We present a mode-coupled weakly nonlinear model for the evolution of perturbations on cylindrical multilayered shells in a decelerating implosion. We show that nonlinear mode-mode interactions among large wave-number fundamental modes are able to induce the growth of small wave number harmonic modes, i.e., forming inverse cascade channels in the wave-number space. When uniform compression and interfacial coupling are taken into consideration, the amplitude of some perturbation modes exhibits an oscillatory growth pattern, which is beyond the intuition that perturbation amplitudes usually have a fast growth tendency in an implosion dominated by the Bell-Plesset effect. Our model accounts well for the previous experiments of Hsing et al. [Hsing et al., Phys. Rev. Lett. 78, 3876 (1997)0031-900710.1103/PhysRevLett.78.3876 and Phys. Plasmas 4, 1832 (1997)1070-664X10.1063/1.872326], which is among the few experiments of multimode multiinterface perturbation development in a cylindrical implosion. In particular, we find that the inverse cascade of modes is the origin of the excitation and growth of the wave number k=2 harmonic mode on the inner interface. The observed decrease of the fundamental modes on the inner interface is mainly attributed to the decreasing period of the oscillatory growth process. These results may afford further insight into the distortion of hot spots in inertial confined fusion implosion near the final stage, and also help to design multimode perturbation experiments in converging geometry in the coming future.