Quasirandom Dynamical Systems. I. Quasirandom Diffeomorphisms

Abstract

In the past decade it has become possible to directly measure the adsorption force of a polymer in contact with a solid surface using single-molecule force spectroscopy. A plateau force in the force-extension curve is often observed in systems of physisorbed or noncovalently bonded polymers. If a molecule is pulled quickly compared to internal relaxation, then nonequilibrium effects can be observed. Here we investigate these effects using statistical mechanical models and experiments with a spider silk polypeptide. We present evidence that most experiments showing plateau forces are done out of equilibrium. We find that the dominant nonequilibrium effect is that the detachment height h max (v) increases with pulling speed v. Based on a nonequilibrium model within a master-equation approach, we show the sigmoidal dependence of the detachment height on the pulling speed of the cantilever, agreeing with experimental data on a spider silk polypeptide. We also show that the slope with which the plateau forces detach is given by the cantilever force constant in both theory and experiment.

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“…The Poincaré map f e , identifying points (t, −z, −ż) and (t, z,ż), has been widely studied, see for instance (Alekseev, 1968), (Moser, 1973) and (Llibre and Simó, 1980). From the results of (Moser, 1973) pages 87-89 we have the following theorem for the map f + e .…”

“…In order to prove the existence of non-symmetric periodic orbits, we will use the results of (Alekseev, 1968) and (Moser, 1973) about the presence of the Bernoulli shift as a subsystem of f e . Remember that (Alekseev, 1968) and (Moser, 1973) identify points (t 0 , 0, −ż 0 ) with (t 0 , 0,ż 0 ) and here we do not use this identification.…”

“…The purpose of this paper is to use the symmetries of the problem to find symmetric periodic solutions of the Sitnikov problem. We do this by means of a Poincaré map, which is essentially the Poincaré map that used Alekseev (1968) and Moser (1973) to analyze the final evolutions of the orbits of the Sitnikov problem. The presence of the Bernoulli shift as a subsystem of that Poincaré map will allow us to prove the existence of non-symmetric periodic orbits.…”

Spatio-Temporal Chaos in a Discrete Turing Model

“…This method showing a given invariant set is hyperbolic is called the cone technique, which is introduced in [2] and constantly developed as in [4], [12], etc. Especially, in [4], the cone criterion was applied to show that the hénon map has hyperbolic sets.…”

Stable oscillating motions in a model of a charged-particle accelerator

“…The concept of an oscillating motion came from celestial mechanics, and the possibility of such motions in the three-body problem was conjectured in 1922 by the French astronomer and mathematician Chazy [11]. In 1959 Sitnikov proved the existence of oscillating motions in the three-body problem for the Kolmogorov model [12], in the cycle of papers [13] Alekseev proved the existence of both two-sided (that is, when t → ±∞, t being the time) and one-sided (that is, when t → +∞) oscillating motions in the Kolmogorov model, and in [8] the author obtained all these results in the n-body problem for n ≥ 3. In [14] oscillating motions were found in a certain billiards problem.…”