Arnold diffusion for a complete family of perturbations with two independent harmonics

Figure 1.  Plane $\varphi \times I$ of inner dynamics for $\mu = 0.75$ and $\varepsilon = 0.01$

Figure 3.  Finding $\tau^*(I,\theta)$ using the straight line $\sigma = \varphi$

Figure 6.  Examples of piecewise smooth global scattering maps. The orbits of scattering maps are represented by the blue lines. In the red zones the values of $I$ on such orbits decrease, in the green one the values of $I$ increase

Figure 2.  $\left|\alpha(I)\right|$ and $\left|\beta(I)\right|$ : Behavior of the crests and tangencies

Figure 4.  Comparison between $\xi_{\text{M}}(I,\varphi)$ and $\eta_{\text{M}}(I,\sigma)$ for $\mu = 0.5$, $I = 0.68$ and $I = 0.72$ respectively

Figure 5.  Different phase space of scattering maps $\mathcal{S}(I,\theta)$ associated to the same horizontal crest $C_{\text{M}}(I)$, for $\mu = 0.6$ and $\varepsilon = 0.01$. The orbits of scattering maps are represented by the blue lines which are, up to $\mathcal{O}(\varepsilon^2)$, level sets of the reduced Poincaré function $\mathcal{L}^*(I,\theta)$. In the red zones the values of $I$ on such orbits decrease, in the green one the values of $I$ increase. The white regions are regions where $\left|\mu\alpha(I)\sin\varphi\right|>1$ is satisfied

Figure 7.  A piecewise smooth global scattering map divided into 3 regions. The vertical black lines are the boundaries of the domains of smooth scattering maps