By Henk Broer, Igor Hoveijn, Gerton Lunter, Gert Vegter
The authors think about functions of singularity conception and laptop algebra to bifurcations of Hamiltonian dynamical platforms. They limit themselves to the case have been the next simplification is feasible. close to the equilibrium or (quasi-) periodic resolution into account 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 method with one measure of freedom. the amount specializes in such aid equipment, the planar aid (or polar coordinates) process and the relief by means of the strength momentum mapping. The one-degree-of-freedom approach then is tackled through singularity thought, the place computing device algebra, particularly, Gröbner foundation recommendations, are utilized. The readership addressed involves complex graduate scholars and researchers in dynamical systems.
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6 we ﬁnd the reparametrizations that induce H c from H u . This step is computationally involved, and is dealt with in the last two chapters. In this step we employ a standard basis and the corresponding division algorithm to compute the required morphisms eﬃciently. Finally, we use the reparametrizations of Sect. 6 to compute the BCKVrestricted normal form H B of our system. 2 Some notation Symmetries and coordinate systems In the sequel, we use Cartesian canonical coordinates xi , yi as well as complex variables zi , z¯i and Hamiltonian polar coordinates Li , φi , because certain transformations take a simple form in one of these coordinates.
The reduction method of this chapter now applies a symplectic transformation that reduces the system to the plane, which gives the method its name. 2. Method I: Planar reduction 23 The last stage consists of normalizing the planar system by arbitrary righttransformations. Such transformations are not symplectic, and therefore do not yield conjugations. , conjugations modulo smooth time-reparametrizations. In particular, bifurcations are preserved, as they involve equilibria in the reduced system.
Points on this periodic orbit have nontrivial stabilizer, Z2 ; in other words, the period of this orbit is (to ﬁrst order) half that of the other periodic orbits. In the literature it is referred to as the short periodic orbit. Since the pole always exists and is always a ﬁxed point, the short periodic orbit always exists. Far from resonance (outside the parabola of Fig. 4) the pole is an extremum of the Hamiltonian, so that this orbit is stable. It is unstable close to resonance. In that situation, the spring-pendulum exhibits two stable periodic trajectories (the long periodic orbits), corresponding to the two extrema existing on the sphere (see the center picture in Fig.
Bifurcations in Hamiltonian Systems: Computing Singularities by Gröbner Bases by Henk Broer, Igor Hoveijn, Gerton Lunter, Gert Vegter