Bifurcation and phase diagram of turbulence constituted from three different scale-length modes

Cases where three kinds of fluctuations having the different typical scale lengths coexist are analysed, and the Statistical theory of strong turbulence in inhomogeneous plasmas is developed. Statistical nonlinear interactions between fluctuations are kept in the analysis as the renormalized drag, statistical noise and the averaged drive. The nonlinear interplay through them induces a quenching or suppressing effect, even if all the modes are unstable when they are analysed independently. Variety in mode appearance takes place: one mode quenches the other two modes, or one mode is quenched by the other two modes, etc. The bifurcation of turbulence is analysed and a phase diagram is drawn. Phase diagrams with cusp-type catastrophe and butterfly-type catastrophe are obtained. The subcritical bifurcation is possible to occur through the nonlinear interplay, even though each one is supercritical turbulence when analysed independently. Analysis reveals that the nonlinear stability boundary (marginal point) and the amplitude of each mode may substantially shift from the conventional results of independent analyses.