Invariant Tori And Its Destruction Mechanisms In Some Non-Smooth Systems

In the evolutions of dynamical systems,the invariant torus is one of the most common invariant sets.The KAM theory created by Kolmogorov,Arnold,Moser in the 60s of the last century and the Aubry-Mather theory by Mather,Aubry in the 80s have provided powerful tools for the study of invariant tori.However,for non-smooth systems,the applications of such theories are still limited.And the mechanisms of tori destructions should be investigated more deeply in such systems.Based on these problems,the existence of invariant tori and its destruction mechanisms for some non-smooth systems are presented in this dissertation.The specific works of this dissertation are as follows:Firstly,we briefly recall the history of KAM theory,Aubry-Mather theory and its applications in one-degree-of-freedom systems.We also summarise the main research results.Being prepared for the subsequent studies,we introduce the definition of twist map and several twist map theorems,including the Moser’s twist theorem,Moser’s small twist theorem,and the Aubry-Mather theorem.The connection between the nearly integrable twist map and the nearly integrable Hamiltonian system is explained.The usual transformations in Hamiltonian systems are summarised,since these transforma-tions are crucial for turning a system with nearly integrable dynamics into an explicit nearly integrable system.We first study the global dynamics of the inverted pendulum with impacts.When there is no exciting force,the phase space can be parted into three regions.With small exciting force,we show that the invariant tori exist in all these regions.Moreover,independent of the size of exciting force,the invariant tori near infinity always persist.At last,we design the numerical algorithm to calculate the invariant manifold under the effect of impact.We find that,using this algorithm,the destruction of tori in the low speed regions is due to the large expands of invariant manifolds of the saddle point.We investigate the existence of invariant tori and escaping orbits for breathing circle billiard under different smoothness conditions.When the system is one of~7class,we establish the twist map in high energy region,and prove the existence of invariant tori near infinity by Moser’s small twist theorem.When the system is piecewise smooth,we derive the normal form near the singularities in detail.Under suitable parameter,the main term of this normal form has an acceleration periodic orbit.We use the non-resonance condition to ensure its stability under perturbations,which implies the existence of escaping orbits and the destruction of invariant tori in high energy region when the smoothness of system is lost.Based on variational method,we study the existence of Aubry-Mather sets for breathing circle billiard when the system is only differentiable.First we explain the connection between the discrete variational problem and the solution of system.Then,under suitable assumptions about generating function,we establish the Aubry-Mather theory in the small twist case when the variational space has boundary.Such theory ensures that for arbitrarily small rotation number,the variational problem has its corre-sponding minimal configuration.At last,we show the generating function of breathing circle billiard satisfies the required assumptions in suitable coordinate,which implies the existence of Aubry-Mather sets.The previous contents are all about the non-smooth Hamiltonian system.Under the effect of dissipation,we study the excited oscillator with dry friction at last.First we give a formal definition of the phase map.Under reasonable assumptions,we prove that the phase map is a small perturbation of a circle rigid rotation.We use the glue technic in topology to show the existence of forward invariant torus.Moreover,we use the circle map theory and KAM theory to discuss the dynamics on the torus.Based on numerical method,we find that when the exciting force increases,the destruction of the torus is due to the presence of grazing orbit.