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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.
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Keywords:
- Helmholtz theorem /
- energy function /
- oscillating system /
- chaotic system
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[1] Wang H Q, Yu L C, Chen Y 2009 Acta Phys. Sin. 58 5070 (in Chinese)[王慧巧, 俞连春, 陈勇2009 58 5070]
[2] Wang C N, Chu R T, Ma J, Huang L 2015 Complexity 21 370
[3] Wu H G, Bao B C, Liu Z, Xu, Q, Jiang P 2016 Nonlinear Dyn. 83 893
[4] Li Q D, Zeng H Z, Li J 2015 Nonlinear Dyn. 79 2295
[5] Liang Y, Yu D S, Chen H 2013 Acta Phys. Sin. 62 158501 (in Chinese)[梁燕, 于东升, 陈昊2013 62 158501]
[6] Li Z J, Zeng Y C, Tang Z P 2014 Acta Phys. Sin. 63 098510 (in Chinese)[李志军, 曾以成, 谭志平2014 63 098501]
[7] Neumann E, Pikovsky A 2003 Eur. Phys. J. B 34 293
[8] Ren G D, Tang J, Ma J, Xu Y 2015 Commun. Nonlinear Sci. Numer. Simulat. 29 170
[9] Wu X Y, Ma J, Yuan L H, Liu Y 2014 Nonlinear Dyn. 75 113
[10] Babacan Y, Kaçar F, Grkan K 2016 Neurocomputing 203 86
[11] Li J J, Tang J, Ma J, Du M M, Wang R, Wu Y 2016 Sci. Rep. 6 32343
[12] Lv M, Ma J 2016 Neurocomputing 205 375
[13] Ma J, Qin H X, Song X L, Chu R T 2015 Int. J. Mod. Phys. B 29 1450239
[14] Song X L, Wang C N, Ma J, Tang J 2015 Sci. China:Technol. Sci. 58 1007
[15] Hindmarsh J L, Rose R M 1982 Nature 296 162
[16] Song X L, Jin W Y, Ma J 2015 Chin. Phys. B 24 128710
[17] Ma J, Tang J 2015 Sci. China:Technol. Sci. 58 2038
[18] Lv M, Ma J 2016 Nonlinear Dyn. 85 1479
[19] Kobe D H 1986 Am. J. Phys. 54 552
[20] Sarasola C, Torrealdea F J, d'Anjou A, Moujahid A, Graña M 2004 Phys. Rev. E 69 011606
[21] Pinto R D, Varona P, Volkovskii A R, Szcs A, Abarbanel H D I, Rabinovich M I 2000 Phys. Rev. E 62 2644
[22] Torrealdea F J, d'Anjou A, Graña M, Sarasola C 2006 Phys. Rev. E 74 011905
[23] Li F, Yao C G 2016 Nonlinear Dyn. 84 2305
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