Double Resonance: Geometric Description

This chapter examines the geometrical structure for the system at double resonance. After describing the normal form near a double resonance, it reduces the system to the slow mechanical system with perturbation. The system is conjugate to a perturbation of a two degrees of freedom mechanical system after a coordinate change and an energy reduction. The chapter then formulates the non-degeneracy conditions and theorems about their genericity. It also considers the normally hyperbolic invariant cylinders, and sketches the proof using local and global maps. The periodic orbits obtained in Theorem 4.4 correspond to the fixed points of compositions of local and global maps, when restricted to the suitable energy surfaces.

Forcing equivalence between kissing cylinders

This chapter formulates and proves the jump mechanism. It constructs a variational problem which proves forcing equivalence for the original Hamiltonian using Definition 6.18. It first constructs a special variational problem for the slow mechanical system. A solution of this variational problem is an orbit “jumping” from one homology class to the other. The chapter then modifies this variational problem for the fast time-periodic perturbation of the slow mechanical system. This is achieved by applying the perturbative results established in Chapter 7. Recall the original Hamiltonian system near a double resonance can be brought to a normal form and this normal form, in turn, is related to the perturbed slow system through coordinate change and energy reduction. The variational problem for the perturbed slow system can then be converted to a variational problem for the original.

Forcing Relation

This chapter describes forcing relations, different diffusion mechanisms, and Aubry-Mather types. The Aubry set can be decomposed into disjoint invariant sets called static classes, which gives important insight into the structure of the Aubry set. The chapter then formulates Theorem 2.2 and shows that it implies the book's main theorem. It utilizes the concept of forcing equivalence. The actual definition is not important for the current discussions, instead, the chapter states its main application to Arnold diffusion. The chapter also looks at symplectic coordinate changes. The definition of exact symplectic coordinate change for a time-periodic system is somewhat restrictive, and in particular, it does not apply directly to the linear coordinate change performed at the double resonance.

Derivation of the slow mechanical system

This chapter proves various normal form results and formulates the coordinate changes that are used to derive the slow system at the double resonance. The discussions here apply to arbitrary degrees of freedom. The results also apply to the proof of the main theorem by restricting to the case n = 2. First, the chapter reduces the system near an n-resonance to a normal form. After that, it performs a coordinate change on the extended phase space, and an energy reduction to reveal the slow system. The chapter then describes a resonant normal form, before explaining the affine coordinate change and the rescaling, revealing the slow system. Finally, it discusses variational properties of these coordinate changes.

Normal Forms and Cohomology Classes at Single Resonances

This chapter discusses the single-resonance non-degeneracy conditions and normal forms. It then formulates Theorem 3.3, which covers the forcing equivalence in the single-resonance regime. The classical partial averaging theory indicates that after a coordinate change, the system has the normal form away from punctures. In order to state the normal form, one needs an anisotropic norm adapted to the perturbative nature of the system. The chapter also uses the idea of Lochak to cover the action space with double resonances. A double resonance corresponds to a periodic orbit of the unperturbed system. Finally, the chapter looks at a lemma which is an easy consequence of the Dirichlet theorem.

Estimate of Head Posture Based on Coordinate Transformation with MP-MTM-LSTM Network

Fatigue monitoring can effectively reduce or even avoid traffic accidents. Head posture estimation is one of the focuses in the field of fatigue monitoring. In this paper, according to the coordinate rotation transformation and neural network theory, a method for predicting the change of head posture with sight-line coordinates is proposed. First, the coordinate rotation transformation theory is used to replace the head posture change amount with the coordinate change amount, and the first-order difference value of the sight-line point coordinate is obtained by the difference method. Then, under the unified Cartesian coordinate system, the MP-MTM-LSTM neural network is established with the input information of first-order difference value and the output information of coordinate change amount. The innovation of this method is that the Cartesian coordinate change is employed instead of the Euler angle transformation. In the model verification phase, the true value of the head pose is collected by the posture meter. The experimental results show that the absolute error between the predicted value and the true value estimated by the new method is less than 15%. In the field of fatigue monitoring, the proposed method can estimate the amount of head posture change effectively, which is suitable for the case where the head center point is not fixed.