Modeling, Dynamic Analysis And Realization Of A Class Of Four-dimensional Rigid-body Chaotic Systems

The 3D Euler rotational equation is the critical equation in rigid body mechanics and mathematics,and the research has been mature.The 4D Euler equation for the rigid body can be the conservative part of a 4D Hamiltonian system and the basis of the generation of 4D conservative and dissipative systems.Therefore,modeling the 4D Euler equation is significant in theory.Because the Lyapunov dimension of the conservative chaotic system is equivalent to the system’s dimension,it has richer ergodicity,which is suitable for generating pseudo-random numbers in encryption.It is necessary to propose a 4D conservative chaotic system via mechanism analysis and reveal energy and force that produces different dynamical behaviors.A practical system is forced-dissipated;it is significant to model a dissipative chaotic rigid-body system based on the 4D Euler equation and give bifurcation and force analyses.The inertial force,damping and external force should be taken into account.The thesis includes the following content:(1)A concrete model of 4D Euler rotational equation for rigid-body system is modeled.The conservations are analyzed,and three constants are uncovered.The mechanism of the system producing periodic orbit is revealed through the integrability principle.The integrability is proved.The simulations verified the correctness of the integrability.The analyses of energy and force for the system is provided using KAM theory and the Kolmogorov model.(2)Based on the 4D Euler equation,by breaking Casimir’s energy conservation and the complete integrability of the system but preserving Hamiltonian conservatism,a Hamiltonian conservative chaotic system is constructed.The equilibrium point of the system is analyzed,and the characteristics of the equilibrium point of the conservative chaotic system are verified.The initial bifurcation diagram of Hamilton energy is used to show the relationship between the different motion state distributions of the system and the Hamilton energy.By comparing the maximum Lyapunov exponent with the distribution of Hamilton energy on the 2D plane,it is explained that Hamilton energy is an essential factor affecting the system dynamics behavior.According to the concept and properties of hidden attractors in dissipative systems,the concept of hidden chaos in conservative systems is given.The Poincaré map analyzes the transition process of the system dynamic behavior and shows the rich coexistence characteristics of different dynamic behaviors in the system.(3)According to Kolmogorov model,a 4D forced dissipative rigid body chaotic system model is constructed based on the proposed 4D Euler equation.The distribution of double-parameter fork bifurcation and classification of equilibrium points with the change of parameters are analyzed.The influence of different structural parameters on the dynamic behavior of the system is studied.The difference between the dynamic behavior of the conservative system and the dissipative system is clarified through parameter bifurcation analysis and orbit simulation.The Poincaré map verifies that conservative chaotic systems have better ergodicity than dissipative chaotic systems.The Casimir energy is given and the influence of Casimir power on system dynamics behavior is explained.The torque field of the dissipative chaotic system is decomposed into conservative torque,dissipative torque and external torque,and the corresponding energy is decomposed.Based on the Hamiltonian energy of the system,the dynamic characteristics of the system under different torques are analyzed.(4)Based on the Multisim circuit simulation platform,the circuit of the Hamiltonian conservative chaotic system and the dissipative chaotic system are realized by using the energy storage element and the reverse multiplier.The circuit simulation results verify the theoretical and numerical results.