The Klein–Gordon equation with cubic nonlinearity is studied in theoretical and numerical ways. The master equation is known to exhibit spontaneous symmetry breaking. Based on the high-precision numerical method [1] consisting of the Fourier spectral method for space and the implicit Runge–Kutta method for time, the detailed structure of dynamical systems around Lyapunov-stable stationary states is studied in Ref. [2]. Here, it is notable that the appearance of breather waves is suggested to be associated with the stability of Lyapunov functions for stationary solutions [3]; more precisely, the breather solution appears in the critical situation when the constant stationary state starts to lose its stability. In this paper, which is also based on the high-precision numerical method [1], finite-dimensional scatter plots are introduced to infinite-dimensional dynamical systems of coupled nonlinear Klein–Gordon equations. The proposed plot is a kind of finite-dimensional representation of originally infinite-dimensional dynamical systems [4]. Scatter plots show a wide variety of geometric shapes, which sometimes have a fractal structure. In particular, the difference between the global existence and finite-time blow-up of solutions is illustrated in a geometric way.

References
[1] Y. Takei, Y. Iwata, Axioms 2022, 11(1), 28
[2] Y. Takei, Y. Iwata, Springer Proceedings in Mathematics and Statistics (SPMS), accepted; arXiv:2509.12272
[3] Y. Iwata and Y. Takei, AIP conf. proc., accepted; arXiv:2309.00822
[4] Y. Iwata, Y. Takei, AIP Conf. Proc.; accepted, arXiv:2309.0082