The Dissipative Nonlocal Oscillator

The most important characteristics of a nonlocal and nonlinear oscillator subject to dissipative forces are extensively studied by means of an asymptotic perturbation method, based upon temporal rescaling and harmonic balance. The conditions under which bifurcations and limit cycles appear are determined. If the parameters satisfy particular conditions, a quasi-periodic motion is predicted, because a second small frequency adds to the natural frequency of the oscillator. The analytical results are validated by numerically solving the original system.     

The Response of an Oscillator Moving along a Parabola to an External Excitation

The behavior of a mass point moving along a parabola under theeffect of an external periodic excitation in resonance with the naturalfrequency of the oscillator is studied. The asymptotic perturbationmethod based on temporal rescaling and balancing of the harmonic termswith a simple iteration is used in order to determine the nonlinearmodulation equations for the amplitude and the phase of the oscillation.External force-response curves are shown and moreover jump phenomena arealso observed. In certain cases a second low frequency appears inaddition to the forcing frequency and then stable two-periodquasi-periodic motions are present with amplitudes depending on theinitial conditions. The value of the low frequency depends on theamplitude of the external excitation. A higher order perturbationanalysis is developed and the validity of the method is highlighted bycomparing the leading order and the higher order approximate analyticsolutions to numerical results.     

Bifurcation Analysis of Parametrically Excited Rayleigh–Liénard Oscillators

A parametrically excited Rayleigh–Liénard oscillator is investigatedby an asymptotic perturbation method based on Fourier expansion and timerescaling. Two coupled equations for the amplitude and the phase ofsolutions are derived and the stability of steady-state periodic solutionsas well as parametric excitation-response and frequency-response curvesare determined. Comparison with the parametrically excited Liénardoscillator is performed and analytic approximate solutions are checkedusing numerical integration. Dulac's criterion, thePoincaré–Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the twocoupled equations. A limit cycle corresponds to a modulated motion forthe Rayleigh–Liénard oscillator. Modulated motion can be also obtainedfor very low values of the parametric excitation, and in this case, anapproximate analytic solution is easily constructed. If the parametricexcitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while themodulation amplitude remains finite and suddenly the attractor settlesdown into a periodic motion. Floquet's theory is used to evaluatethe stability of the periodic solutions, and in certain cases,symmetry-breaking bifurcations are predicted. Numerical simulationsconfirm this scenario and detect chaos and unbounded motions in theinstability regions of the periodic solutions.     

Parametric Excitation for Two Internally Resonant van der Pol Oscillators

The response of a system of two nonlinearly coupled van der Poloscillators to a principal parametric excitation in the presence ofone-to-one internal resonance is investigated. The asymptoticperturbation method is applied to derive the slow flow equationsgoverning the modulation of the amplitudes and the phases of the twooscillators. These equations are used to determine steady-stateresponses, corresponding to a periodic motion for the starting system(synchronisation), and parametric excitation-response andfrequency-response curves. Energy considerations are used to studyexistence and characteristics of limit cycles of the slow flowequations. A limit cycle corresponds to a two-period amplitude- andphase-modulated motion for the van der Pol oscillators. Two-periodmodulated motion is also possible for very low values of the parametricexcitation and an approximate analytic solution is constructed for thiscase. If the parametric excitation increases, the oscillation period ofthe modulations becomes infinite and an infinite-period bifurcationsoccur. Analytical results are checked with numerical simulations.     

Vibration control by nonlocal feedback and jerk dynamics

Jerk dynamics is used for a new method for the suppression of self-excited vibrations in nonlinear oscillators. Two cases are considered, the van der Pol equation and nonlinear oscillator with quadratic and cubic nonlinearities. A nonlocal control force is introduced in such a way to obtain a third order nonlinear differential equation (jerk dynamics). Using the asymptotic perturbation method, two slow flow equations on the amplitude and phase of the response are obtained, and subsequently the performance of the control strategy is investigated. The feedback gains are connected with the stability and response of the system under control. Uncontrolled and controlled systems are compared and the appropriate choices for the feedback gains are found in order to reduce the amplitude peak of the self-excitations. Numerical simulation confirms the validity of the new method.     

