numerical simulations with two-triad clusters (fixed con- nection type, various coupling coefficients) can be found in [2]. As one can see from (17), the time evolution of a res- onant triad does not depend on the individual phases of the resonant modes but on the dynamical phase ϕ cor- responding to the resonance conditions. In this case, the closed expression for the evolution of dynamical phase, first found in [31], reads where and while ϕ$_{0}$ is the initial dynamical phase and the constants µ, K ( µ ) , τ are known explicitly as functions of I$_{13}$, I$_{23}$ and H$_{T}$ . The time evolution of a generic cluster again does not depend on the individual phases but on the dynamical phases; the corresponding equations for the dynamical phases have a form similar to the last equation of (17). For instance, in case of two triads a and b connected via then this range is substantially diminished: depending on the initial energy distribution within a triad it can de- crease 10 times and more, being minimal for ϕ$_{in}$ = π/ 2 . For the cluster shown in Fig.5 the maximal range of the amplitudes is observed for ϕ$_{a,in}$ = ϕ$_{b,in}$ = 0 . An interesting phenomenon has been uncovered in the case when at least one of the phases ϕ$_{a}$ and ϕ$_{b}$ is initially non-zero: a new time-scale is clearly observable where the amplitudes of the resonant modes are modulated by the evolution of the dynamical phases. The characteristic form of evolution is shown in Fig.6 for ϕ$_{a}$ = ϕ$_{b}$ = π/ 2 , [3]. This is a manifestation of one more crucial distinction between discrete and statistical wave turbulent regimes: while in the first regime phases are locked, in the latter they are supposed to be random. The effect of phase randomization among three-wave interactions of capillary waves has been studied recently experimentally in [40]. The transition from a coherent-phase to a random- phase system occurs above some excitation threshold when coherent wave harmonics broaden spectrally. The waves were excited parametrically in the range of the shaker frequencies 40 ÷ 3500 Hz and the modulation in- stability of capillary waves, which can be approximated by the squared secant function sech $^{2}$, has been estab- H = C$_{1}$ ( Z$_{a}$C$_{2}$$_{|}$$_{a}$C$_{3}$$_{|}$$_{a}$ sin ϕ$_{a}$ + Z$_{b}$C$_{2}$$_{|}$$_{b}$C$_{3}$$_{|}$$_{b}$ sin ϕ$_{b}$ ) . (24) ˙ ϕ$_{a}$ = − H ( C − 2 1 + C − 2 2 | a − C − 2 3 | $_{a}$) , (22) ˙ ϕ$_{b}$ = − H ( C − 2 1 + C − 2 2 | b − C − 2 3 | $_{b}$) . (23) ϕ ( t ) = sign ( ϕ$_{0}$ ) arccot ( k$_{1}$ F ( k$_{2}$ ( t − t$_{0}$ ) , µ )) , (20) k$_{1}$ = − µ | H$_{T}$ $_{|}$( 2 K ( µ ) Z τ ) 3 , k$_{2}$ = 2 K ( µ ) τ F ( x, µ ) ≡ sn ( x, µ ) cn ( x, µ ) dn ( x, µ ) , B. Evolution of the dynamical phases FIG. 6: Upper panel . Color online. Plots of the dynamical phases as functions of time, for each frame: ϕ$_{a}$ ( t ) is (red) solid and ϕ$_{b}$ ( t ) is (black) dashed, with ϕ$_{a,in}$ = ϕ$_{b,in}$ = π/ 2 . Lower panel . Plots of the squared amplitudes, as functions of time. The connecting mode C 1 is blue (the upper one), C 2 | a and C 3 | a are green (in the middle), while C 2 | b and C 3 | b are purple (at the bottom). To facilitate the view, C 2 2 | $_{b}$, C 2 3 | b and C 2 1 are shifted upwards. The horizontal axis stands for non-dimensional time and the vertical axis for the phase in the upper panel and for the squared amplitudes in the lower panel. FIG. 5: NR -diagram for a two-triad cluster with one PP - connection. 7 0 20 40 60 80 100 120 0 Π GLYPH<FractionBarExt>GLYPH<FractionBarExt>GLYPH<FractionBarExt>GLYPH<FractionBarExt> GLYPH<FractionBarExt> 2 Π 0 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 one PP -connection (the NR -diagram is shown in Fig.5), the equations for the dynamical phases read where the joint mode is denoted by C$_{1}$ and H is the Hamiltonian The effect of the dynamical phases on the evolution of the amplitudes has been studied numerically in [3]. It turned out that if initially ϕ$_{in}$ = 0 for a triad, then it stays zero and the changing range of amplitudes is max- imal. If ϕ$_{in}$ is very small but non-zero, say, ϕ$_{in}$ = 0 . 01 , ϕ$_{a}$ = θ$_{1}$$_{|}$$_{a}$ + θ$_{2}$$_{|}$$_{a}$ − θ$_{3}$$_{|}$$_{a}$, ϕ$_{b}$ = θ$_{1}$$_{|}$$_{b}$ + θ$_{2}$$_{|}$$_{b}$ − θ$_{3}$$_{|}$$_{b}$ (21)