Breaking of integrability and conservation leading to Hamiltonian chaotic system and its energy-based coexistence analysis

Abstract

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.

A class of 5D Hamiltonian conservative hyperchaotic systems with symmetry and multistability

“…In Table . 4, the equilibrium point types of HCCSs-3.1 are only the center point and saddle point. This is consistent with the related theory in the existing literature expressing that a system with a center point or a saddle point can produce a conservative chaotic orbit [48,49].…”

Modeling method of a class of 5D Hamiltonian conservative hyperchaotic systems with adjustable signal amplitude

“…Where C is the Casimir function, it serves as a motion constant in Hamiltonian systems [33]. According to the definition, we can determine all the Casimir functions of system Σ 12 [34,35]:…”

Construction of a novel five-dimensional Hamiltonian conservative hyperchaotic system and its application in image encryption

“…In 2020, Gu et al introduced a novel four-dimensional non-Hamiltonian conservative hyperchaotic system and conducted a comprehensive analysis of its dynamics [16].In 2022, Jie et al proposed a straightforward method for constructing a class of four-dimensional HCCSs with circuit simulation implementation [17]. However, most of the systems represented by Qi's construction idea are limited to four dimensions [18][19][20], with only a few discussions on five-dimensional systems [21], greatly constraining the richness of conservative chaos theory. In fact, chaotic systems in dimensions higher than or equal to five [22][23][24] exhibit a wider range of diverse and less predictable dynamical behaviors, making them highly promising for various applications.…”