This tutorial presents an in-depth introduction to chaos in dynamical systems, and presents several practical techniques for recognizing and classifying chaotic behavior. These techniques include the poincaré map, Lyapunov exponents, capacity, information dimension, correlation dimension, Lyapunov dimension, and the reconstruction of attractors from a single time series.
This paper provides a rigorous mathematical proof that the double scroll is indeed chaotic. Our approach is to derive a linearly equivalent class of piecewise-linear differential equations which includes the double scroll as a special case. A necessary and sufficient condition for two such piecewise-linear vector fields to be linearly equivalent is that their respective eigenvalues be a scaled version of each other. In the special case where they are identical, we have exact equivalence in the sense of linear conjugacy. An explicit normalform equation in the context of global bifurcation is derived and parametrized by their eigenvalues. Analytical expressions for various Poincaré maps are then derived and used to characterize the birth and the death of the double scroll, as well as to derive an approximate one-dimensional map in analytic form which is useful for further bifurcation analysis. In particular, the analytical expressions characterizing various half-return maps associated with the Poincaré map are used in a crucial way to prove the existence of a Shilnikov-type homoclinic orbit, thereby establishing rigorously the chaotic nature of the double scroll. These analytical expressions are also fundamental in our in-depth analysis of the birth (onset of the double scroll) and death (extinction of chaos) of the double scroll. The unifying theme throughout this paper is to analyze the double scroll system as an unfolding of a large family of piecewise-linear vector fields in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^3</tex> . Using this approach, we were able to prove that the chaotic dynamics of the double scroll is quite common, and is robust because the associated horseshoes predicted from Shilnikov's theorem are structurally stable. In fact, it is exhibited by a large family (in fact, infinitely many linearlyequivalent circuits) of vector fields whose associated piecewise-linear differential equations bear no resemblance to each other. It is therefore remarkable that the normalized eigenvalues, which is a local concept, completely determine the system's global qualitative behavior.
The authors prove that the piecewise-linear Lorenz circuit is chaotic in the sense of Shilnikov. They first prove the existence of a heteroclinic orbit, and then prove an inequality among the eigenvalues. In addition to a detailed analysis of the piecewise linear dynamics, interval analysis is utilized to prove various inequalities. This novel approach can be applied to many other problems that require proving rigorously the existence of either a homoclinic or a heteroclinic orbit. With this proof, the piecewise-linear Lorenz circuit becomes one of the very few real physical systems where chaos has been observed by laboratory measurement, confirmed by simulation, and proved mathematically.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
Nanoscale tip-induced polymerization of a sulfur thin film affords a simple negative tone resist that can be used as mask for substrate etching in fluorinated solutions or as a chemical template for the directed self-assembly of gold nanocrystals. Detailed facts of importance to specialist readers are published as "Supporting Information". Such documents are peer-reviewed, but not copy-edited or typeset. They are made available as submitted by the authors. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.
Read moreWe have investigated the transport properties in the direction perpendicular to the substrate of regioregular poly(3-hexyl-thiophene) of different molecular weights (MW) in a diode geometry. In these devices, which exhibit space-charge-limited behavior, we find that the mobility values at room temperature increase from 1.33×10−5cm2∕Vsto3.30×10−4cm2∕Vs as the MW is increased from 2.9to31.1kg∕mol. The mobility is found to be field independent for high MW films, but field dependent for the low MW films. The current–voltage characteristics of the diodes are also studied as a function of temperature from 160Kto300K. The activation energy for carrier transport, extracted from the Arrhenius plot, is found to decrease gradually from 143meVto126meV as the MW is increased.
Read moreIt is shown by experiment and by computer simulation that two mutually coupled PLLs (phase-locked loops) can become chaotic when they are operating in a marginal out-of-lock condition. i.e. when they are out-of-lock but very close to lock-in. The system is described by a fourth-order nonlinear autonomous equation. The power spectrum of the out-of-lock solution (close to lock-in) is observed to be a continuous broadband similar to that of a random signal. Moreover, the largest Lyapunov exponent is found to be positive. This chaotic regime has been observed over a large area of typical damping coefficients used in practical coupled PLL systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
Read moreDextran, a biocompatible, water-soluble polysaccharide, was modified at its hydroxyls with acetal moieties such that it became insoluble in water but freely soluble in common organic solvents enabling its use in the facile preparation of acid-sensitive microparticles. These particles degrade in a pH-dependent manner: FITC-dextran was released with a half-life at 37 degrees C of 10 h at pH 5.0 compared to a half-life of approximately 15 days at pH 7.4. Both hydrophobic and hydrophilic cargoes were successfully loaded into these particles using single and double emulsion techniques, respectively. When used in a model vaccine application, particles loaded with the protein ovalbumin (OVA) increased the presentation of OVA-derived peptides to CD8+ T-cells 16-fold relative to OVA alone. Additionally, this dextran derivative was found to be nontoxic in preliminary in vitro cytotoxicity assays. Owing to its ease of preparation, processability, pH-sensitivity, and biocompatibility, this type of modified dextran should find use in numerous drug delivery applications.
