Scatter plots on the dynamical system of coupled nonlinear Klein–Gordon equations

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