Chaotic dynamics in Parkfield region, California and characteristics earthquakes

Based on about 75000 earthquakes in the California region detected through Parkfield network during the years 1969-1987, the occurrence of chaos was examined by two different approaches, namely, strange at tractor dimension and the Lyapunov exponent. The strange at tractor dimension was found as 6.3 in this region suggesting atleast 7 parameters for earthquake predictability. Small positive Lyapunov exponent of 0.045 provided further evidence for deterministic chaos in the region which showed strong dependence on the initial conditions. Implications of chaotic dynamics on characteristic Parkfield earthquakes has been discussed. The strange at tractor dimension in the region could be representative for the Transform type of plate boundary which is lower than that reported for continent collision type of plate boundary which is lower than that reported for continent collision type of plate boundary near Hindukush northwest Himalayan region.

Positive Lyapunov Exponent for Some Schrödinger Cocycles Over Strongly Expanding Circle Endomorphisms

We show that for a large class of potential functions and big coupling constant $$\lambda $$ λ the Schrödinger cocycle over the expanding map $$x\mapsto bx ~( \text{ mod } 1)$$ x ↦ b x ( mod 1 ) on $$\mathbb {T}$$ T has a Lyapunov exponent $$>(\log \lambda )/4$$ > ( log λ ) / 4 for all energies, provided that the integer $$b\ge \lambda ^3$$ b ≥ λ 3 .

Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles

We consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates, which lead to refined Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.

Hyperchaos and its control in two-level quantum oscillators lattice

The behavior of the square lattices of coupled two-level quantum oscillators is investigated. As quantum oscillators, we use the model of Rydberg atoms obtained using the approximation of a fully factorized density matrix. To investigate the behavior of the system the Lyapunov exponents spectra are calculated. Chaos and hyperchaos in the systems are revealed. It was shown that the number of positive Lyapunov exponents almost linearly depends on the number of atoms in the system, at a rate suggesting that adding three atoms leads to the appearance of an additional positive Lyapunov exponent. Using an external parametric effect and continuous feedback is suggested to control the complicated dynamics in the system. Using continuous feedback allows reducing the number of positive Lyapunov exponents from 3 to only 2 while introducing external parametric influence into the system allows reducing their number down to 0 and completely suppresses hyperchaos.