Construction, analysis and DSP implementation of Hamiltonian conservative chaotic system based on permutation group rotation multiplication method

Conservative chaotic systems with high ergodicity and attractor-free advantages are advantageous in the field of data security. In addition, the methods for constructing the conservative chaotic system are fewer and more restrictive, leading to a limited number of constructed systems. Therefore, this paper proposes a method for constructing a skew-symmetric matrix within a Hamiltonian vector field using the rotation multiplication of the permutation group. By using the properties of skew-symmetric matrices, the conservation and non-conservation of the Hamiltonian and Casimir energies of the equations of the system are proved. To verify the validity and universality of the method, Hamiltonian conservative chaotic systems with multiple dimensions and different linear combinations are constructed. The dynamics of the 5-dimensional bivariate Hamiltonian chaotic system are analyzed, with the presence of symmetric and asymmetric multi-orbital coexistence and with a high complexity property that can be maintained around a complexity of 0.94 over a wide range of initial value intervals. Interestingly, the 5D Hamiltonian conservative system we constructed exhibits a wide range of parameters in its hyperchaotic and chaotic states, and with a large range of initial values. Moreover, the maximum Lyapunov exponent increases as certain initial values increase, the maximum Lyapunov exponent can exceed 20 within the initial value interval [0,50], exhibiting strong stability and complex chaotic characteristics. Finally, by using the NIST test and DSP hardware implementation, the system is further verified to have better pseudo-random and physical realizability, which lays the foundation for the application of the conservative chaotic system.