Existence of invariant tori in three dimensional maps with degeneracy (Q441804)

Consider the mapping \(M:(x,y,z)\mapsto (x_1,y_1,z_1)\) defined by NEWLINE\[NEWLINE\begin{cases} x_1 = x + z + \epsilon X(x,y,z), \\ y_1 = y + \epsilon g_0(z) + \epsilon Y(x,y,z), \\ z_1 = z + \epsilon Z(x,y,z), \end{cases} NEWLINE\]NEWLINE for \(x,y \in [0, 2\pi)\) and \(z \in [a,b]\), and where \(X,Y\) and \(Z\) are real analytic functions, \(2\pi\)-periodic in \(x,y\), and the function \(g_0\) is analytic and satisfies \(|g_0| \leq 1\), \(g_0'' \geq c_1 >0\), and \(\epsilon > 0\) is a small parameter. Moreover, \(\int_0^{2\pi} Z dx =0\), and \(\int_0^{2\pi} Y dx\) could depend on \(z\) only and \(M\) satisfies the intersection property: for each torus \(z=\gamma (x,y)\), \(x,y \in [0,2\pi)\), \(M\) intersects its image. The authors establish the persistence of two--dimensional invariant tori in the perturbations of integrable action--angle--angle maps with a degenerate angle:NEWLINENEWLINE{ Theorem}. There exists a positive number \(\epsilon_0\), such that for all \(\epsilon \in (0,\epsilon_0)\) the mapping \(M\) admits a family of invariant tori NEWLINE\[NEWLINE x = \xi + u(\xi,\zeta,\omega),\;y = \zeta + v(\xi,\zeta,\omega),\;z = w(\xi,\zeta,\omega), NEWLINE\]NEWLINE where \(u,v\) and \(w\) are real analytic functions of period \(2\pi\) in \(\xi,\zeta\), while \(\omega \in S_{\omega} \subset [a,b]\), \(S_{\omega}\) being a Cantor set with positive Lebesgue measure. The mapping can be parameterized so that the induced mapping on the tori is given by \(\xi_1=\xi+\omega\), \(\zeta_1=\zeta+\epsilon g_0(\omega)+g^*(\omega,\epsilon)\) with \(g^*(\omega,\epsilon)\) an analytic function, \(g^*(\omega,0)=0\).NEWLINENEWLINEThe first step of averaging coordinate transformation reduces all three perturbations to order \(\epsilon^2\). Next, a finite sequence of transformations reduces the action perturbations to order \(\epsilon^3\). At last, an infinite sequence of coordinate transformations finishes the proof of this classical KAM theorem. Let us note that some modifications of the proof technics (compared with the general KAM theory case) appear.NEWLINENEWLINEUsing computer simulations, the authors illustrate their theory with the swirling Hill's vortex with strong swirl NEWLINE\[NEWLINE\begin{cases} \dot{r} = rz + \sqrt{2r} \sin\theta \,\sin\frac{t}{\epsilon} \\ \dot{z} = 1 - 2r^2 - z^2 - \frac{z}{\sqrt{2r}} \sin\theta\,\sin\frac{t}{\epsilon} \\ \dot{\theta} = \frac{2}{\epsilon r^2} + \sqrt{2r} \cos\theta \, \sin\frac{t}{\epsilon}. \end{cases}NEWLINE\]NEWLINE Rescale the time \(t=\epsilon\tau\), and change the cylindrical coordinates \((r,z,\theta)\) to action--angle--angle coordinates; then the Poincaré map corresponds to the mapping \(M\).