Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach

Abstract

In this paper, we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: Given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities, we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer-assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to $\varepsilon =0.9716$), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.

Appendix: A Heuristic Selection of Parameters to Validate Invariant Tori

In this appendix, we describe a direct method to select the implementation parameters $\rho$, $\delta$, $\sigma$, $\rho _{\infty }$, $d_{{{\mathcal {B}}}}$ and ${{\hat{\rho }}}$, required in Input 4 of Algorithm 5.1 to rigorously validate an invariant torus. The idea consists in using the structure of the constants that appear in Step 4 (c.f. Sect. 5.6). We do not claim that the procedure described below is optimal, but allows us to obtain suitable values of the parameters at a moderate computational cost.

We assume that the term $2(a_3)^{3\tau +1} \gamma ^3 \rho ^{3 \tau -1} C_3$ does not contribute to the constant ${\mathfrak {C}}_1$, and we look for parameters such that the constants in (108) satisfy ${\mathfrak {C}}_3={\mathfrak {C}}_4={\mathfrak {C}}_5$. The neglected term is, in general, very small (it stands for the control of the approximate Lagrangian character of the approximated torus), and this assumption allows us to simplify the heuristic analysis of the constants.

We proceed in analogy with the procedure described in [37]. We observe that the dependence on $a_2$ is very simple: It appears only in the expression $a_3 = 3 \tfrac{a_1}{a_1-1} \tfrac{a_2}{a_2-1}$ and in the final strip of analyticity $\rho _{\infty }=\rho /a_2$. In order to look for the limit condition, we take $a_2=\infty$ (so $\rho _\infty =0$) in all subsequent computations.

Now we can describe a very simple algorithm to obtain suitable values of the parameters $\rho$, $\delta$, $\sigma$, $d_{{\mathcal {B}}}$ and ${{\hat{\rho }}}$ for a given parameterization K in a grid of size ${{N}_{\scriptscriptstyle {\mathrm{F}}}}=({N}_{\scriptscriptstyle {\mathrm{F},1}},\ldots ,{N}_{\scriptscriptstyle {\mathrm{F},n}})$:

• I. We take $\rho _0=- \log (||E||_{F,0})/(2 \pi N)$, where E is the error of invariance and $N=\max _{i} \{ {N}_{\scriptscriptstyle {\mathrm{F},i}}\}$. This will be the initial value of $\rho$.

• II. For a given value $\rho$, we consider values $\delta \in [\tfrac{\rho }{6.5},\tfrac{\rho }{4.5}]$ (recall that $a_3=\tfrac{\rho }{\delta }$ and $a_1= \tfrac{a_3}{a_3-3}$). For any of these values $(\rho ,\delta )$, we compute $\sigma$ and $d_{{\mathcal {B}}}$ solving the equations ${\mathfrak {C}}_4={\mathfrak {C}}_5$ and ${\mathfrak {C}}_3={\mathfrak {C}}_4$, which are, respectively, written as follows:

  $$\begin{aligned}&\sigma _*(1-a_1^{-2\tau }) d_{{\mathcal {B}}}-(\sigma -1) \delta (1-a_1^{1-2\tau })=0,\end{aligned}$$

  (116)
  $$\begin{aligned}&\sigma _* (a_3)^{2\tau +1} \gamma ^2 \rho ^{2\tau -1} {{\hat{C}}}_2 - (\sigma -1)(1-a_1^{1-2\tau })(a_1a_3)^{4 \tau }{{\hat{C}}}_5 =0. \end{aligned}$$

  (117)

  Let us recall that $\sigma _*$, ${{\hat{C}}}_2$ and ${{\hat{C}}}_5$ depend on $\rho$, $\delta$, $\sigma$, $d_{{\mathcal {B}}}$ and ${{\hat{\rho }}}$. In order to avoid the dependence on ${{\hat{\rho }}}$ in the above expression, we take $C_{{N}_{\scriptscriptstyle {\mathrm{F}}}}(\rho , {{\hat{\rho }}})=0$ when computing these constants. A suitable value of ${{\hat{\rho }}}$ is fixed later. Then, we select the value of $\delta$ that minimizes the expression ${\mathfrak {C}}_1 \gamma ^{-4} \rho ^{-4\tau } ||E||_\rho$.

• III. If ${\mathfrak {C}}_1 \gamma ^{-4} \rho ^{-4\tau } ||E||_\rho \ge 1$, we decrease the value of $\rho$ and repeat step II, thus obtaining a new value of ${\mathfrak {C}}_1$. We proceed until we find that ${\mathfrak {C}}_1 \gamma ^{-4} \rho ^{-4\tau } ||E||_\rho <1$. If at any point we reach a minimum of the function ${\mathfrak {C}}_1 \gamma ^{-4} \rho ^{-4\tau } ||E||_\rho$, then we stop the computations. If this condition is not satisfied at the minimum, then we need a better approximation of the invariant torus.

• IV. Assume that we have obtained values of $\rho$, $\delta$, $\sigma$ and $d_{{\mathcal {B}}}$ as above. Then, we take a sequence of increasing values of ${{\hat{\rho }}}$ (starting at a value slightly greater than $\rho$) and compute the constant $C_{{N}_{\scriptscriptstyle {\mathrm{F}}}}(\rho ,{{\hat{\rho }}})$. Then, we compute again the constant ${\mathfrak {C}}_1$ and the bound ${b}_{\scriptscriptstyle {E}}$ (see Sect. 5.3). We select a value of ${{\hat{\rho }}}$ that minimizes the expression ${\mathfrak {C}}_1 \gamma ^{-4} \rho ^{-4\tau } {b}_{\scriptscriptstyle {E}}$.