The reducible-tree conjecture for time-periodic solutions

Consider the nonlinear wave equations

$$u_{tt}-u_{xx}\pm u^3=0,$$

and rewrite their time-periodic solutions as solutions of the associated infinite Galerkin algebraic system for the Fourier coefficients of $u$ and the oscillation frequency $\Omega$. A reducible tree is the structure obtained from a smaller subsystem of this Galerkin system, with trunks and branches, and its rescalings are taken according to the scaling in the source.

Reducible-tree conjecture. The infinite reducible tree and its rescalings according to the relevant scaling provide an effective description for the structure of time-periodic solutions to the equations above.

This conjecture proposes that the reducible-tree structures capture the overall bifurcation pattern of time-periodic solutions, including the way solutions develop across frequency gaps. The paper motivates it through detailed analysis of reducible systems and comparison with truncated Galerkin systems; the supplied text gives no resolution, so its status remains open.