论文标题
在耗散系统模型中的Arnold扩散
Arnold diffusion in a model of dissipative system
论文作者
论文摘要
对于由旋转器和摆的机械系统，通过一个小的，时间周期性的哈密顿扰动耦合，Arnold扩散问题断言“扩散轨道”的存在，旋转器的能量通过耦合参数的大小而成的旋转器的能量就可以通过耦合参数的大小而成。关于建立此类系统的Arnold扩散有大量文献。在这项工作中，我们考虑了与耦合一起添加额外的耗散扰动的情况。因此，获得的系统不是符合性的，而是共态的。我们在耗散参数上提供明确的条件，以使所得系统仍然表现出能量增长。 Chirikov猜想Arnold扩散可能在较小耗散的系统中发挥作用。在这项工作中，耦合是仔细选择的，但是我们提出的机制可以适应一般耦合，我们将在将来的工作中处理一般案例。
For a mechanical system consisting of a rotator and a pendulum coupled via a small, time-periodic Hamiltonian perturbation, the Arnold diffusion problem asserts the existence of `diffusing orbits' along which the energy of the rotator grows by an amount independent of the size of the coupling parameter, for all sufficiently small values of the coupling parameter. There is a vast literature on establishing Arnold diffusion for such systems. In this work, we consider the case when an additional, dissipative perturbation is added to the rotator-pendulum system with coupling. Therefore, the system obtained is not symplectic but conformally symplectic. We provide explicit conditions on the dissipation parameter, so that the resulting system still exhibits energy growth. The fact that Arnold diffusion may play a role in systems with small dissipation was conjectured by Chirikov. In this work, the coupling is carefully chosen, however the mechanism we present can be adapted to general couplings and we will deal with the general case in future work.
