Abstract. We consider a π-mode solution of the Fermi-Pasta-Ulam β system. By perturbing it, we study the system as a function of the energy density from a regime where the solution is stable to a regime, where is unstable, first weakly and then strongly chaotic. We introduce, as indicator of stochasticity, the ratio ρ (when is defined) between the second and the first moment of a given probability distribution. We will show numerically that the transition between weak and strong chaos can be interpreted as the symmetry breaking of a set of suitable dynamical variables. Moreover, we show that in the region of weak chaos there is a numerical evidence that the thermostatistic is governed by the Tsallis distribution.
The stability of the one-mode nonlinear solutions of the Fermi-Pasta-Ulam beta system is numerically investigated. No external perturbation is considered for the one-mode exact analytical solutions, the only perturbation being that introduced by computational errors in the numerical integration of motion equations. The threshold energy for the excitation of the other normal modes and the dynamics of this excitation are studied as a function of the parameter micro characterizing the nonlinearity, the energy density epsilon and the number N of particles of the system. The results achieved confirm in part previous ones, obtained with a linear analysis of the problem of the stability, and clarify the dynamics by which a one-mode exchanges energy with the other modes with increasing energy density. In a range of energy density near the threshold value and for various values of the number of particles N, the nonlinear one-mode exchanges energy with the other linear modes for a very short time, immediately recovering all its initial energy. This sort of recurrence is very similar to Fermi recurrences, even if in the Fermi recurrences the energy of the initially excited mode changes continuously and only periodically recovers its initial value. A tentative explanation for this intermittent behavior, in terms of Floquet's theorem, is proposed. Preliminary results are also presented for the Fermi-Pasta-Ulam alpha system which show that there is a stability threshold, for large N, independent of N.
An algebraic method is devised to look for non-local symmetries of the pseudopotential type of nonlinear field equations. The method is based on the use of an infinite-dimensional subalgebra of the prolongation algebra L associated with the equations under consideration. Our approach, which is applied by way of example to the Dym and the Korteweg-de Vries equation, allows us to obtain a general formula for the infinitesimal operator of non-local symmetries expressed in terms of elements of L. The method could be exploited to investigate the symmetry properties of other nonlinear field equations possessing nontrivial prolongations. * )
We present a detailed numerical and analytical study of the stability properties of the N/4 (pi/2-mode) one-mode nonlinear solution of the Fermi-Pasta-Ulam-beta system. The numerical analysis is made as a function of the number N of the particles of the system and of the product lambda=epsilonbeta , where epsilon is the energy density and beta is the parameter characterizing the nonlinearity. It is shown that, both for beta>0 and beta<0 , the instability threshold value |lambda(t)(N)| converges, with increasing N , to the same value 2pi(2)(3N(2)) , that for beta>0 |lambda(t)N(2)| is a decreasing function of N as in the pi-mode, whereas, for beta<0 , it is an increasing one. The asymptotic behavior of |lambda(t)| for large values of N is analytically obtained in both cases with a Floquet analysis of the stability.
We apply the Bogoliubov-Krilov method of averaging to the study of the stability of the pi -mode solution (N/2 one-mode nonlinear solution) of the Fermi-Pasta-Ulam- beta system, with negative values of the nonlinearity parameter beta in the Hamiltonian of the system. The analysis is made as a function of the number N of the particles and of the product lambda = epsilon|beta|, where epsilon is the energy density. The results of this application are in excellent agreement with those obtained by the direct integration of motion equation.
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