Nonlinear dynamics plays a very important role in undergraduate physics education, and the power spectrum, as an important frequency domain analysis method, is often used to analyze and predict the nonlinear dynamic behavior of systems. This paper first derives the specific form of the power spectrum using the Fourier transform, and then applies it to the important model for nuclear fusion confinement mode transition—the ZCD model, observing the dynamic behavior of the system. By adjusting the amplitude of the heating power in the ZCD model and plotting the corresponding power spectrum and phase diagrams, we observe and compare the changes in the nonlinear dynamic properties of the model under different parameters. The study found that as the amplitude of the heating power increases, the turbulence intensity in the system enters chaos through a period-doubling route, making the system behavior unpredictable. During this transition, the position of the system's period-doubling bifurcation points matches the pattern described by the Feigenbaum constant.

Nonlinear dynamics plays an active role in university physics such as double pendulum and resonant circuit. As an important concept in nonlinear dynamics, Jacobian matrix is often adopted to analyze the properties of fixed points. In this paper, we derive the specific form of the Jacobian matrix in a classic two-dimensional system, and apply it to the L-H mode transition model of magnetic confinement fusion, then solve the eigenvalues of the Jacobian matrix at each fixed point in the ZCD model. We adjust the external heating power in the model and analyze the nonlinear dynamic properties of the model by the Jacobian matrix. It is found that the variance of external heating power changes the size of limit cycle in ZCD model; and the higher the power is, the larger the limit cycle will be. The expansion of the limit cycle radius leads itself to intersect with the saddle point into a homologous orbit, which induces the system to occur homoclinic bifurcation.