Periodic motions with impact chatters in an impact Duffing oscillator.

The periodic motions of discontinuous nonlinear dynamical systems are very difficult problems to solve in engineering and physics. Until now, except for numerical studies, one cannot find a better way to solve such problems. In fact, one still has difficulty obtaining periodic motions in continuous nonlinear dynamical systems. In this paper, a method is presented systematically for periodic motions in discontinuous nonlinear dynamical systems. The stability and grazing bifurcations of such periodic motions are studied. Such a method is presented through discussion on a periodically forced, impact Duffing oscillator. Thus, periodic motions with impact chatters in a periodically forced Duffing oscillator with one-sidewall constraint are studied. The analytical conditions for motion grazing at the boundary are developed from discontinuous dynamical systems. The impact Duffing oscillator is discretized to generate subimplicit mappings. With impact, the mapping structures are employed to construct specific impact periodic motions for an impact Duffing oscillator. The bifurcation trees of impact chatter periodic motions are achieved semi-analytically. The grazing and period-doubling bifurcations are obtained, and the grazing bifurcations are for the appearing and disappearance for an impact chatter periodic motion. The impact chatter periodic motions with and without grazing are presented for illustration of impact periodic motion complexity in the impact Duffing oscillator.

Limit cycles and homoclinic networks in two-dimensional polynomial systems.

In this paper, the properties of equilibriums in planar polynomial dynamical systems are studied. The homoclinic networks of sources, sinks, and saddles in self-univariate polynomial systems are discussed, and the numbers of sources, sinks, and saddles are determined through a theorem, and the first integral manifolds are determined. The corresponding proof of the theorem is completed, and a few illustrations of networks for source, sinks, and saddles are presented for a better understanding of the homoclinic networks. Such homoclinic networks are without any centers even if the networks are separated by the homoclinic orbits. The homoclinic networks of positive and negative saddles with clockwise and counterclockwise limit cycles in crossing-univariate polynomial systems are studied secondly, and the numbers of saddles and centers are determined through a theorem, and the first integral manifolds are determined through polynomial functions. The corresponding proof of the theorem is given, and a few illustrations of networks of saddles and centers are given to show the corresponding geometric structures. Such homoclinic networks of saddles and centers are without any sources and sinks. Since the maximum equilibriums for such two types of planar polynomial systems with the same degrees are discussed, the maximum centers and saddles should be obtained, and maximum sinks, sources, and saddles should be achieved. This paper may provide a different way to determine limit cycles in the Hilbert 16th problem.

Constructed complex motions and chaos.

Constructed motions and dynamic topology are new trends in solving nonlinear systems or system interactions. In nonlinear engineering, it is significant to achieve specific complex motions to satisfy expected dynamical behaviors (e.g., nonlinear motions, singularities, bifurcations, chaos, etc.), and complex motion application and control. To achieve such expected motions and global dynamical behaviors, mapping dynamics, constructed networks, random/discontinuous dynamic theorems, etc., are applied to quantitatively determine the complex motions. These theories adopt the symbolic dynamic abstracts and topological structures with nonlinear dynamics to investigate constructed complex motions to satisfy expected dynamical behaviors.

Period-3 motions to chaos in a periodically forced nonlinear-spring pendulum.

In this paper, the complete bifurcation dynamics of period-3 motions to chaos are obtained semi-analytically through the implicit mapping method. Such an implicit mapping method employs discrete implicit maps to construct mapping structures of periodic motions to determine complex periodic motions. Analytical bifurcation trees of period-3 motions to chaos are determined through nonlinear algebraic equations generated through the discrete implicit maps, and the corresponding stability and bifurcations of periodic motions are achieved through eigenvalue analysis. To study the periodic motion complexity, harmonic amplitudes varying with excitation amplitudes are presented. Once more, significant harmonic terms are involved in periodic motions, and such periodic motions will be more complex. To illustrate periodic motion complexity, numerical and analytical solutions of periodic motions are presented for comparison, and the corresponding harmonic amplitudes and phases are also presented for such periodic motions in the bifurcation trees of period-3 motions to chaos. Similarly, other higher-order periodic motions and bifurcation dynamics for the nonlinear spring pendulum can be determined. The methods and analysis presented herein can be applied for other nonlinear dynamical systems.

Periodic motions and homoclinic orbits in a discontinuous dynamical system on a single domain with multiple vector fields.

In this paper, periodic motions and homoclinic orbits in a discontinuous dynamical system on a single domain with two vector fields are discussed. Constructing periodic motions and homoclinic orbits in discontinuous dynamical systems is very significant in mathematics and engineering applications, and how to construct periodic motions and homoclinic orbits is a central issue in discontinuous dynamical systems. Herein, how to construct periodic motions and homoclinic orbits is presented through studying a simple discontinuous dynamical system on a domain confined by two prescribed energies. The simple discontinuous dynamical system has energy-increasing and energy-decreasing vector fields. Based on the two vector fields and the corresponding switching rules, periodic motions and homoclinic orbits in such a simple discontinuous dynamical system are studied. The analytical conditions of bouncing, grazing, and sliding motions at the two energy boundaries are presented first. Periodic motions and homoclinic orbits in such a discontinuous dynamical system are determined through the specific mapping structures, and the corresponding stability is also presented. Numerical illustrations of periodic motions and homoclinic orbits are given for constructed complex motions. Through this study, using discontinuous dynamical systems, one can construct specific complex motions for engineering applications, and the corresponding mathematical methods and computational strategies can be developed.

