simplification is feasible. close to the equilibrium or (quasi-) periodic answer into account the linear half permits approximation via a normalized Hamiltonian approach with a torus symmetry. it truly is assumed that aid through this symmetry results in a process with one measure of freedom. the quantity specializes in such aid equipment, the planar aid (or polar coordinates) approach and the aid by means of the power momentum mapping. The one-degree-of-freedom procedure then is tackled through singularity conception, the place desktop algebra, particularly, Gr?bner foundation recommendations, are utilized. The readership addressed involves complicated graduate scholars and researchers in dynamical systems.

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1 Overview The starting point is a Hamiltonian H 0 with an equilibrium at the origin. It is supposed to be close to some resonance of the form p : q, and to depend on several coefficients ai . Optionally, the system may be invariant (or reversing) under some symmetry group Γ , which is supposed to respect the symplectic structure. ) The first step is to apply the Birkhoff procedure around the resonance, resulting in a system H n which has acquired a (formal) S1 -symmetry. This step singles out a detuning parameter denoted by b1 , which measures the deviation from the resonance around which the Birkhoff procedure is performed.

G. [AM78, Bro79, CS85, CB97, Mee85, Tak74b]. The idea is to divide out the symmetry, and regard the associated conserved quantities as parameters (also called integrals), a procedure known as orbit space reduction. Sometimes this reduction is done on the entire phase space, and sometimes on each leaf of the foliation defined by the levels of the integrals. , when some points have nontrivial isotropy group); see [AM78, CS85] for details. The result is a reduced Hamiltonian system with one degree of freedom, and whose dynamics coincides with the projection of the dynamics of the original system onto the orbit space.

In Sect. 1 we give a necessary and sufficient condition for a deformation to be versal. It amounts to solvability of the well-known infinitesimal stability equation2 adapted to our equivariant context. In the case of the 1 : 2 resonance, the central singularity is isomorphic to x(x2 + y 2 ), with a symmetry group Z2 acting on R2 via (x, y) → (x, −y). 4) g(x, y) = α1 (x, y)x ∂f ∂f ∂f + α2 (x, y)y 2 + α3 (x, y)y + u1 x + u2 y 2 . ∂x ∂x ∂y Here f = x(x2 + y 2 ) is the central singularity. For this f the condition is indeed satisfied; see Sect.

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