We numerically study the time evolution of a system which consists of two attractors connected by a one-dimensional Fermi-Pasta-Ulam (FPU) chain. It is found that there exists a self-consistent large-scale structure in the system even after a sufficiently long time. The wavelet transform is envoked to separate the modes of different scales. We discover that the existence of a nonlinear long-wavelength mode in the system assists the energy transport, which leads to the violation of the Fourier heat law in a one-dimensional FPU model as observed by other authors.