Download Bifurcations in Hamiltonian Systems: Computing Singularities by Henk Broer, Igor Hoveijn, Gerton Lunter, Gert Vegter PDF

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

The authors give some thought to purposes of singularity idea and machine 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 relief by way of this symmetry ends up in a procedure with one measure of freedom. the amount specializes in such relief equipment, the planar aid (or polar coordinates) technique and the relief via the power momentum mapping. The one-degree-of-freedom approach then is tackled via singularity idea, the place machine algebra, particularly, Gröbner foundation concepts, are utilized. The readership addressed comprises complicated graduate scholars and researchers in dynamical systems.

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Method I: Planar reduction 41 in Sect. 2, where we said that S3 divided out by an S1 -action gives S2 . After this division, the singular circle collapses to a single point on S2 , and is referred to as the pole. More precise information can be obtained by looking at the algebra. , they form a Hilbert basis). 2. The invariants satisfy the relation ρ1 ρ22 = (ψ 2 + χ2 )/4. Moreover, reality conditions imply ρ1 ≥ 0 and ρ2 ≥ 0. The quadratic part H2 is an integral of H n , and without loss of generality we may reduce to H2 = 2ρ1 + ρ2 = , where is some small positive number.

The transformation from xi , yi to complex coordinates (with this symplectic form) is symplectic with multiplier 2i. Hence, real Hamiltonians in Cartesian (or Hamiltonian polar) coordinates correspond to purely imaginary Hamiltonians in complex coordinates. This explains why H20 is purely imaginary in the propositions above. 2. 4 Reduction to planar 1 degree-of-freedom system We now discuss the reduction to the plane, for a system in p : q resonance. The normalized system H n has an additional (formal) S1 symmetry, with action (z1 , z2 ) → (eiqξ z1 , eipξ z2 ) for ξ ∈ S1 = R/2πZ, and corresponding conserved quantity H20 = z1 z¯1 + ωz2 z¯2 .

4. For the case of the symmetry groups considered above, we have the following result. g. ) 2. 1. Let H2 = iz1 z¯1 + iωz2 z¯2 , where ω = 0, and assume Γ is either {Id}, {Id, T } or {Id, S, T, ST }. If ω ∈ / Q, a Hilbert basis for the algebra ker adH2 in RΓ is {z1 z¯1 , z2 z¯2 }. In the case that ω = p/q = P/Q, where p, P = 0, q, Q > 0, Q even and gcd(p, q) = gcd(P, Q/2) = 1, the Hilbert basis for ker adH2 additionally contains a) Γ = {Id}: b) Γ = {Id, T }: c) Γ = {Id, S, T, ST }: z1p z¯2q , z¯1p z2q (p ≥ 0) or or z1p z¯2q + z¯1p z2q (p ≥ 0) z1P z¯2Q + z¯1P z2Q (p ≥ 0) or z¯1−p z¯2q , z1−p z2q (p ≤ 0) z¯1−p z¯2q + z1−p z2q (p ≤ 0) z¯1−P z¯2q + z1−P z2q (p ≤ 0) A finite set of invariants generating the invariant ring of a Lie group is called a Hilbert basis; the terminology above is justified since ker adH2 is the invariant ring of the Lie group consisting of exponentials of the vector field associated to H2 ; see [Gat00, Hil93].

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