Modelling of both energy and volume conservative chaotic systems and their mechanism analyses

Abstract 4D Euler rotational equation is essential in providing the symplectic (skew symmetric) structures for the dynamics of rigid-body and fluid mechanics and generalized Hamiltonian systems. In this paper, a 4D Euler equation is proposed by combining the two generalized sub-Euler equations with two common axes. The conservations of both Hamiltonian and Casimir energies are proved theoretically for the proposed 4D Euler equation. Based on the 4D Euler equation, three different types of conservative chaotic systems are proposed. Firstly, through breaking the conservation of Casimir energy, but preserving Hamiltonian, a Hamiltonian conservative chaotic system is proposed. Conversely, by altering Hamiltonian, but preserving the Casimir energy, a Casimir conservative chaotic system is proposed. The initial bifurcation diagram demonstrates the richness of dynamics of the conservative chaotic system. The maximum of Lyapunov exponent reaches to 2097 indicating the randomness, the property of full space of Poincare map and wide bandwidth of power spectrum exhibit the ergodicity which is greatly useful in chaos-based cryptography. Furthermore, by breaking the conservations of both Hamiltonian and Casimir energy, but keeping the volume in phase space, a volume conservative chaotic system is proposed. The breaking either Hamiltonian or Casimir energy is taken as a method leading a system to producing chaos. The mechanics are studied in terms of the torque and energy for 4D rigid body, which reveals that the force twisting and energy flow and exchange are the causes of chaos production. The supremum bounds of both Hamiltonian conservative chaotic system and Casimir conservative chaotic system are analytically provided and verified. The Casimir power and Hamiltonian power methods are proposed to be an analytical measuring indexes of orbital mode.