Trees, trunks, and branches -- bifurcation structure of time-periodic solutions to $u_{tt}-u_{xx}\pm u^{3}=0$

TL;DR

This work develops a Galerkin-based framework to analyze time-periodic solutions of the 1D cubic wave equation with Dirichlet boundaries, focusing on the intricate network of bifurcations revealed by sparse multi-mode interactions. By isolating reducible subsystems, the authors construct an infinite reducible tree of trunks and branches that captures the energy–frequency organization and predicts solutions for every frequency $\Omega>0$ with potentially unbounded energy. The approach complements small-amplitude existence proofs and illustrates fractal-like self-similarity and rescaling symmetries, linking trunks, branches, and higher-order interactions to the full Galerkin dynamics. Perturbations by extra modes modify this tree into loops but preserve the core structure, suggesting robustness of the reducible-tree picture under non-sparse couplings and supporting the conjecture of dense, multi-scale families of time-periodic solutions.

Abstract

We propose a systematic approach to analysing the complex structure of time-periodic solutions to the cubic wave equation on an interval with Dirichlet boundary conditions first reported in arXiv:2407.16507. The analysis we present is based on a detailed study of sparse mode interactions suggested by the previous numerical work. Our results complement prior rigorous existence proofs and suggest that solutions exist for any frequency, however, they may be arbitrarily large.

Figures (5)

• Figure 1: A primary branch (black), spanned by modes $A \cos{\tau}\,\sin{x}$ and $B \cos (2m+1)\tau\, \sin (2n+1) x$, connecting the primary trunk (blue) with a secondary trunk (red). Here we show the $(m,n)=(1,2)$ case, corresponding to the secondary trunk bifurcating from frequency $\Omega=5/3$. On the right plot we show mode amplitudes, $A$ and $B$, along the respective elements of the reducible tree. Note that due to the symmetry of the problem the four solutions (four quarters), see Eq. \ref{['eq:24.06.18_01']}, yield the same curves on the $E-\Omega$ diagram. The blue and red dots indicate bifurcation points on primary ($B=0$) and secondary ($A=0$) trunks, respectively.
• Figure 2: Illustration of higher order structures appearing in the reducible tree: solutions spanned by modes the $A\cos{\tau}\sin{x}$, $B_1\cos (2m_1+1) \tau\, \sin (2n_1+1) x$, and $B_2 \cos (2m_2+1) \tau\, \sin (2n_2+1) x$, with $(m_1,n_1)=(1,3)$ and $(m_2,n_2)=(3,8)$. The primary (blue) and secondary (light and dark red) trunks are joined by the two-modes solutions: the primary branches (black and grey) and branch of type $\{(1,3),(3,8)\}$ (orange). In addition, there exists a three-modes solution (green) connecting one of the primary branches with the branch of order two. The bifurcations points are the end points of the curves corresponding to branches while trunks extend to infinity, cf. \ref{['eq:24.05.11_07']} and \ref{['eq:24.05.11_05']}. The bifurcation structure is additionally visualised on the 3d diagram of the solution space $(A_{1},B_{1},B_{2})$; for clarity we show only one octant.
• Figure 3: Comparison of solutions to the truncated Galerkin system (left column) and the corresponding $N$-reducible tree (middle column) for increasing truncations $N=3$ (top row), $N=4$ (middle row), and $N=9$ (bottom row). $N$-reducible trees reproduce qualitatively the structure of solutions to the truncated Galerkin system. This is illustrated in the right column, where plots from the left and middle columns are superimposed for comparison, refer also to Fig. \ref{['fig:EOmegaZoomN4N9']} for further details. Moreover, these plots show how complexity of the reducible tree increases with truncation.
• Figure 4: Zoom on one of the branches shown in Fig. \ref{['fig:EOmegaN349']}, for truncations $N=4$ (left) and $N=9$ (right). In blue we plot solutions of the truncated Galerkin system, while other colours indicate the elements of the $N$-reducible tree. When increasing $N$, new primary branches appear. Besides, new features, described by branches of higher order (in red) as well as branches not connected to the primary trunk (in dark red) emerge. The $N=9$ case is the lowest truncation for which a branch of order $3$ appears as a part of the reducible tree.
• Figure 5: Solutions of perturbed reducible system \ref{['eqn:pert_system']}. The solution curves of the corresponding reducible system on the $E-\Omega$ diagram (dashed black) resolve into a loop (solid blue). The bottom plots illustrate how the perturbation affects the structure coming from the reducible systems (dashed lines), see Fig. \ref{['fig:2modePlot']}.

Theorems & Definitions (9)

• Conjecture 1
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Remark 1
• Theorem 1
• proof