3 research outputs found
Farey sequence in the appearance of subharmonic Shapiro steps
Largest Lyapunov exponent has been examined in the dynamical-mode locking
phenomena of the ac+dc driven dissipative Frenkel-Kontorova model with
deformable substrate potential. Due to deformation, large fractional and higher
order subharmonic steps appear in the response function of the system.
Computation of the largest Lyapunov exponent as a way to verify their presence
led to the observation of the Farey sequence. In the standard regime, between
the large harmonic steps, the appearance of halfinteger and subharmonic steps,
and their relative sizes follow the Farey construction. In the nonstandard
regime, though halfinteger steps are larger than harmonic ones, Farey
construction is still present in the appearance of higher order subharmonic
steps. The examination of Lyapunov exponents also shows that there is no chaos
in the system.Comment: 7 pages, 9 figure
Random Matrix Ensembles in Hyperchaotic Classical Dissipative Dynamical Systems
We study the statistical fluctuations of Lyapunov exponents in the discrete
version of the non-integrable perturbed sine-Gordon equation, the dissipative
ac+dc driven Frenkel-Kontorova model. Our analysis shows that the fluctuations
of the exponent spacings in the strictly overdamped limit, which is nonchaotic,
conforms to the \textit{uncorrelated} Poisson distribution. By studying the
spatiotemporal dynamics we relate the emergence of the Poissonian statistics to
Middleton's no-passing rule. Next, by scanning over the dc driving and particle
mass we identify several parameter regions for which this one-dimensional model
exhibits hyperchaotic behavior. Furthermore, in the hyperchaotic regime where
roughly fifty percent of exponents are positive, the fluctuations exhibit
features of the \textit{correlated} universal statistics of the Gaussian
Orthogonal Ensemble (GOE). Due to the dissipative nature of the dynamics, we
find that the match, between the Lyapunov spectrum statistics and the universal
statistics of GOE, is not complete. Finally, we present evidence supporting the
existence of the Tracy-Widom distribution in the fluctuation statistics of the
largest Lyapunov exponent.Comment: 16 pages, 12 figure
Application of largest Lyapunov exponent analysis on the studies of dynamics under external forces
Dynamics of driven dissipative Frenkel-Kontorova model is examined by using
largest Lyapunov exponent computational technique. Obtained results show that
besides the usual way where behavior of the system in the presence of external
forces is studied by analyzing its dynamical response function, the largest
Lyapunov exponent analysis can represent a very convenient tool to examine
system dynamics. In the dc driven systems, the critical depinning force for
particular structure could be estimated by computing the largest Lyapunov
exponent. In the dc+ac driven systems, if the substrate potential is the
standard sinusoidal one, calculation of the largest Lyapunov exponent offers a
more sensitive way to detect the presence of Shapiro steps. When the amplitude
of the ac force is varied the behavior of the largest Lyapunov exponent in the
pinned regime completely reflects the behavior of Shapiro steps and the
critical depinning force, in particular, it represents the mirror image of the
amplitude dependence of critical depinning force. This points out an advantage
of this technique since by calculating the largest Lyapunov exponent in the
pinned regime we can get an insight into the dynamics of the system when
driving forces are applied.Comment: 10 pages, 8 figure