By Henk Broer, Igor Hoveijn, Gerton Lunter, Gert Vegter
The authors reflect on purposes of singularity thought and laptop algebra to bifurcations of Hamiltonian dynamical structures. 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 via a normalized Hamiltonian process with a torus symmetry. it truly is assumed that aid through this symmetry results in a process with one measure of freedom. the amount specializes in such aid equipment, the planar aid (or polar coordinates) procedure and the relief by means of the power momentum mapping. The one-degree-of-freedom method then is tackled by means of singularity thought, the place laptop algebra, specifically, Gröbner foundation thoughts, are utilized. The readership addressed includes complicated graduate scholars and researchers in dynamical systems.
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Extra info for Bifurcations in Hamiltonian Systems: Computing Singularities by Gröbner Bases
19. 13) is negative. In the system H 0 , therefore, the corresponding bifurcation does not occur. 13) the singular circle disappears, whereas H r exhibits this singularity for all parameter values (as long as λ > 0). The second solution does deﬁne a bifurcation, however. We continue with its description. 3 see [BCKV93, Sect. 3. Spring-pendulum in 1:2-resonance Fig. , small balloons) through the singular point in ρ2 , ψ, χ-space, for three values of the detuning parameter. Bifurcations and dynamical implications First we discuss the bifurcation of the reduced system H c in the plane.
The agreement of numerical data and the pulled-back bifurcation curve is good, especially for small excitations. The reduction methods, used here and in the next chapter, consist of three parts: Birkhoﬀ normalization, symmetry reduction, and singularity theory. In each of these stages the coordinate transformations are explicitly computed, and especially in the ﬁrst and ﬁnal stage this is rather involved. For these calculations we summon the computer’s help, using algorithms described and developed in later chapters.
The transformation φ was computed using the algorithm outlined above. , at the central singularity b1 = λ = 0 it reduces to the normal form x(x2 + y 2 ). , independent of parameters. It does depend on the coeﬃcients b2 , b3 , . . however, since H r |λ=b1 =0 also depends on those. Some leading order parameters and coeﬃcients of the deformation H c are: 2 c1 = − b1 b63 b4 , c2 = b1 b23 , 3 1 1 d1 = − 2 , d2 = b23 (b4 + 3b5 ), b3 3 d3 = b23 (b4 + b5 ). 12 in Chap. 5 it follows that H u = x(x2 + y 2 ) + u1 x + u2 y 2 is a versal deformation of the hyperbolic umbilic x(x2 + y 2 ) (in the context of Z2 -symmetric potential functions).