Scaling and synchronization in a ring of diffusively coupled nonlinear oscillators

Chaos synchronization in a ring of diffusively coupled nonlinear oscillators driven by an external identical oscillator is studied. Based on numerical simulations we show that by introducing additional couplings at $(mN_c+1)$-th oscillators in the ring, where $m$ is an integer and $N_c$ is the maximum number of synchronized oscillators in the ring with a single coupling, the maximum number of oscillators that can be synchronized can be increased considerably beyond the limit restricted by size instability. We also demonstrate that there exists an exponential relation between the number of oscillators that can support stable synchronization in the ring with the external drive and the critical coupling strength $\epsilon_c$ with a scaling exponent $\gamma$. The critical coupling strength is calculated by numerically estimating the synchronization error and is also confirmed from the conditional Lyapunov exponents (CLEs) of the coupled systems. We find that the same scaling relation exists for $m$ couplings between the drive and the ring. Further, we have examined the robustness of the synchronous states against Gaussian white noise and found that the synchronization error exhibits a power-law decay as a function of the noise intensity indicating the existence of both noise-enhanced and noise-induced synchronizations depending on the value of the coupling strength $\epsilon$. In addition, we have found that $\epsilon_c$ shows an exponential decay as a function of the number of additional couplings. These results are demonstrated using the paradigmatic models of R\"ossler and Lorenz oscillators.Comment: Accepted for Publication in Physical Review

Cardiovascular magnetic resonance of scar and ischemia burden early after acute ST elevation and non-ST elevation myocardial infarction

<p>Abstract</p> <p>Background</p> <p>The acute coronary syndrome diagnosis includes different classifications of myocardial infarction, which have been shown to differ in their pathology, as well as their early and late prognosis. These differences may relate to the underlying extent of infarction and/or residual myocardial ischemia. The study aim was to compare scar and ischemia mass between acute non-ST elevation myocardial infarction (NSTEMI), ST-elevation MI with Q-wave formation (Q-STEMI) and ST-elevation MI without Q-wave formation (Non-Q STEMI) in-vivo, using cardiovascular magnetic resonance (CMR).</p> <p>Methods and results</p> <p>This was a prospective cohort study of twenty five consecutive patients with NSTEMI, 25 patients with thrombolysed Q-STEMI and 25 patients with thrombolysed Non-Q STEMI. Myocardial function (cine imaging), ischemia (adenosine stress first pass myocardial perfusion) and scar (late gadolinium enhancement) were assessed by CMR 2–6 days after presentation and before any invasive revascularisation procedure. All subjects gave written informed consent and ethical committee approval was obtained. Scar mass was highest in Q-STEMI, followed by Non-Q STEMI and NSTEMI (24.1%, 15.2% and 3.8% of LV mass, respectively; p < 0.0001). Ischemia mass showed the reverse trend and was lowest in Q-STEMI, followed by Non-Q STEMI and NSTEMI (6.9%, 14.7% and 19.9% of LV mass, respectively; p = 0.012). The combined mass of scar and ischemia was similar between the three groups (p = 0.17). The ratio of scar to ischemia was 3.5, 1.0 and 0.2 for Q-STEMI, Non-Q STEMI and NSTEMI, respectively.</p> <p>Conclusion</p> <p>Prior to revascularisation, the ratio of scar to ischemia differs between NSTEMI, Non-Q STEMI and Q-STEMI, whilst the combined scar and ischemia mass is similar between these three types of MI. These results provide in-vivo confirmation of the diverse pathophysiology of different types of acute myocardial infarction and may explain their divergent early and late prognosis.</p

Analysis of the shearing instability in nonlinear convection and magnetoconvection

Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasising how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced

Symbolic Dynamics Analysis of the Lorenz Equations

Recent progress of symbolic dynamics of one- and especially two-dimensional maps has enabled us to construct symbolic dynamics for systems of ordinary differential equations (ODEs). Numerical study under the guidance of symbolic dynamics is capable to yield global results on chaotic and periodic regimes in systems of dissipative ODEs which cannot be obtained neither by purely analytical means nor by numerical work alone. By constructing symbolic dynamics of 1D and 2D maps from the Poincare sections all unstable periodic orbits up to a given length at a fixed parameter set may be located and all stable periodic orbits up to a given length may be found in a wide parameter range. This knowledge, in turn, tells much about the nature of the chaotic limits. Applied to the Lorenz equations, this approach has led to a nomenclature, i.e., absolute periods and symbolic names, of stable and unstable periodic orbits for an autonomous system. Symmetry breakings and restorations as well as coexistence of different regimes are also analyzed by using symbolic dynamics.Comment: 35 pages, LaTeX, 13 Postscript figures, uses psfig.tex. The revision concerns a bug at the end of hlzfig12.ps which prevented the printing of the whole .ps file from page 2

