## 4 The Phase Space Structure of the Main Terms
### The Resonant Dynamics  
A non-autonomous dynamical system can be converted into an autonomous one by adding one dimension to the phase space. Therefore, without loss of generality, in what follows, we assume to have an autonomous 3-degree-of-freedom nearly integrable Hamiltonian system. Referring to the classical theory presented in [16], let us take into account as a concrete example, useful to our purpose, the resonant Hamiltonian:  
\[\mathcal{H}_{\textit{res}}(\mathbf{I},\boldsymbol{\psi})=\mathcal{H}_{0}( \mathbf{I})+\varepsilon f(\mathbf{I})\cos(\mathbf{k}\cdot\  
After performing a coordinate change \(\Phi\), the new one-degree-of-freedom (1-DOF) Hamiltonian  
\[\mathcal{H}^{\prime}_{res}(J_{1},\theta_{1})=\mathcal{H}^{\prime}_{0}(\mathbf{J} )+\varepsilon f^{\prime}(\mathbf{J})\cos\theta_{1}, \tag{22}\]  
describes a two-dimensional motion taking place along the level curves \(J_{2}=J_{2}^{*}\) and \(J_{3}=J_{3}^{*}\) in the \((J_{1},\theta_{1})\) plane, where \(J_{1}\) is the action conjugate to the critical angle \(\theta_{1}\) and \(J^{*}=\Phi(\mathbf{I}^{*})\). According to the Standard Resonance Model (SRM), also known as _integrable approximation_ in [16], the Hamiltonian \(\mathcal{H}^{\prime}_{res}\) can be developed in Taylor series of \(J_{1}\) around \(J_{1}^{*}\), which, as a first approximation, represents a pendulum-like dynamics in the neighborhood of the exact resonance. Following [16], the resonant region is the libration region around the elliptic equilibrium point and its _maximum libration width_ measured at the apex of the separatrix is given by  
\[|J_{1}-J_{1}^{*}|\leq 2\sqrt{\left|\frac{c}{\beta}\right|}. \tag{23}\]  
If there are two or more resonances, then we can separately study the dynamics corresponding to each one, making the assumption that they are isolated. The resulting motion is pendulum-like with an appropriate coordinate change for every single resonance and the pendulum-like model may give a good approximation as long as the libration regions remain isolated. If any resonance overlap occurs, then the pendulum-like model breaks down. The separatrices of different resonances are connected if two or more resonances overlap; therefore, an initial condition in this region may produce jumps from one libration region to one other, showing chaotic diffusion [16]. Another scenario in which the classical approach does not provide a reliable description of the real resonant dynamics occurs when the exact resonance is not a good approximation of the real resonance. The real equilibria arising from the suitable Hamiltonian \(\mathcal{H}^{\prime}_{res}\) are solutions of:  
\[\left\{ \begin{array}{l}J_{1}=\varepsilon f^{\prime}(\mathbf{J})\sin\theta_{1 }=0,\\ \dot{\theta}_{1}=\frac{\partial\mathcal{H}^{\prime}_{0}}{\partial J_{1}}( \mathbf{J})+\varepsilon\frac{\partial f^{\prime}}{\partial J_{1}}(\mathbf{J} )\cos\theta_{1}=0.\end{array} \right. \tag{24}\]  
As in the pendulum case, the condition \(J_{1}=0\) implies \(\theta_{1}=n\pi\) with \(n\in\mathbb{Z}\). By replacing this solution in the second equation of (24), then the latter splits into two different equations, namely  
\[\left\{ \begin{array}{l}\frac{\partial\mathcal{H}^{\prime}_{0}}{\partial J_ {1}}(\mathbf{J})+\varepsilon\frac{\partial f^{\prime}}{\partial J_{1}}( \mathbf{J})=0,\\ \frac{\partial\mathcal{H}^{\prime}_{0}}{\partial J_{1}}(\mathbf{J})- \varepsilon\frac{\partial f^{\prime}}{\partial J_{1}}(\mathbf{J})=0.\end{array} \right. \tag{25}\]  
Hence, the solutions of (25) are not necessarily the same: the stronger the perturbing effects, controlled by \(\varepsilon\), the more the solutions of the system (25) are separated in the phase space. In such a case, the [PERSON] approximation, which characterizes the classical approach, fails to capture a _deep asymmetry_.