Nonlinear piezoelectric energy harvesting induced through the Duffing oscillator.

In this paper, nonlinear piezoelectric energy harvesting induced by a Duffing oscillator is studied, and the bifurcation trees of period-1 motions to chaos for such a piezoelectric energy-harvesting system are obtained analytically. Distributed-parameter electromechanical modeling of a piezoelectric energy harvester is presented first, and the electromechanically coupled circuit equation excited by infinitely many vibration modes is developed. The governing electromechanical equations are reduced to ordinary differential equations in modal coordinates, and eventually an infinite set of algebraic equations is obtained for the complex modal vibration responses and the complex voltage responses of the energy harvester beam. One single mode case is considered in this paper, and periodic motions with bifurcation trees are obtained through an implicit discrete mapping method. The frequency-amplitude characteristics of periodic motions are obtained for the nonlinear piezoelectric energy-harvesting systems, which provide a better understanding of where and how to achieve the best energy harvesting. This study describes about how the nonlinear oscillator induces piezoelectric energy harvesting through a beam system. The nonlinear piezoelectric energy harvesting is presented through a nonlinear oscillator. Due to the nonlinear oscillator, chaotic piezoelectric energy-harvesting states can get more energy compared to the linear piezoelectric energy-harvesting system.

On an origami structure of period-1 motions to homoclinic orbits in the Rössler system.

In this paper, an origami structure of period-1 motions to spiral homoclinic orbits in parameter space is presented for the Rössler system. The edge folds of the origami structure are generated by the saddle-node bifurcations. For each edge, there are two layers to form the origami structure. On one layer of the origami structure, there is a pair of period-doubling bifurcations inducing periodic motions from period-1 to period-2n motions (n=1,2,…,∞). On such a layer, the unstable period-1 motion goes to the homoclinic orbits with a mapping eigenvalue approaching negative infinity. However, on the corresponding adjacent layers, no period-doubling bifurcations exist, and the unstable period-1 motion goes to the homoclinic orbit with a mapping eigenvalue approaching positive infinity. To determine the origami structure of the period-1 motions to homoclinic orbits, the implicit map of the Rössler system is developed through the discretization of the corresponding differential equations. The Poincaré mapping section can be selected arbitrarily. Before construction of the origami structure, the bifurcation diagram of periodic motions varying with one parameter is developed, and trajectories of stable periodic motions on the bifurcation diagram to homoclinic orbits are illustrated. Finally, the origami structures of period-1 motions to homoclinic orbits are developed through a few layers. This study provides the mathematical mechanisms of period-1 motions to homoclinic orbits, which help one better understand the complexity of periodic motions near the corresponding homoclinic orbit. There are two types of infinitely many homoclinic orbits in the Rössler system, and the corresponding mapping structures of the homoclinic orbits possess positive and negative infinity large eigenvalues. Such infinitely many homoclinic orbits are induced through unstable periodic motions with positive and negative eigenvalues accordingly.

On infinite homoclinic orbits induced by unstable periodic orbits in the Lorenz system.

In this paper, infinite homoclinic orbits existing in the Lorenz system are analytically presented. Such homoclinic orbits are induced by unstable periodic orbits on bifurcation trees through period-doubling cascades. Each unstable periodic orbit ends at its corresponding homoclinic orbit. Traditional computational methods cannot obtain homoclinic orbits from the corresponding unstable periodic orbits. This is because unstable periodic orbits in the Lorenz system cannot be achieved in numerical simulations. Herein, the stable and unstable periodic motions to chaos on the period-doubling cascaded bifurcation trees are determined through a discrete mapping method. The corresponding homoclinic orbits induced by the unstable periodic orbits are predicted analytically. A period-doubling bifurcation tree of period-1, period-2, and period-4 motions are generated as an example. The homoclinic orbits relative to unstable period-1, period-2, and period-4 motions are determined. Illustrations of homoclinic orbits and periodic orbits are given. This study presents how to determine infinite homoclinic orbits through unstable periodic orbits in three-dimensional or higher-dimensional nonlinear systems.

Identification of lysophosphatidylcholine-chlorohydrin in human atherosclerotic lesions.

Lysophosphatidylcholine (LysoPtdCho) levels are elevated in sera in patients with atherosclerosis and in atherosclerotic tissue. Previous studies have shown that reactive chlorinating species attack plasmalogens in human coronary artery endothelial cells (HCAEC), forming lysoPtdCho and lysoPtdCho-chlorohydrin (lysoPtdCho-ClOH). The results herein demonstrate for the first time that lysoPtdCho-ClOH is elevated over 60-fold in human atherosclerotic lesions. In cultured HCAEC, 18:0 lysoPtdCho-ClOH led to a statistically significant increase in P-selectin cell-surface expression, but unlike 18:1 lysoPtdCho did not lead to cyclooxygenase-2 protein expression. These data show that 18:0 lysoPtdCho-ClOH is elevated in atherosclerotic tissue and may have unique pro-atherogenic properties compared to lysoPtdCho.

