Interplay between dynamics and geometry in integrable systems and engineering problems

The geometry of the phase space imposes restrictions on the dynamics of the system, and the system’s dynamics reflect the geometric properties of the space. In this thesis, we study several geometric objects/structures which either come from the phase space or arise from certain dynamical systems and look into their influences on the dynamics.In Chapter 2 we study interactions between Hamiltonian monodromy and Maslov indices, and generalize these interactions to nonHamiltonian integrable systems. In Chapter 3 we explore the bundle structures (e.g. the bundle of Lagrangian planes, the Maslov S1 bundles) over symplectic manifolds behind/related to the concept of Maslov indices and study the implications of the structural properties of the bundles on the symplectic dynamics on the manifolds. In Chapters 4 and 5, we discuss a particular type of singular fibers in integrable systems and the foliation structures in a vicinity of the fibers. We show how such a fiber fails to be homeomorphic to S2 (and hence can only be a pinched torus), and provide detailed discussions on the construction of a small neighbourhood of a pinched torus, and, the structure of the restricted Maslov S1 bundle over such a fiber. In Chapter 6 we discuss the topology of domains of attraction of stable manifolds with uniform asymptoticity and In Chapter 7 we discuss a particular case in which the communicationnetwork of a robotic swarm changes its topology in the movement