On the recurrence and robust properties of Lorenz'63 model

Lie-Poisson structure of the Lorenz'63 system gives a physical insight on its dynamical and statistical behavior considering the evolution of the associated Casimir functions. We study the invariant density and other recurrence features of a Markov expanding Lorenz-like map of the interval arising in the analysis of the predictability of the extreme values reached by particular physical observables evolving in time under the Lorenz'63 dynamics with the classical set of parameters. Moreover, we prove the statistical stability of such an invariant measure. This will allow us to further characterize the SRB measure of the system.Comment: 44 pages, 7 figures, revised version accepted for pubblicatio

X-ray absorption spectroscopy systematics at the tungsten L-edge

A series of mononuclear six-coordinate tungsten compounds spanning formal oxidation states from 0 to +VI, largely in a ligand environment of inert chloride and/or phosphine, has been interrogated by tungsten L-edge X-ray absorption spectroscopy. The L-edge spectra of this compound set, comprised of [W&lt;sup&gt;0&lt;/sup&gt;(PMe&lt;sub&gt;3&lt;/sub&gt;)&lt;sub&gt;6&lt;/sub&gt;], [W&lt;sup&gt;II&lt;/sup&gt;Cl&lt;sub&gt;2&lt;/sub&gt;(PMePh&lt;sub&gt;2&lt;/sub&gt;)&lt;sub&gt;4&lt;/sub&gt;], [W&lt;sup&gt;III&lt;/sup&gt;Cl&lt;sub&gt;2&lt;/sub&gt;(dppe)&lt;sub&gt;2&lt;/sub&gt;][PF&lt;sub&gt;6&lt;/sub&gt;] (dppe = 1,2-bis(diphenylphosphino)ethane), [W&lt;sup&gt;IV&lt;/sup&gt;Cl&lt;sub&gt;4&lt;/sub&gt;(PMePh&lt;sub&gt;2&lt;/sub&gt;)&lt;sub&gt;2&lt;/sub&gt;], [W&lt;sup&gt;V&lt;/sup&gt;(NPh)Cl&lt;sub&gt;3&lt;/sub&gt;(PMe&lt;sub&gt;3&lt;/sub&gt;)&lt;sub&gt;2&lt;/sub&gt;], and [W&lt;sup&gt;VI&lt;/sup&gt;Cl&lt;sub&gt;6&lt;/sub&gt;] correlate with formal oxidation state and have usefulness as references for the interpretation of the L-edge spectra of tungsten compounds with redox-active ligands and ambiguous electronic structure descriptions. The utility of these spectra arises from the combined correlation of the estimated branching ratio (EBR) of the L&lt;sub&gt;3,2&lt;/sub&gt;-edges and the L&lt;sub&gt;1&lt;/sub&gt; rising-edge energy with metal Z&lt;sub&gt;eff&lt;/sub&gt;, thereby permitting an assessment of effective metal oxidation state. An application of these reference spectra is illustrated by their use as backdrop for the L-edge X-ray absorption spectra of [W&lt;sup&gt;IV&lt;/sup&gt;(mdt)&lt;sub&gt;2&lt;/sub&gt;(CO)&lt;sub&gt;2&lt;/sub&gt;] and [W&lt;sup&gt;IV&lt;/sup&gt;(mdt)&lt;sub&gt;2&lt;/sub&gt;(CN)&lt;sub&gt;2&lt;/sub&gt;]&lt;sup&gt;2–&lt;/sup&gt; (mdt&lt;sup&gt;2–&lt;/sup&gt; = 1,2-dimethylethene-1,2-dithiolate), which shows that both compounds are effectively W&lt;sup&gt;IV&lt;/sup&gt; species. Use of metal L-edge XAS to assess a compound of uncertain formulation requires: 1) Placement of that data within the context of spectra offered by unambiguous calibrant compounds, preferably with the same coordination number and similar metal ligand distances. Such spectra assist in defining upper and/or lower limits for metal Z&lt;sub&gt;eff&lt;/sub&gt; in the species of interest; 2) Evaluation of that data in conjunction with information from other physical methods, especially ligand K-edge XAS; 3) Increased care in interpretation if strong π-acceptor ligands, particularly CO, or π-donor ligands are present. The electron-withdrawing/donating nature of these ligand types, combined with relatively short metal-ligand distances, exaggerate the difference between formal oxidation state and metal Z&lt;sub&gt;eff&lt;/sub&gt; or, as in the case of [W&lt;sup&gt;IV&lt;/sup&gt;(mdt)&lt;sub&gt;2&lt;/sub&gt;(CO)&lt;sub&gt;2&lt;/sub&gt;], add other subtlety by modulating the redox level of other ligands in the coordination sphere