Symmetry-breaking in local Lyapunov exponents

Integrable dynamical systems, namely those having as many independent conserved quantities as freedoms, have all Lyapunov exponents equal to zero. Locally, the instantaneous or finite time Lyapunov exponents are nonzero, but owing to a symmetry, their global averages vanish. When the system becomes nonintegrable, this symmetry is broken. A parallel to this phenomenon occurs in mappings which derive from quasiperiodic Schr?dinger problems in 1-dimension. For values of the energy such that the eigenstate is extended, the Lyapunov exponent is zero, while if the eigenstate is localized, the Lyapunov exponent becomes negative. This occurs by a breaking of the quasiperiodic symmetry of local Lyapunov exponents, and corresponds to a breaking of a symmetry of the wavefunction in extended and critical states. Received 25 October 2001 / Received in final form 8 December 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: r.ramaswamy@mail.jnu.ac.in     

Essential dynamics/factor analysis for the interpretation of molecular dynamics trajectories

Subject of this work is the analysis of molecular dynamics (MD) trajectories of neurophysins I (NPI) and II (NPII) and their complexes with the neurophyseal nonapeptide hormones oxytocin (OT) and vasopresssin (VP), respectively, simulated in water. NPs serve in the neurosecretory granules as carrier proteins for the hormones before their release to the blood. The starting data consisted of two pairs of different trajectories for each of the (NPII/VP)2 and (NPI/OT)2 heterotetramers and two more trajectories for the NPII2 and NPI2 homodimers (six trajectories in total). Using essential dynamics which, to our judgement, is equivalent to factor analysis, we found that only about 10 degrees of freedom per trajectory are necessary and sufficient to describe in full the motions relevant for the function of the protein. This is consistent with these motions to explain about 90% of the total variance of the system. These principal degrees of freedom represent slow anharmonic motional modes, clearly pointing at distinguished mobility of the atoms involved in the protein's functionality.     

Fuzzy dynamical systems

A fuzzy dynamical system on an underlying complete, locally compact metric state space X is defined axiomatically in terms of a fuzzy attainability set mapping on X. This definition includes as special cases crisp single and multivalued dynamical systems on X. It is shown that the support of such a fuzzy dynamical system on X is a crisp multivalued dynamical system on X, and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of X. In addition fuzzy trajectories are defined, their existence established and various properties investigated.     

What’s Wrong with Einstein’s 1927 Hidden-Variable Interpretation of Quantum Mechanics?

Einsteins unpublished 1927 deterministic trajectory interpretation of quantum mechanics is critically examined, in particular with regard to the reason given by Einstein for rejecting his theory. It is shown that the aspect Einstein found objectionable—the mutual dependence of the motions of particles when the (many-body) wavefunction factorises—is a generic attribute of his theory but that this feature may be removed by modifying Einsteins method in either of two ways: using a suggestion of Grommer or, in a physically important special case, using a simpler technique. It is emphasized though that the presence or absence of the interdependence property does not determine the acceptability of a trajectory theory. It is shown that there are other grounds for rejecting Einsteins theory (and the two modified theories), to do with its domain of applicability and compatibility with empirical predictions. That Einsteins reason for rejection is not a priori grounds for discarding a trajectory theory is demonstrated by reference to an alternative deterministic trajectory theory that displays similar particle interdependence yet is compatible with quantum predictions.PACS: 03.65.Bz.Essay written in memory of J.T. Cushing.     

The Reality in Bohmian Quantum Mechanics or Can You Kill with an Empty Wave Bullet?

Several situations, in which an empty wave causes an observable effect, are reviewed. They include an experiment showing surrealistic trajectories proposed by Englert et al. and protective measurement of the density of the quantum state. Conditions for observable effects due to empty waves are derived. The possibility (in spite of the existence of these examples) of minimalistic interpretation of Bohmian quantum mechanics in which only Bohmian positions supervene on our experience is discussed.     

Quantum Trajectories, State Diffusion, and Time-Asymmetric Eventum Mechanics

We show that the quantum stochastic Langevin model for continuous in time measurements provides an exact formulation of the von Neumann uncertainty error-disturbance principle. Moreover, as it was shown in the 1980s, this Markov model induces all stochastic linear and nonlinear equations of the phenomenological informational dynamics such as quantum state diffusion and spontaneous localization by a simple quantum filtering method. Here we prove that the quantum Langevin equation is equivalent to a Dirac-type boundary-value problem for the second quantized input offer waves from future in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the trajectories of the output particles. This result supports the wave-particle duality in the form of the thesis of Eventum Mechanics that everything in the future is constituted by quantized waves, everything in the past by trajectories of the recorded particles. We demonstrate how this time arrow can be derived from the principle of quantum causality for nondemolition continuous in time measurements.     

Impact Oscillator with Hertz's Model of Contact

Dynamic properties of simple mechanical 1 DOF system containing soft stop is described and analyzed. The proposed general dynamical impact model respects the nonlinearity of the restoring contact force between solid bodies as function of deformation and velocity. It describes the real behavior of impacting system more exact than the piecewise linear model or the Kelvin–Voigt model and even model based on application of constant coefficient of restitution. Free and forced vibrations of system with Hertz's contact are investigated and domains of various types of impact motion, response curves and phase-plane trajectories are presented.     

Ising-Type and Other Transitions in One-Dimensional Coupled Map Lattices with Sign Symmetry

We consider a one-dimensional lattice of expanding antisymmetric maps [–1, 1][–1, 1] with nearest neighbor diffusive coupling. For such systems it is known that if the coupling parameter is small there is unique stationary (in time) state, which is chaotic in space-time. A disputed question is whether such systems can exhibit Ising-type phase transitions as grows beyond some critical value _c. We present results from computer experiments which give definite indication that such a transition takes place: the mean square magnetization appears to diverge as approaches some critical value, with a critical exponent around 0.9. We also study other properties of the coupled map system.     

一类参数不确定统一混沌系统的脉冲滞后同步

针对一类具有传输信道时间延迟且参数不确定的统一混沌系统，提出了一种脉冲控制同步方法.该方法将两个非自治混沌系统之间脉冲同步的实用稳定性问题转化为同步误差系统在原点的实用稳定性问题，基于脉冲微分方程的相关理论，给出了同步误差系统在原点的实用稳定判据，仿真结果验证了方法的有效性. 关键词： 统一混沌系统 混沌同步 脉冲控制 实用稳定性