Calculation of Hamilton energy function of dynamical system by using Helmholtz theorem

The Helmholtz theorem confirms that any vector field can be decomposed into gradient and rotational field. The supply and transmission of energy occur during the propagation of electromagnetic wave accompanied by the variation of electromagnetic field, thus the dynamical oscillators and neurons can absorb and release energy in the presence of complex electromagnetic condition. Indeed, the energy in nonlinear circuit is often time-varying when the capacitor is charged or discharged, and the occurrence of electromagnetic induction is available. Those nonlinear oscillating circuits can be mapped into dynamical systems by using scale transformation. Based on mean field theory, the energy exchange and transmission between electronic field and magnetic field can be estimated by appropriate nonlinear dynamical equations for oscillating circuits. In this paper, we investigate the calculation of Hamilton energy for a class of dimensionless dynamical systems based on Helmholtz's theorem. Furthermore, the scale transformation can be used to develop dynamical equations for the realistic nonlinear oscillating circuit, so the Hamilton energy function could be obtained effectively. These results can be greatly useful for self-adaptively controlling dynamical systems.