6 amplitudes B$_{j}$, j = 1 , 2 , 3 ) and only 3 conservation laws (15), (16) which, generally speaking, are not enough for integrability. However, if we rewrite it in the standard amplitude-phase representation B$_{j}$ = C$_{j}$ exp( iθ$_{j}$ ) as cluster is shown in Fig.4: the cluster is a 3-chain with one PA - and one PP -connections. The dynamical system cor- responding to the cluster has been solved numerically: for all calculations the dynamical phase of the initially excited triad was set to π, while during the experiments the phases of the modes have not been measured (D. Henderson, private communication). This yielded quali- tative agreement of the results of numerical simulations with measured data but higher magnitudes of observed amplitudes. The evolution of the amplitudes and phases in the dis- crete turbulent regime is discussed in Section III A and in Section III B correspondingly. it becomes clear immediately that in fact we only have 4 independent variables: three real amplitudes C$_{j}$ and one dynamical phase according to the choice of resonance conditions. In this section we will concentrate on the wave amplitudes while the importance of dynamical phases will be discussed in the Section III B. Making use of the addition theorems sn $^{2}$+ cn 2 = 1 and dn + µ 2 sn = 1 (here µ is a known func- tion of conservation laws, µ = µ ( H, I$_{23}$, I$_{13}$ ) and of the amplitude-phase representation B$_{j}$ = C$_{j}$ exp( iθ$_{j}$ ) , one can easily get explicit expressions for the squared ampli- tudes: with the coefficients α$_{1}$$_{j}$, ..., α$_{5}$$_{j}$ being explicit functions of I$_{13}$, I$_{23}$ and H$_{T}$ , for j = 1 , 2 , 3 , [31]. The energy of each mode E$_{j}$ is proportional to the square of its amplitude, E$_{j}$ ( t ) ∼ C 2 $_{j}$( t ) ∼ dn $^{2}$, i.e. its changing range is finite and defined by the initial conditions as 0 < dn $^{2}$( t ) ≤ 1 . There is no analytical expression for the amplitudes (energies) of a cluster formed by two or more triads. On the contrary, as it was shown in [2, 49, 50], the corre- sponding dynamical systems are integrable only in a few exceptional cases. For instance, clusters formed by N triads all having one common mode and with all con- nections of AA - or PP -type are integrable for arbitrary initial conditions if Z$_{i}$/Z$_{j}$ = 1 / 2 , 1 or 2 . Some clusters are known to be integrable for arbitrary coupling coeffi- cients but only for some specific initial conditions. On the other hand, already the smallest possible cluster of two connected triads can present chaotic behavior [30], though the modes energies are still bounded as the cor- responding dynamical systems are energy conserving. In the case of a generic cluster, the time evolution of the amplitudes has to be studied numerically and de- pends crucially on the initial conditions. In order to de- crease the number of degrees of freedom one has to use the fact that each triad has 3 integrals of motion given by (15)(16), i.e. N isolated triads have 2 N conservation laws and N Hamiltonians. A cluster formed by N triads with n connections has (2 N − n ) conservation laws and one Hamiltonian. The integrability of a generic cluster depends on the connection types within a cluster and on the ratios of coupling coefficients Z$_{j}$ . Some results of providing the integrability of (12) in terms of the Jaco- bian elliptic functions sn , cn and dn [55]. Notice that (12) has 6 real variables (real and imaginary parts of the and two Manley-Rowe constants of motion [35] in the form The analogy with a string is quite helpful for the un- derstanding of the dynamics of resonant modes. A string can be compressed or pulled (without destruction) only in a finite range of the applied forces; similarly, the range of amplitudes’ changing is also finite. Indeed, the sim- plest way to show it is just to look at the smallest possible resonant cluster — a triad. The dynamical system for a resonant triad (12) has the Hamiltonian In Fig.2 we show the difference in dynamics of discrete modes of capillary waves with and without vorticity. The energies of fluxless modes in the frozen turbulence regime for irrotational capillary waves are shown as (blue) bold vertical lines and the energies of resonant modes appear- ing due to the non-zero constant vorticity are depicted by (red) bold lines with a spring part inside. The "springs" show symbolically that the magnitudes of energies can change in time, similarly to the process of compressing or pulling an elastic string. Notice that this situation is quite general and does not depend on the wave type (see [25] for numerical simulations of atmospheric plan- etary waves — in Fig.1 the periodical time evolution of resonant modes is shown and in Fig.3 the fluxless modes demonstrate no time changes in the magnitudes of am- plitudes). FIG. 4: Color online. NR -diagram of the resonance cluster ap- pearing in the laboratory experiments with gravity-capillary water waves reported in [4]. A. Evolution of the amplitudes      ˙ C$_{1}$ = ZC$_{2}$C$_{3}$ cos ϕ, ˙ C$_{2}$ = ZC$_{1}$C$_{3}$ cos ϕ, ˙ C$_{3}$ = − ZC$_{1}$C$_{2}$ cos ϕ, ˙ ϕ = Z H$_{T}$ ( C − 2 1 + C − 2 2 − C − 2 3 ) , (17) ϕ = θ$_{1}$ + θ$_{2}$ − θ$_{3}$ (18) C 2 $_{j}$( t ) = α$_{1}$$_{j}$ + α$_{2}$$_{j}$ · dn 2 ( α$_{3}$$_{j}$ · t · Z, α$_{4}$$_{j}$ ) − α$_{5}$$_{j}$ (19) H$_{T}$ = Im( B$_{1}$B$_{2}$B ∗ $_{3}$) , (15) I$_{23}$ = | B$_{2}$ | 2 + | B$_{3}$ | $^{2}$, I$_{13}$ = | B$_{1}$ | 2 + | B$_{3}$ | $^{2}$, (16)