基于亥姆霍兹定理计算动力学系统的哈密顿能量函数

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

亥姆霍兹定理表明任意空间矢量场可以分解为涡旋场和梯度场的叠加.由于电磁场变化和电磁波传播则导致电磁场能量的迁移,动力学振子和神经元处于复杂电磁环境下必然伴随能量的吸收和释放.在非线性混沌电路、电容器充电放电以及电感线圈感应过程中都伴随着能量的转换和迁移.包含量纲的非线性振荡电路可利用标度变换方法转换为无量纲的动力学方程.利用平均场理论,电场能量和磁场能量的转换可用若干非线性振荡电路的动力学方程来刻画.基于亥姆霍兹定理来研究一类无量纲非线性动力学系统的哈密顿能量计算问题,对于实际的非线性振荡电路,通过标度变换可快速计算其能量函数.该结果对于动力学系统自适应控制有重要的参考价值.

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.