Qi Guoyuan, Gou Ting, Hu Jianbing, Chen Guanrong
Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy, School of Electrical Engineering and Automation, Tiangong University, Tianjin 300387, People's Republic of China.
School of Mechanical Engineering, Tiangong University, Tianjin 300387, People's Republic of China.
Chaos. 2021 Jan;31(1):013101. doi: 10.1063/5.0012236.
In this paper, a four-dimensional conservative system of Euler equations producing the periodic orbit is constructed and studied. The reason that a conservative system often produces periodic orbit has rarely been studied. By analyzing the Hamiltonian and Casimir functions, three invariants of the conservative system are found. The complete integrability is proved to be the mechanism that the system generates the periodic orbits. The mechanism route from periodic orbit to conservative chaos is found by breaking the conservation of Casimir energy and the integrability through which a chaotic Hamiltonian system is built. The observed chaos is not excited by saddle or center equilibria, so the system has hidden dynamics. It is found that the upgrade in the Hamiltonian energy level violates the order of dynamical behavior and transitions from a low or regular state to a high or an irregular state. From the energy bifurcation associated with different energy levels, rich coexisting orbits are discovered, i.e., the coexistence of chaotic orbits, quasi-periodic orbits, and chaotic quasi-periodic orbits. The coincidence between the two-dimensional diagram of maximum Lyapunov exponents and the bifurcation diagram of Hamiltonian energy is observed. Finally, field programmable gate array implementation, a challenging task for the chaotic Hamiltonian conservative system, is designed to be a Hamiltonian pseudo-random number generator.
本文构建并研究了一个产生周期轨道的四维欧拉方程保守系统。保守系统常产生周期轨道的原因鲜有研究。通过分析哈密顿函数和卡西米尔函数，找到了保守系统的三个不变量。证明了完全可积性是系统产生周期轨道的机制。通过打破卡西米尔能量守恒和可积性，找到了从周期轨道到保守混沌的机制路径，由此构建了一个混沌哈密顿系统。观测到的混沌并非由鞍点或中心平衡点激发，因此该系统具有隐藏动力学。发现哈密顿能级的提升违反了动力学行为的顺序，从低或规则状态转变为高或不规则状态。从与不同能级相关的能量分岔中，发现了丰富的共存轨道，即混沌轨道、准周期轨道和混沌准周期轨道的共存。观测到最大李雅普诺夫指数的二维图与哈密顿能量的分岔图之间的一致性。最后，针对混沌哈密顿保守系统的一项具有挑战性的任务——现场可编程门阵列实现，被设计为一个哈密顿伪随机数发生器。