Read moreWe report chaotic phenomena observed from a simple nonlinear switched-capacitor circuit. The experimentally measured bifurcation tree diagram reveals a period-doubling route to chaos. This circuit is described by a first-order discrete equation which can be transformed into the logistic map whose chaotic dynamics is well known.
Read moreAcid-degradable particles containing a model protein antigen, ovalbumin, were prepared from a polyurethane with acetal moieties embedded throughout the polymer, and characterized by dynamic light scattering and transmission electron microscopy. The small molecule degradation byproduct of the particles was synthesized and tested in vitro for toxicity indicating an LC 50 of 12,500 microg/mL. A new liquid chromatography-mass spectrometry technique was developed to monitor the in vitro degradation of these particles. The degradation byproduct inside RAW macrophages was at its highest level after 24 h of culture and was efficiently exocytosed until it was no longer detectable after 4 days. When tested in vitro, these particles induced a substantial increase in the presentation of the immunodominant ovalbumin-derived peptide SIINFEKL in both macrophages and dendritic cells. In addition, vaccination with these particles generated a cytotoxic T-lymphocyte response that was superior to both free ovalbumin and particles made from an analogous but slower-degrading acid-labile polyurethane polymer. Overall, we present a fully degradable polymer system with nontoxic byproducts, which may find use in various biomedical applications including protein-based vaccines.
Read moreA "generalized brightness theorem" is derived that describes the thermodynamic limitations of the fluorescent planar concentrator. The maximum brightness concentration ratio allowed by thermodynamics is exp (hΔν/kT) where Δν is the Stokes shift in fluorescence.
Read moreWe cast the problem of very-low-threshold, daylight-pumped lasers in a general thermodynamic framework. We calculate that the requirements to reach threshold are that the Stokes shift of such a laser be greater than 13.3kT and that the absorption ratio of the pump/emission band and the geometrical aspect ratio both be greater than exp (13.3). We describe some luminescent material systems that might be able to satisfy these rigid requirements.
Read moreAbstract A new simple circuit model for the MOS (metal‐oxide‐semiconductor) structure is presented. the model consists of three elements, namely, a linear capacitor, a non‐linear capacitor and a C‐dynamic element. Each component bears a simple relationship to the physical operating mechanism inside the MOS structure. the model can be used for simulating arbitrary MOS structure circuits under all operating conditions. In particular, it is capable of reproducing the structure's frequency‐dependent small‐signal characteristics. the model is also shown to exhibit many important and interesting dynamic behaviours under forward, reverse and sinusoidal operating modes. The model is based mainly upon the device's physical operating principles. But perhaps the most significant implication of this model is that it is the first ever to use a dynamic element to model the MOS structure from a physical approach.
Read moreWe have investigated the temperature-dependent behavior of thiolated azobenzene molecules on Au(111) using scanning tunneling microscopy. The addition of a thiol functional group to azobenzene molecules leads to increased surface anchoring of single azobenzene molecules to gold. Thiolated azobenzene shows diverse surface morphology and does not form well-ordered structures at low coverage. At elevated temperatures, anchored molecules are observed to spin in place via hindered rotation. By measuring the number of rotating molecules as a function of temperature and using a simple model, we are able to estimate the energy barrier and attempt frequency for thermally induced hindered rotation to be 102±3 meV and 110±2 GHz, respectively.
Read moreA classic pump–probe experiment is performed on SF6. Using 30 psec pulses from two independently tunable CO2 lasers, we have studied the intramolecular energy transfer processes occurring in a collisionless environment. Distinct from nanosecond pulse experiments, we find an extremely narrow and deep hole burning feature at the pump wavelength. Experimental results exhibit both discrete state bottlenecking effects and quasicontinuum absorption. New evidence of an extremely rapid collisional relaxation is found, which may be related to rotational hole filling.
Read moreThe synthesis and utilization of polymeric catalyst 1 bearing proline moieties as active sites is reported. This catalyst aggregate was applied successfully in aldol reactions of aldehydes and ketones in aqueous media. In a self-aldol process of butanal, mainly reversibly formed β-hydroxyaldehyde 2 was obtained, whereas the formation of aldol condensation product 3 could be suppressed. These results were transferred to more complex dynamic catalytic cross-aldol reactions, giving rise to α,β-unsaturated ketones 4, favored over formation of self-condensation product 5. The likewise expected ketones 6 were not observed.
Read moreAbstract The in situ preparation of monolithic capillary columns comprising copolymers of butyl methacrylate with ethylene dimethacrylate, diethylene glycol dimethacrylate, triethylene glycol dimethacrylate, and pentaerythritol tetraacrylate using thermal polymerization within 250 μm ID capillaries and their application for μ‐HPLC separations of proteins has been studied. For all crosslinkers, optimization of the porogenic mixture consisting of 1‐propanol and 1,4‐butanediol yielded monoliths with pore sizes above 1 μm suitable for rapid separations at low back pressure. Very good separations were achieved for a protein mixture consisting of ribonuclease A, cytochrome c, myoglobin, and ovalbumin with all tested columns.
Read moreThe double scroll attractor has been experimentally observed from an extremely simple circuit using an op amp. The purpose of this brief note is to give an alternate realization of the circuit using only two transistors as the active elements.
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