Analog Circuit Implementation and Geometric Structure Analysis of Phase Portrait in Planar Polynomial Dynamic System

The study of planar polynomial dynamical systems in Hilbert’s 16th problem offers critical insights into nonlinear dynamical systems. However, current research rarely conducts quantitative analysis of the system from the perspective of phase portrait geometric structures, nor does it feature physical implementations. To address this gap, this paper will investigate planar polynomial systems with four, six, and nine equilibrium points from the perspective of geometric structures, revealing intrinsic characteristics such as equilibrium point distributions and homoclinic/heteroclinic orbits. Then, the system is investigated via Hamiltonian energy to explain the fundamental causes behind the geometric structure formation. Most importantly, a method for hardware circuit implementation of geometric structures is successfully developed for the first time. Oscilloscope observations show consistency between experimental results and theoretical predictions, and the hardware implementation lays a foundation for engineering applications. Overall, this study contributes to the interdisciplinary integration of geometric topology, nonlinear dynamics, analog circuit design, and engineering applications, which provides a new methodological framework for exploring complex dynamical systems.