Chaotic and fractal patterns for interacting nonlinear waves

Using an appropriate reduction method, a quite general new integrable system of equations 2 + 1 dimensions can be derived from the dispersive long-wave equation. Various soliton and dromion solutions are obtaining by selecting some types of solutions appropriately. The interaction between the localized solutions is completely elastic, because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. The arbitrariness of the functions included in the general solution implies that approximate lower dimensional chaotic patterns such as chaotic–chaotic patterns, periodic–chaotic patterns, chaotic line soliton patterns and chaotic dromion patterns can appear in the solution. In a similar way, fractal dromion patterns and stochastic fractal excitations also exist for appropriate choices of the boundary conditions and/or initial conditions.     

Capillary electrophoresis of complex natural polysaccharides

Complex natural polysaccharides, glycosaminoglycans (GAGs), are a class of ubiquitous macromolecules that exhibit a wide range of biological functions and participate and regulate multiple cellular events and (patho)physiological processes. They are generally present either as free chains (hyaluronic acid and bacterial acidic polysaccharides) or as side chains of proteoglycans (PGs; chondroitin/dermatan sulfate, heparin/heparan sulfate, and keratan sulfate) and are most often found in cell membranes and in the extracellular matrix. The recent emergence of modern analytical tools for their study has produced a virtual explosion in the field of glycomics. CE, due to its high resolving power and sensitivity, has been useful in the analysis of intact GAGs and GAG-derived oligosaccharides and disaccharides affording concentration and structural characterization data essential for understanding the biological functions of GAGs. In this review, novel off-line and on-line CE-MS and MS/MS methods for screening of GAG-derived oligosaccharides and disaccharides will be discussed.     

Electrophoresis for the analysis of heparin purity and quality

The adulteration of raw heparin with oversulfated chondroitin sulfate (OSCS) in 2007-2008 produced a global crisis resulting in extensive revisions to the pharmacopeia monographs and prompting the FDA to recommend the development of additional methods for the analysis of heparin purity. As a consequence, a wide variety of innovative analytical approaches have been developed for the quality assurance and purity of unfractionated and low-molecular-weight heparins. This review discusses recent developments in electrophoresis techniques available for the sensitive separation, detection, and partial structural characterization of heparin contaminants. In particular, this review summarizes recent publications on heparin quality and related impurity analysis using electrophoretic separations such as capillary electrophoresis (CE) of intact polysaccharides and hexosamines derived from their acidic hydrolysis, and polyacrylamide gel electrophoresis (PAGE) for the separation of heparin samples without and in the presence of its relatively specific depolymerization process with nitrous acid treatment.     

Configurational assignments of the diastereomers of 3,3′‐(1,2‐ethanediyl)bis[2‐(3‐fluorophenyl)‐5‐methyl‐4‐thiazolidinone] derivative with four stereogenic centers

Diastereomers of antiinflammatory/analgesic and antihistaminic 3,3′[(1,2‐ethanediyl)bis(2‐aryl‐4‐thiazo‐lidinone)] derivatives possessing two stereogenic centers (indicated as BIS 2*C) have been widely investigated in recent years. The 5,5′‐dimethyl analogues (BIS 4*C), now reported, have been synthesized by reaction of (±) α‐mercaptopropionic acid and N,N'‐di(3‐fluorobenzyliden)ethylenediamine. Because the 2 and 2′carbons bear the same groups and similarly the 5 and 5′ carbons, and the latter groups are different from the former, four enantiomeric pairs and two meso forms exist in this situation. These diastereomers were identified by the concerted use of nmr spectroscopy and hplc on chiral stationary phase.