Physical equations provide good access to explore correlation between physical variables in some realistic systems such as nonlinear circuits and electromechanical devices, and further dynamical analysis requires consistence of physical units on the involved variables and parameters. Therefore, proper nondimensionalization is essential for revealing the intrinsic dynamics of the system by converting the physical equations into equivalent dimensionless theoretical models. For example, for a RLC (C: capacitor, L: inductor, R: resistor) circuit, both [RC] and [LC]1/2 can be used to time scale for obtaining dimensionless time variable as τ = t/RC, or τ = t/(LC)1/2, while a mechanical oscillator requires τ = t/[m/K]1/2, in which K and m denote elasticity coefficient for the spring and mss of the oscillator, respectively. This paper investigates an FitzHugh-Nagumo (FHN) neuron coupled with an electromechanical arm device, which involves two inherent time scales associated with the [RC] and the [m/K]1/2. By introducing different time bases, two nondimensional formulations are derived and compared. The results show that the two theoretical models are related through a linear time-scaling transformation, indicating their dynamical equivalence and consistence. Numerical simulations based on bifurcation diagrams and the largest Lyapunov exponent confirm that the two formulations exhibit identical qualitative dynamical behaviors. Furthermore, energy analysis reveals distinct distribution patterns under different dynamical regimes. Finally, simulation on Multisim is verified for clarifying the influence of time scale on the theoretical models and the dynamical characteristic. This study provides a systematic understanding of time-scale effects in electromechanical nonlinear systems and offers guidance for interpreting models under different nondimensional frameworks.
Graphical abstract
Schematic diagram for electromechanical device coupled by a neural circuit. The red box is the electromechanical arm/leg branch, see Ref.[13]. Time scale can be selected with τ = t/RC, or τ = t/(LC)1/2, τ = t/[m/K]1/2, in which K and m denote elasticity coefficient for the spring and mss of the oscillator, respectively, the theoretical model will be presented in different forms.
