Integrable approximation of regular islands: The iterative canonical transformation method

Generic Hamiltonian systems have a mixed phase space, where classically disjoint regions of regular and chaotic motion coexist. We present an iterative method to construct an integrable approximation $H_{\mathrm{reg}}$, which resembles the regular dynamics of a given mixed system $H$ and extends it into the chaotic region. The method is based on the construction of an integrable approximation in action representation which is then improved in phase space by iterative applications of canonical transformations. This method works for strongly perturbed systems and arbitrary degrees of freedom. We apply it to the standard map and the cosine billiard.

Figure 1

(a) Phase space of the standard map, Eq. (22) at $K=1.25$, with regular orbits (black lines) and chaotic orbits (black dots) and (b) of its integrable approximation $H_{\mathrm{reg}}$ (thin red lines) following from the iterative canonical transformation method.

Figure 2

Numerically determined frequency $\omega$ vs action $J$ (dots) fitted by Eq. (25) with $\mathcal{K}=5$ (red line) and extrapolated beyond the border of the regular island.

Figure 3

The phase space of the standard map, Eq. (22), at $K=1.25$ (thick gray lines and dots) compared to the tori (thin colored lines) of (a) the initial integrable approximation $H_{\mathrm{reg}}^0$, (b) its transformation after the first iteration step $T_1$, and (c) after the final iteration step $T_N$, $N=60$. (d) Magnification of the border torus of $H_{\mathrm{reg}}^0$ and $H_{\mathrm{reg}}^N$ (thin green lines) and the standard map (thick gray line) together with the individual points ${\mathbf{x}}_t^{\tau ,n}$ (small green dots) of $H_{\mathrm{reg}}^n$ approaching the reference points ${\mathbf{x}}_t^{\tau }$ (big black dots) of the standard map. Straight lines (blue) indicate the initial and final distances that contribute to the cost function, Eq. (12).

Figure 4

Cost function $L$ vs iteration step $n$.

Figure 5

(a) Phase space of the standard map, Eq. (22), at $K=2.9$ (thick gray lines and dots) and its integrable approximation (thin red lines) obtained using $N=30$ steps with the parameters ${\mathcal{L}}_q=2.86$, ${\mathcal{L}}_p=1.33$, ${\mathcal{N}}_q=1$, ${\mathcal{N}}_p=2$ in Eq. (35) and a damping factor $\eta =0.1$. (b) Cost function $L$ vs iteration step $n$.

Figure 6

(a) Cosine billiard in position space for $h=0.2$ and $w=0.066$ with a regular trajectory (dark red line), a chaotic trajectory (dashed blue line), and the positions of the Poincaré sections ${\Sigma }_1$ and ${\Sigma }_2$ (light green lines). (b) Dynamics on ${\Sigma }_1$ with a chaotic orbit (blue dots) and regular orbits (lines). (c) Same as (b) for ${\Sigma }_2$, zoomed to $p_y\in [0.5,1]$.

Figure 7

(a) Numerically determined actions ${\mathbf{J}}^{\tau }=(J_1^{\tau },J_2^{\tau })$ of the cosine billiard (dots) and contour line $E=1$ (red) of the integrable Hamiltonian ${\mathcal{H}}_{\mathrm{reg}}\left(\mathbf{J}\right)$, Eq. (46). (b) Relative error ${\Delta }_0$ of the energy (solid) and the frequencies ${\Delta }_1$ (dashed) and ${\Delta }_2$ (dotted).

Figure 8

(a) Cosine billiard and a trajectory (red line) in the coordinates (a) $\mathbf{q}=(x,y)$, (b) $\overline{\mathbf{q}}=(\overline{x},\overline{y})$, and (c) $\mathbf{Q}=(X,Y)$ using the transformations ${\mathcal{T}}_1$, Eq. (50), and ${\mathcal{T}}_2$, Eqs. (51) to (54). The colored dots correspond to the reflection points of the trajectory in (a). The gray grid shows lines of constant $X$ and lines of constant $Y$.

Figure 9

Momentum components (a) $p_x$, (b) $p_y$ of the trajectory from Fig. 8a and (c) $P_x$, (d) $P_y$ of the transformed trajectory from Fig. 8c.

Figure 10

Poincaré sections ${\Sigma }_1$ and ${\Sigma }_2$ for the cosine billiard $H(\mathbf{Q},\mathbf{P})$ [(a) and (d)], its initial integrable approximation $H_{\mathrm{reg}}^0(\mathbf{Q},\mathbf{P})$ [(b) and (e)], and its final integrable approximation $H_{\mathrm{reg}}(\mathbf{Q},\mathbf{P})$ [(c) and (f)].

Figure 11

Cost function $L$ vs iteration step $n$.

Figure 12

Visualization of the point transformation $(x,y)\mapsto (\overline{x},\overline{y})$ for the cosine billiard. The billiard domain is filled with a family of curves (vertical lines) that represent the contour lines of $\overline{x}(x,y)$ and intersect the upper boundary $y=r\left(x\right)$ at $x=\overline{x}$.