Stability properties of theN∕4(π∕2-mode) one-mode nonlinear solution of the Fermi-Pasta-Ulam-βs

Abstract

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.

Thermostatistics in the neighbourhood of the π-mode solution for the Fermi–Pasta–Ulam β system: from weak to strong chaos

“…Beyond a certain threshold value ǫ t of the energy density ǫ, the nonlinear mode becomes unstable. In [26]- [28], this mechanism has been extensively used to analyze the stability properties of the π-mode and π/2-mode (n = N/4), both for positive and negative values of the nonlinearity parameter β.…”

Stability of nonlinear normal modes in the Fermi-Pasta-Ulamβchain in the thermodynamic limit

“…The above discussed threshold function A c (N ) for N → ∞ can be calculated analytically. For the nonlinear normal mode B[a 4 , iu], such calculation has been recently performed in [2]. The similar analytical results for the πmode j 0 = N 2 were earlier obtained in [1,14,[16][17][18][19][20].…”

“…For studying stability of NNMs in the thermodynamic limit (N → ∞), we use a method which is similar, in some sense, to that developed in [2]. However, our method is more general and, as a consequence, it can be applied to all nonlinear normal modes in the FPU-β chain for the case N → ∞.…”

“…In general case, parametric interaction with the original (active) NNM leads to exciting not only one sleeping mode, but a certain set of such modes (for the FPU-β this set can consist of 1, 2 or 3 sleeping modes). For example, the variational system for NNM B[a 4 , iu] can be decomposed into two-dimensional subsystems L (2) j [see Eq. (33) and the discussion below this equation].…”

The zone boundary mode in periodic nonlinear electrical lattices

“…This work is similar to ours in that the eigenvectors of the linearized problem are used to change variables and simplify the original N-dimensional problem. Continuing in this vein, we find [20] uses symplectic numerical methods to study the nonlinear stability of all NNMs found by [19]; [21] studies ZBM stability in the case β < 0 using the method of averaging; and [22] studies NNM stability with wavelength equal to four times the lattice spacing, the so-called ''π /2 mode''. Note that these authors refer to NNMs as one-mode solutions (OMSs).…”