Boundary effects in the stepwise structure of the Lyapunov spectra for quasi-one-dimensional systems

Abstract

Boundary effects in the stepwise structure of the Lyapunov spectra and corresponding wavelike structure of the Lyapunov vectors are discussed numerically in quasi-one-dimensional systems of many hard disks. Four different types of boundary conditions are constructed by combinations of periodic boundary conditions and hard-wall boundary conditions, and each leads to different stepwise structures of the Lyapunov spectra. We show that for some Lyapunov exponents in the step region, the spatial y component of the corresponding Lyapunov vector deltaq(yj), divided by the y component of momentum p(yj), exhibits a wavelike structure as a function of position q(xj) and time t. For the other Lyapunov exponents in the step region, the y component of the corresponding Lyapunov vector deltaq(yj) exhibits a time-independent wavelike structure as a function of q(xj). These two types of wavelike structure are used to categorize the type and sequence of steps in the Lyapunov spectra for each different type of boundary condition.

The maximum number of highly localized Lyapunov vectors at low density

“…In these systems, the value of W (n) usually increases with the Lyapunov index n, and this implies that Lyapunov vectors for the largest exponent region are the most localized. The quantity W (n) has a minimum value of 2, as a minimum of two particles are involved in each collision, and it has been shown numerically that the value of W (n) for the largest Lyapunov exponent approaches 2 as the density approaches zero [23]. It was also shown that W (n) can detect not only the localized behavior of Lyapunov vectors, but also the de-localized behavior observed in the Lyapunov modes [23].…”

“…One of the most significant points of the Lyapunov spectrum is that each Lyapunov exponent indicates a time scale given by the inverse of the Lyapunov exponent so we can consider the Lyapunov spectrum as a spectrum of time-scales. The smallest positive Lyapunov exponent region of the spectrum is dominated by macroscopic time and length scale behavior, and here some delocalized mode-like structures (the Lyapunov modes) have been observed in the Lyapunov vectors [5,6,7,8,9,10,11,12,13,14]. On the other hand, the largest Lyapunov exponent region of Lyapunov spectrum is dominated by short time scale behavior, and in this region the Lyapunov vectors are localized (Lyapunov localization).…”

“…Here the radius of the particles R = 1, the mass of each particle m = 1, the total energy of system E = N , and the system size L y = 2R(1+10 −6 ) and L x = N L y (1+d) with parameter d, related to the density by ρ = πR 2 /[(1 + d)L 2 y ]. In this system the particles remain ordered because the vertical size L y prohibits the exchange of particles [8,12,23]. In Fig.…”

“…The quantity W (n) satisfies the inequality 1 ≤ W (n) ≤ N , and can be interpreted as the effective number of particles contributing to the non-zero components of the Lyapunov vector. In a Hamiltonian system it satisfies the conjugate relation W (D−j+1) = W (j) for any j with the phase space dimension D, because of the symplectic structure.The set of quantities {W (n) } n which we call the Lyapunov localization spectrum, has been calculated previously in many-particle systems with hard-core interactions [23] and with soft-core interactions [13]. In these systems, the value of W (n) usually increases with the Lyapunov index n, and this implies that Lyapunov vectors for the largest exponent region are the most localized.…”

“…The quantity W (n) has a minimum value of 2, as a minimum of two particles are involved in each collision, and it has been shown numerically that the value of W (n) for the largest Lyapunov exponent approaches 2 as the density approaches zero [23]. It was also shown that W (n) can detect not only the localized behavior of Lyapunov vectors, but also the de-localized behavior observed in the Lyapunov modes [23].One of the important characteristics of the Lyapunov localization spectrum is its bending behavior at low density [23]. In Fig.…”