Nekhoroshev Theory

Article Outline

Glossary

Definition of the Subject

Introduction

Exponential Stability of Constant Frequency Systems

Nekhoroshev Theory (Global Stability)

Applications

Future Directions

Appendix: An Example of Divergence Without Small Denominators

Bibliography

Appendix: An Example of Divergence Without Small Denominators

We would like to develop the following example of Neishtadt [60] where the phenomenons of divergence without small denominators and summation up to an exponentially small remainder appear in its simplest setting (see also [76] for another example).

Consider the quasi integrable system governed by the Hamiltonian

$$ \mathcal{H}(I_1,I_2;\theta_1,\theta_2)=I_1-\varepsilon[I_2 -\cos (\theta_1) f(\theta_2)]\\\text{defined over}\enskip \mathbb{R}^2\times\mathbb{T}^2\:, $$

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with

$$ f(\theta)=\sum_{m\geq 1}\frac{\alpha_m}{m}\cos (m\theta)\:, $$

where ${\alpha_m =\text{e}^{-\alpha m}}$ for ${0<\alpha\leq 1}$, this last choice corresponds to the exponential decay of the Fourier coefficients for a holomorphic function.

Indeed, ${f^{\prime}(\theta)=\mathrm{Im}(g(-\alpha+i\theta))}$ where $g(z)=(\exp (z)-1)^{-1}$ and the complex pole of f which is closest to the real axis is located at ${-i\alpha}$.

Conversely, all real function which admits a holomorphic extension over a complex strip of width α (i. e.: ${\vert\mathrm{Im}(z)\vert\leq\alpha}$) has Fourier coefficients bounded by ${(C\alpha_m)_{m\in\mathbb{N}^*}}$ for some constant ${C > 0}$.

As in the Sect. “Hamiltonian Perturbation Theory”, it is possible to eliminate formally completely the fast angle θ₁ in the perturbation without the occurrence of any small denominators.

Indeed, one can consider a normalizing transformation generated by ${X(\theta_1,\theta_2)=\sum_{n\geq 1}\varepsilon^n X_n (\theta_1,\theta_2)}$ where the functions X _n satisfy the homological equations:

$$ \begin{aligned} \partial_{\theta_1}X_1(\theta_1,\theta_2) &= \cos (\theta_1) f(\theta_2)\enskip\text{and}\\ \partial_{\theta_1} X_n(\theta_1,\theta_2) &=\partial_{\theta_2}X_{n-1}(\theta_1,\theta_2)\:. \end{aligned}$$

The solutions can be written $X_n(\theta_1,\theta_2) = \cos(\theta_1- n\frac{\pi}{2}) f^{(n-1)}(\theta_2)$, hence:

$$ X\left(\theta_1,\theta_2\right) = \sum_{n\in\mathbb{N}^*}\varepsilon^n\sum_{m \in\mathbb{N}^*}\frac{\alpha_m}{m^2}m^n\\\cdot\cos\left(\theta_1-n\frac{\pi}{2}\right) \sin\left(m\theta_2+n\frac{\pi}{2}\right) $$

and, for instance,

$$ X\left(\frac{\pi}{2},0\right) = \sum_{m\in\mathbb{N}^*}\sum_{k\in\mathbb{N}} \frac{\text{e}^{-\alpha m}}{m^2}(\varepsilon m)^{2k+1}$$

which is divergent for all ${\varepsilon > 0}$ since ${\varepsilon\geq 1/m}$ for m large enough.

We see that divergence comes from coefficients arising with successive differentiations.

On the other hand, if one considers the transformation generated by the truncated Hamiltonian $\sum\nolimits_{n\geq 1}^N\varepsilon^n X_n (\theta_1,\theta_2)$ then the transformed Hamiltonian becomes

$$ \mathcal{H}(I_1,I_2;\theta_1,\theta_2)=I_1-\varepsilon I_2 + \varepsilon^{N+1}\partial_{\theta_2}X_N (\theta_1,\theta_2)\:, $$

where ${\partial_{\theta_2}X_N (\theta_1,\theta_2) = \cos\left(\theta_1-N\frac{\pi}{2}\right) f^{(N)}(\theta_2)}$.

Especially, with ${g(z)={1}/{\exp (z)-1}}$, we have:

$$ \partial_{\theta_2}X_{2N} (\theta_1,\theta_2) = -\cos(\theta_1) \mathrm{Re}\left(g^{(2N-1)}(-\alpha +i\theta_2)\right) \\\shoveright{\qquad\text{for}\enskip N > 0\:,} \\ \partial_{\theta_2}X_{2N+1}(\theta_1,\theta_2) = \sin(\theta_1) \mathrm{Im}\left(g^{(2N)}(-\alpha +i\theta_2)\right)\\\qquad \text{for}\enskip N\geq 0\:. $$

Now, around the real axis, the main term in the asymptotic expansion of ${g^{(n)}(z)}$ as n goes to infinity is given by the derivative of the polar term ${1/z}$ in the Laurent expansion of g at 0.

Indeed, we consider the function ${h(z)=g(z)-1/z}$ which is real analytic and admits an analyticity width of size ${2\pi}$ around the real axis. Hence, Cauchy estimates ensure that for n large enough and z in the strip of width π around the real axis, the derivative ${h^{(n)}(z)}$ becomes negligible with respect to the derivative of ${1/z}$.

Consequently, ${g^{(n)}(z)}$ is equivalent to ${(-1)^n n!/z^{n+1}}$ as n goes to infinity for z in the strip of width π around the real axis.

Hence, the remainder ${\partial_{\theta_2}X_N (\theta_1,\theta_2)}$ has a size of order ${{(N-1)!}/{\alpha^N}}$ for N large. This latter estimate cannot be improved, since ${\partial_{\theta_2}X_{2N}(\pi, 0)}$ and $\partial_{\theta_2}X_{2N+1}({\pi}/{2},\alpha\tan({\pi}/{4N+2}))$ admit the same size of order ${(N-1)!}/{\alpha^N}$ for N large.

Now, we can make a “summation at the smallest term” as in Sect. “The Case of a Single Frequency System” to obtain an optimal normalization with an exponentially small remainder.