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Higher order normal modes

  • * Corresponding author: Giuseppe Gaeta

    * Corresponding author: Giuseppe Gaeta 
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  • Normal modes are intimately related to the quadratic approximation of a potential at its hyperbolic equilibria. Here we extend the notion to the case where the Taylor expansion for the potential at a critical point starts with higher order terms, and show that such an extension shares some of the properties of standard normal modes. Some symmetric examples are considered in detail.

    Mathematics Subject Classification: Primary: 37J06, 34A05; Secondary: 15A69.

    Citation:

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  • Figure 1.  The potential $ W (\theta) $ as in eq.(11) for different values of $ {\beta} $; here $ {\beta} = -1, -0.5,0.5,1 $. The exchange of stability takes place at $ {\beta} = 0 $

    Figure 2.  Numerical integration of the motion generated by the potential (10) with the choice $ {\beta} = 1 $ for initial conditions near to normal modes. In all cases, initial data correspond to zero speed and position at $ r = 1 $ along eigenvectors, with an offset of 0.001 from the latter. The simulation show the outcome, for $ t \in (0,100) $, for initial data: (a) near the eigenvector $ \theta = 0 $, (b) near the eigenvector $ \theta = \pi $, (c) near the eigenvector $ \theta = \pi/4 $, (d) near the eigenvector $ \theta = - \pi/4 $

    Figure 3.  The potential $ W(\theta) $, see (13), for $ {\alpha} = 1/4 $ and various choices of $ {\beta} $. Here $ \theta $ is measured in units of $ \pi $

    Table 1.  Different possibilities for the number and type of critical points in the case of a cubic potential in three dimensions; here "Max"and "Min" represent the number of maxima and minima, while "$ S_k $" represents the number of saddle points of index $ - k $. Finally, "NCP" is the total number of critical points

    Max Min $ S_1 $ $ S_2 $ $ S_3 $ NCP
    1 1 0 0 0 2
    2 2 2 0 0 6
    3 3 4 0 0 10
    3 3 0 2 0 8
    4 4 6 0 0 14
    4 4 2 2 0 12
    4 4 0 0 2 10
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