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In this paper, a dynamical systems (DS) approach is proposed for the analysis and design of bio-inspired silicon central pattern generator (CPG) systems. Based on this approach, a new leaky-integrate-and-leaky-discharge oscillator circuit is proposed that has dynamical properties closer to biological half-center oscillators while being power and area efficient. The membrane potential charges and discharges through a single resistor eliminating mismatch in charging and discharging phases. Switched-capacitor (SC) and floating-gate wide input linear range operational transconductance amplifier (FGOTA) based approaches have been proposed to implement the resistor. Both approaches enable controllable and large resistances in a small area. Oscillation frequency can be easily controlled by the frequency of switching in SC based and bias current in FGOTA based implementations, which are very useful for global change of oscillation frequency in an array of oscillators. Dynamical systems analysis has shown that when it is used as a single oscillator, the proposed circuit is able to produce a phase response curve (PRC) close to that of a lamprey CPG system. By applying averaging theory to a system of coupled oscillators, the averaged H and G functions for unidirectional and bidirectional coupling cases are obtained. Analysis of these functions shows our circuit's superior capability to achieve entrainment when driven by a periodic input (e.g., from sensory feedback) and reach equilibrium even with high frequency mismatch.