# American Institute of Mathematical Sciences

September  2020, 12(3): 421-434. doi: 10.3934/jgm.2020026

## Higher order normal modes

 1 Dipartimento di Matematica, Università degli Studi di Milano, v. Saldini 50, I-20133 Milano, Italy 2 SMRI, I-00058 Santa Marinella, Italy 3 Mathematik A, RWTH Aachen, D-52056 Aachen, Germany

* Corresponding author: Giuseppe Gaeta

Dedicated to James Montaldi on his 25+something anniversary

Received  September 2019 Revised  July 2020 Published  September 2020

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.

Citation: Giuseppe Gaeta, Sebastian Walcher. Higher order normal modes. Journal of Geometric Mechanics, 2020, 12 (3) : 421-434. doi: 10.3934/jgm.2020026
##### References:

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Dedicated to James Montaldi on his 25+something anniversary

##### References:
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 ${\beta} = 0$
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 $\theta = - \pi/4$
The potential $W(\theta)$, see (13), for ${\alpha} = 1/4$ and various choices of ${\beta}$. Here $\theta$ is measured in units of $\pi$
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
 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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