Hamilton energy dependence and quasi-synchronization behaviors of non-identical dynamic systems

It becomes crucial to calculate the Hamilton energy and its evolution when exploring dynamics characters in nonlinear systems. Based on the Helmholtz's theorem, the Hamilton energy of the HR system in an electric field and the modified Chua's circuit are calculated and analyzed. Then the energy feedback is used to control the system to the desired states. The results show that this method can not only effectively suppress the chaotic behavior of the system, but also induce abundant discharge behavior. In addition, the synchronization behavior of the two HR systems or Chua's circuits is explored respectively using the adaptive synchronization control method. Taking into account that different Lyapunov functions will yield different results, we propose the idea that the coefficient ratio of each state variable in the Hamilton energy function can be utilized to design the Lyapunov function for the dynamic systems in this paper. It is found that the controller and parameter adaptive law obtained by this method can make the drive system and response system realize quasi-synchronization. This result provides an effective method for further exploring adaptive synchronization between neurons.