Figure 2

Numerical solutions of the system for the case \(N=2\). (a) Plots of the Fourier amplitudes \(F_{1}(s)\) and \(F_{2}(s)\) (red and red-dashed), \(G_{1}(s)\) and \(G_{2}(s)\) (blue and blue-dashed), and \(H_{1}(s)\) and \(H_{2}(s)\) (green and green-dashed). The solution selects \(\nu = 1.10091\), \(H_{1}^{(0)}= 3.13151 \), and \(H_{2}^{(0)} = 0.0993008\). (Inset) The plot of the function \(L(s)\) shows that it never crosses the critical points defined by condition (40). From \(L(s)\), \(F_{1}(s)\), and \(G_{1}(s)\), we built the 3D heteroclinic orbit of Fig. 1. (b) Same plot as (a) but in semilog scale, \(\log |F_{n}(s)|\), \(\log |G_{n}(s)|\), and \(\log |H_{n}(s)|\) vs. s. One notices readily the exponential behavior for the inner behavior of the amplitudes \(F_{n}(s) \sim e^{2s}\) and \(G_{n}(s) \sim e^{2s} \). Additionally, we observe the right slopes \(1/\nu \) and \(2/\nu \) in the large s limit