Fig. 3: Cobweb plots and spatial distributions of zero modes in the nonlinear SSH model.

a–d Cobweb plots at different strengths of nonlinearity a are shown. The blue curves represent the absolute values of the nonlinear functions F(Ψ) in Eq. (7) with Ψ corresponding to a component of an eigenvector. The orange lines represent ∣F(Ψ)∣ = Ψ. The red lines show the dynamics of Eq. (7), which correspond to the blue polylines. a Weakly nonlinear topological case. If the nonlinear winding number is ν = 1 in the linear limit, localized edge modes converging to zero are obtained for smaller initial amplitudes than that represented by the red dotted line. We use the parameter a = 0.8. b Nonlinearity-induced topological phase. When the nonlinearity-induced topological phase transition from a trivial phase to a topological phase occurs, localized zero modes are obtained if the initial amplitude is in the region sandwiched by the red dotted lines. We use the parameter a = 1.8. c Periodic zero mode. We consider a = 2.1 and obtain a stable periodic orbit. d Chaotic zero mode. At large a (a = 2.5 in this panel), we obtain a chaotic dynamics of a zero mode. e–h Zero modes at each parameter are shown. The colored polylines show the spatial distributions of zero modes, where the color shows the difference in the initial condition Ψ_A(1). The red dashed lines show the critical amplitude of the nonlinear winding number. We use the parameters a = 0.8, 1.8, 2.1, 2.5 in e–h for each, which correspond to the upper panels (a–d).