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Register a new userScars on quantum networks ignore the Lyapunov exponent
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We show that enhanced wavefunction localization due to the presence of short unstable orbits and strong scarring can rely on completely different mechanisms. Specifically we find that in quantum networks the shortest and most stable orbits do not support visible scars, although they are responsible for enhanced localization in the majority of the eigenstates. Scarring orbits are selected by a criterion which does not involve the classical Lyapunov exponent. We obtain predictions for the energies of visible scars and the distributions of scarring strengths and inverse participation ratios.
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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.
We consider families of piecewise linear maps in which the moduli of the two slopes take different values. In some parameter regions, despite the variations in the dynamics, the Lyapunov exponent and the topological entropy remain constant. We provide numerical evidence of this fact and we prove it analytically for some special cases. The mechanism is very different from that of the logistic map and we conjecture that the Lyapunov plateaus reflect arithmetic relations between the slopes.
In a recent publication, J. Phys.: Condens. Matt. 14 13777 (2002), Kuzovkov et. al. announced an analytical solution of the two-dimensional Anderson localisation problem via the calculation of a generalised Lyapunov exponent using signal theory. Surprisingly, for certain energies and small disorder strength they observed delocalised states. We study the transmission properties of the same model using well-known transfer matrix methods. Our results disagree with the findings obtained using signal theory. We point to the possible origin of this discrepancy and comment on the general strategy to use a generalised Lyapunov exponent for studying Anderson localisation.
The well-known Vicsek model describes the dynamics of a flock of self-propelled particles (SPPs). Surprisingly, there is no direct measure of the chaotic behavior of such systems. Here, we discuss the dynamical phase transition present in Vicsek systems in light of the largest Lyapunov exponent (LLE), which is numerically computed by following the dynamical evolution in tangent space for up to one million SPPs. As discontinuities in the neighbor weighting factor hinder the computations, we propose a smooth form of the Vicsek model. We find that there is chaotic behavior in the disordered phase, which supports the claim that the LLE can be useful as an indicator of phase transitions even for this out-of-equilibrium system.
Out-of-time-order correlator (OTOC) $langle [x(t),p]^2 rangle $ in an inverted harmonic oscillator (IHO) in one-dimensional quantum mechanics exhibits remarkable properties. The quantum Lyapunov exponent computed through the OTOC precisely agrees with the classical one. Besides, it does not show any quantum fluctuations for arbitrary states. Hence, the OTOC may be regarded as ideal indicators of the butterfly effect in the IHO. Since IHOs are ubiquitous in physics, these properties of the OTOCs might be seen in various situations too. In order to clarify this point, as a first step, we investigate the OTOCs in one dimensional quantum mechanics with polynomial potentials, which exhibit butterfly effects around the peak of the potential in classical mechanics. We find two situations in which the OTOCs show exponential growths reproducing the classical Lyapunov exponent of the peak. The first one, which is obvious, is using suitably localized states near the peak and the second one is taking a double scaling limit akin to the non-critical string theories.
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