Nonlinear Dynamics

, Volume 95, Issue 3, pp 2063–2077 | Cite as

Modelings and mechanism analysis underlying both the 4D Euler equations and Hamiltonian conservative chaotic systems

  • Guoyuan QiEmail author
Original Paper


Four sub-Euler equations for four sub-rigid bodies are generalized by extending the dimension of the state space from 3D to 4D. Six integrated 4D Euler equations are proposed by combining any two of the four sub-Euler equations with two common axes. These 4D equations are essential in providing the symplectic structures for the dynamics of rigid body and fluid mechanics and generalized Hamiltonian systems. The conservation of both the Hamiltonian and Casimir energies is proved for the six 4D Euler equations. Conservative chaos is more advantageous than dissipative chaos regarding ergodicity, the distribution of probability, and fractional dimensions in the application of chaos-based secure communications and generation of pseudo-random numbers. Six 4D Hamiltonian chaotic systems are proposed through breaking of the conservation of Casimir energies and preserving of the Hamiltonian energies, one of which is analyzed in detail. This system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 80 K), a large state amplitude and energy, and power spectral density with a wide bandwidth. The system passed the NIST tests performed on it. Therefore, strong pseudo-randomness of this Hamiltonian conservative chaotic system is confirmed. The Casimir power method is verified as an alternative analytical measuring index of orbital mode to the Lyapunov exponent. The force interaction and exchange in Casimir energy are the causes of chaos production. The mechanism underlying the transition from regular orbits to irregular orbits to stronger irregular orbits is studied using the Casimir power and the variability of physical parameters of the chaotic system. The supremum is also found using the property of Hamiltonian conservation.


4D Euler equations Hamiltonian conservative chaotic system Rigid body Casimir power NIST tests Chaos bound 



This work is supported by the National Natural Science Foundation of China (61873186) and the Tianjin Natural Science Foundation (17JCZDJC38300). We thank Richard Haase, Ph.D., from Liwen Bianji, Edanz Group China (, for editing the English text of a draft of this manuscript.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy, School of Electrical Engineering and AutomationTianjin Polytechnic UniversityTianjinChina

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