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 think of functions of singularity idea and laptop 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 consideration the linear half permits approximation by way of a normalized Hamiltonian process with a torus symmetry. it's assumed that relief by way of this symmetry results in a process with one measure of freedom. the amount specializes in such relief equipment, the planar relief (or polar coordinates) process and the relief via the strength momentum mapping. The one-degree-of-freedom approach then is tackled via singularity thought, the place computing device algebra, specifically, Gröbner foundation strategies, are utilized. The readership addressed involves complex graduate scholars and researchers in dynamical systems.

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

The authors think of functions of singularity idea and laptop 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 consideration the linear half permits approximation by way of a normalized Hamiltonian process with a torus symmetry. it's assumed that relief by way of this symmetry results in a process with one measure of freedom. the amount specializes in such relief equipment, the planar relief (or polar coordinates) process and the relief via the strength momentum mapping. The one-degree-of-freedom approach then is tackled via singularity thought, the place computing device algebra, specifically, Gröbner foundation strategies, are utilized. The readership addressed involves complex graduate scholars and researchers in dynamical systems.

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AM78] for the Hamiltonian version), the 1-parameter continuous symmetries are related to conserved quantities, in this context also called momenta, see [CS85]. After applying the Birkhoff procedure, n − 1 of such independent conserved quantities can be found explicitly. The large symmetry group and related conserved quantities imply that the system is integrable, but generically an n degree of freedom system (n ≥ 2) is not [BT89]. In fact, the formal transformation lifts to smooth coordinate transformations by a theorem of Borel and Schwarz [Bro81, Dui84, GSS88], but these do not form conjugations.

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.

9). t. from H r . This morphism should respect the Z2 symmetry (x, y) → (x, −y). 11 in Sect. 2. Armed with the knowledge that φ exists we set out to compute it, using the following iterative approach. 11) H r ◦ φk = x(x2 + y 2 ) + O(|x, y|k+3 ) for some k. To find φk with k = k + 1 we set φk = φk + αi ti , where {ti } span the space of Z2 -equivariant terms in x, y of degree k . 3. Spring-pendulum in 1:2-resonance set of linear equations for the real numbers αi . By existence of the normalizing transformation, this set of equations is not over-determined, in fact it is usually under-determined.

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