Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic,chaos and strong chaos |
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| Affiliation: | 1. Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy, School of Electrical Engineering and Automation, Tianjin Polytechnic University, Tianjin 300387, China;2. School of Mechanical Engineering, Tianjin Polytechnic University, Tianjin 300387, China;1. Department of Product Design, Tianjin University of Science and Technology, Tianjin 300222, China;2. Tianjin Key Laboratory of Integrated Design and On-line Monitoring for Light Industry & Food Machinery and Equipment, College of Mechanical Engineering, Tianjin University of Science and Technology, Tianjin 300222, China;3. School of Electronic Information and Automation, Tianjin University of Science and Technology, Tianjin 300222, China;4. Department of Electrical Engineering, University of South Africa, Florida 1710, South Africa;5. College of Artificial Intelligence, Nankai University, Tianjin 300350, China;1. Jiangsu Key Laboratory of Meteorological Observation and Information Processing, Nanjing University of Information Science & Technology, Nanjing 210044, China;2. Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science & Technology, Nanjing 210044, China;3. Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA;4. Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, Lodz 90-924, Poland |
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| Abstract: | In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10 K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos. |
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