Heterogeneity in oscillator networks: are smaller worlds easier to synchronize?

Small-world and scale-free networks are known to be more easily synchronized than regular lattices, which is usually attributed to the smaller network distance between oscillators. Surprisingly, we find that networks with a homogeneous distribution of connectivity are more synchronizable than heterogeneous ones, even though the average network distance is larger. We present numerical computations and analytical estimates on synchronizability of the network in terms of its heterogeneity parameters. Our results suggest that some degree of homogeneity is expected in naturally evolved structures, such as neural networks, where synchronizability is desirable.     

Numerical and experimental investigation of the effect of filtering on chaotic symbolic dynamics

Motivated by the practical consideration of the measurement of chaotic signals in experiments or the transmission of these signals through a physical medium, we investigate the effect of filtering on chaotic symbolic dynamics. We focus on the linear, time-invariant filters that are used frequently in many applications, and on the two quantities characterizing chaotic symbolic dynamics: topological entropy and bit-error rate. Theoretical consideration suggests that the topological entropy is invariant under filtering. Since computation of this entropy requires that the generating partition for defining the symbolic dynamics be known, in practical situations the computed entropy may change as a filtering parameter is changed. We find, through numerical computations and experiments with a chaotic electronic circuit, that with reasonable care the computed or measured entropy values can be preserved for a wide range of the filtering parameter.     

Differential equations having rapidly changing solutions: Analytic methods for weakly nonlinear systems

Initial value problems for weakly nonlinear systems of differential equations are studied here. A perturbation result is obtained which is shown to give a valid approximation to the solution. The approximation is determined by solving two associated problems, and it is especially derived for problems which have rapidly oscillating and rapidly decaying solutions.     

Asymptotic stability in singular perturbation problems. II: Problems having matched asymptotic expansion solutions

The stability of systems of ordinary differential equations of the form

$\text{dx}\text{dt}\text{= f(t, x, y, ?), ?}\text{dy}\text{dt}\text{= g(t, x, y, ?)}$

, where ? is a real parameter near zero, is studied. It is shown that if the reduced problem

$\text{dx}\text{dt}\text{= f(t, x, y, 0), 0 = g(t, x, y, 0)}$

, is stable, and certain other conditions which ensure that the method of matched asymptotic expansions can be used to construct solutions are satisfied, then the full problem is asymptotically stable as t → ∞, and a domain of stability is determined which is independent of ?. Moreover, under certain additional conditions, it is shown that the solutions of the perturbed problem have limits as t → ∞. In this case, it is shown how these limits can be calculated directly from the equations

$\text{f(∞, x, y, ?) = 0 g(∞, x, y, ?) = 0}$

as expansions in powers of ?.     

Numerical solution of Volterra integro-differential equations modelling thalamo-cortical systems

Our study concerns thalamo-cortical systems which are modelled by nonlinear systems of Volterra integro-differential equations of convolution type. The thalamo-cortical systems describe a new architecture for a neurocomputer. Such a computer employs principles of human brain. It consists of oscillators which have different frequencies and are weakly connected via a common medium forced by an external input. Since a neurocomputer consists of many interconnected oscillators (referred also as neurons), the thalamo-cortical systems include large numbers of Volterra integro-differential equations. Solving such systems numerically is expensive not only because of their large dimensions but also because of many kernel evaluations which are needed over the whole interval from the initial point, where the initial condition is imposed, up to the present point, where the computations are currently executed. Moreover, the whole computed history of the solution has to be stored in the memory of the computing machine. Therefore, robust and efficient numerical algorithms are needed for computer simulations for the solutions to the thalamocortical systems. In this paper, we illustrate an iteration technique to solve the thalamo-cortical systems. The proposed successive iterates are vector functions of time, which change the original problems into systems of easier and separated equations. Such separated equations can then be solved in parallel computing environments. Results of numerical experiments are presented for large numbers of oscillators. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discrete time semigroup transformations with random perturbations

Let (X, ) and (Y,C) be two measurable spaces withX being a linear space. A system is determined by two functionsf(X): X X and:X×YX, a (small) positive parameter and a homogeneous Markov chain {y _n } in (Y,C) which describes random perturbations. States of the system, say {x _n X, n=0, 1,}, are determined by the iteration relations:x _n+1 =f(x _n )+(x _n ,Y_n+1) forn0, wherex ₀ =x ₀ is given. Here we study the asymptotic behavior of the solutionx _n as 0 andn under various assumptions on the data. General results are applied to some problems in epidemics, genetics and demographics.Supported in part by NSF Grant DMS92-06677.Supported in part by NSF Grant DMS93-12255.     

Host-guest interactions at self-assembled monolayers of cyclodextrins on gold

We have developed synthesis routes for the introduction of short and long dialkylsulfides onto the primary side of alpha-, beta-, and gamma-cyclodextrins. Monolayers of these cyclodextrin adsorbates were characterized by electrochemistry, wettability studies, X-ray photoelectron spectroscopy (XPS), time-of-flight secondary ion mass spectrometry (TOF-SIMS), and atomic force microscopy (AFM). The differences in thickness and polarity of the outerface of the monolayers were measured by electro-chemistry and wettability studies. On average about 70% of the sulfide moieties were used for binding to the gold, as measured by XPS. Tof-SIMS measurements showed that the cyclodextrin adsorbates adsorb without any bond breakage. AFM measurements revealed for beta-cyclodextrin monolayers a quasi-hexagonal lattice with a lattice constant of 20.6 A, which matches the geometrical size of the adsorbate. The alpha-cyclodextrin and gamma-cyclodextrin monolayers are less ordered. Interactions of the anionic guests 1-anilinonaphthalene-8-sulfonic acid (1,8-ANS) and 2-(p-toluidinyl)naphthalene-6-sulfonic acid (2,6-TNS) and the highly ordered monolayers of heptapodant beta-cyclodextrin adsorbates were studied by surface plasmon resonance (SPR) and electrochemical impedance spectroscopy. The SPR measurements clearly showed interactions between a beta-cyclodextrin monolayer and 1,8-ANS. Electrochemical impedance spectroscopy measurements gave high responses even at low guest concentrations (< or = 5 microM). The association constant for the binding of 1,8-ANS (K = 289,000 +/- 13,000M-1) is considerably higher than the corresponding value in solution. (Partial) methylation of the secondary side of the beta-cyclodextrin strongly decreases the binding.