This paper extends the analysis started in Pierini (2006) where a nonlinear reduced gravity model forced by a constant and idealized wind and with moderately realistic coastal boundaries seems to reproduce well the bimodality of the Kuroshio extension (KE) for a sufficiently weak value of horizontal dissipation. The analysis uses various tools from dynamical systems: bifurcation curves, trajectories of the system in phase space, finite-time Lyapunov exponents, etc. Their conclusion is that internal nonlinear dynamics could suffice to explain the bimodality of the KE.