Integrable approximation of regular regions with a nonlinear resonance chain

Generic Hamiltonian systems have a mixed phase space where regions of regular and chaotic motion coexist. We present a method for constructing an integrable approximation to such regular phase-space regions including a nonlinear resonance chain. This approach generalizes the recently introduced iterative canonical transformation method. In the first step of the method a normal-form Hamiltonian with a resonance chain is adapted such that actions and frequencies match with those of the nonintegrable system. In the second step a sequence of canonical transformations is applied to the integrable approximation to match the shape of regular tori. We demonstrate the method for the generic standard map at various parameters.

Figure 1

(a) Phase space of the standard map, Eq. (3), at $\kappa =3.4$, with regular orbits (gray lines) and chaotic orbits (gray dots) and (b) its integrable approximation (thin red lines).

Figure 2

Phase space of the standard map, Eq. (3), at $\kappa =3.4$ with a chaotic orbit (gray dots), regular tori (gray lines) including the dominant 6:2 resonance chain and a chaotic layer (inset). The arrows indicate one iteration step when applying the maps $U$ and $U^r$.

Figure 3

(a) Phase space of the normal-form Hamiltonian ${\mathcal{H}}_{r:s}(\theta ,I)$, Eq. (8), with the areas $A_{r:s}$ of the resonance regions and the area $A_1$ below the resonance regions. (b) The corresponding areas ${\overline{A}}_{r:s}$ and ${\overline{A}}_1$ for the standard map, Eq. (3), at $\kappa =3.4$.

Figure 4

(a) Phase space of the standard map, Eq. (3), at $\kappa =3.4$ (gray lines and dots) with regular orbits ${\overline{\mathbf{x}}}_{\ell r}^{\tau }$ (black) on noble tori. (b) Frequencies ${\overline{\omega }}_{\tau }$ of these orbits (black dots) and the fitted function ${\mathcal{H}}_0^{\prime }\left(J\right)$ (red solid line).

Figure 5

Comparison of the frequency function $\omega \left(J\right)$ of the determined integrable approximation ${\mathcal{H}}_{r:s}(\theta ,I)$ (red thin lines) to frequencies of $U^r$ (dots): (a) the frequencies ${\overline{\omega }}_{\tau }$ (black dots), frequencies close to the resonance region (light blue or gray dots) and (b) frequencies inside the resonance region (green or gray dots).

Figure 6

(a) Phase space of the normal-form Hamiltonian ${\mathcal{H}}_{r:s}(\theta ,I)$, Eq. (8), with ${\mathcal{H}}_0\left(I\right)$ and $\mathcal{V}\left(I\right)$ as determined in Sec. 4a (thin colored lines). (b–d) Phase space of the standard map, Eq. (3), at $\kappa =3.4$ (light gray lines and dots) and tori (thin colored lines) of the integrable approximations $H_{r:s}^n(q,p)$ obtained from the transformation $T_n\circ \cdots \circ T_1\circ T_0$ (b) $H_{r:s}^0(q,p)$, (c) $H_{r:s}^1(q,p)$, and (d) $H_{r:s}^N(q,p)$, $N=15$. The magnifications show the improvement of the integrable approximations.

Figure 7

Cost function $L$, Eq. (21), vs iteration step $n$.

Figure 8

Integrable approximations for the standard map, Eq. (3), at different parameters (a) $\kappa =2.9$, (b) $\kappa =3.3$, and (c) $\kappa =3.5$. Top: Phase space of the standard map (light gray lines and dots) and tori of the integrable approximation (thin colored lines). Bottom: Cost function $L$, Eq. (21), vs iteration step $n$.