Hypoglossal nerve transfer in obstetric brachial plexus palsy.

A cost-benefit analysis was performed of hypoglossal nerve transfer in six patients with obstetric brachial palsy taking into account the factors donor site morbidity and extent of recovery. Hypoglossal nerve transfer was employed in four children for elbow flexion only; in two patients for elbow flexion as well as for elbow extension. The transfer was part of an extended brachial plexus reconstruction for treatment of obstetric brachial plexus palsy. After a mean post-operative interval of 52 months (SD+/-8.1), two professional speech therapists investigated late donor site morbidity by analyzing elementary and communicative functions. The functional result for the arm was assessed using the Mallet scale and by performing a physical examination. Following hypoglossal nerve transfer, early donor site morbidity was significant causing great anxiety in the parents. Late donor site morbidity consisted of serious oral problems in a number of the children. They also showed clear associated movements in the arm during mouth/tongue activity. Recovery of powerful volitional elbow flexion was achieved in four cases only. We do not believe that the sacrifice of such an important function as exerted by the hypoglossal nerve is balanced by the gain demonstrated in our series.

Transfer of pectoral nerves to the musculocutaneous nerve in obstetric upper brachial plexus palsy.

OBJECTIVE: To investigate the results of transfer of pectoral nerves to the musculocutaneous nerve for treatment of obstetric brachial palsy. METHODS: In 25 cases of obstetric brachial palsy (20 after breech deliveries), branches of the pectoral nerve plexus were transferred directly to the musculocutaneous nerve. For all patients, the nerve transfer was part of an extended brachial plexus reconstruction. Results were tested both clinically and with the Mallet scale, at a mean follow-up time of 70 months (standard deviation, 34.3 mo). RESULTS: There were two complete failures, which were attributable to disconnection of the transferred nerve endings. The results after transfer were excellent in 17 cases and fair in 5 cases. Steindler flexorplasty improved elbow flexion for three patients. CONCLUSION: Transfer of pectoral nerves to the musculocutaneous nerve for treatment of obstetric upper brachial palsy may be effective, if the specific anatomic features of the pectoral nerve plexus are sufficiently appreciated.

Reactive chlorinating species produced by myeloperoxidase target the vinyl ether bond of plasmalogens: identification of 2-chlorohexadecanal.

Plasmalogens contain a vinyl ether bond linking the sn-1 aliphatic chain to the glycerol backbone of this predominant phospholipid molecular subclass, which is found in many mammalian tissues. The present study demonstrates that the vinyl ether bond of plasmalogens is a molecular target of the reactive chlorinating species produced by myeloperoxidase. Analysis by thin layer chromatography revealed that reactive chlorinating species produced by myeloperoxidase target the vinyl ether bond of the plasmalogen, lysoplasmenylcholine (1-O-hexadec-1'-enyl-sn-glycero-3-phosphorylcholine), resulting in the production of a neutral lipid. Capillary gas chromatographic analyses demonstrated that the neutral lipid generated from lysoplasmenylcholine was neither hexadecanal nor did it contain masked hexadecanal (i.e. the vinyl ether) because the dimethyl acetal of hexadecanal produced by acid methanolysis derivatization was no longer present. Electrospray ionization mass spectrometry of the myeloperoxidase-generated neutral lipid product was consistent with the production of a 16-carbon fatty aldehyde containing one chlorine atom. Furthermore, proton NMR analysis indicated that this neutral lipid product was a 2-chloro-fatty aldehyde. Additional structural analysis of this neutral lipid by gas chromatography-mass spectrometry of the underivatized product as well as its pentafluorobenzyl oxime-derivative product was consistent with the neutral lipid being 2-chlorohexadecanal. The reactive chlorinating species, hypochlorous acid and chlorine gas, both attacked the vinyl ether bond of lysoplasmenylcholine resulting in the production of 2-chlorohexadecanal. The production of 2-chlorohexadecanal was dependent on the presence of the plasmalogen masked aldehyde (i.e. the vinyl ether) in the substrate because the free fatty aldehyde, hexadecanal, was not converted to 2-chlorohexadecanal by the reactive chlorinating species generated by myeloperoxidase. Taken together, the present studies demonstrate for the first time the targeting of the vinyl ether bond of plasmalogens by the reactive chlorinating species produced by myeloperoxidase resulting in the production of novel chlorinated fatty aldehydes.

Protein kinase C translocation and PKC-dependent protein phosphorylation during myocardial ischemia.

The present study demonstrates that the alpha, epsilon, and iota isozymes of protein kinase C (PKC) are translocated to particulate fractions from the cytosol during brief intervals of global ischemia as well as reperfusion of ischemic rat myocardium. In contrast, phorbol ester treatment of perfused hearts resulted in the translocation of the alpha, delta, and epsilon isozymes of PKC to particulate fractions. Additionally, the alpha, delta, and epsilon isozymes of PKC are translocated to particulate fractions in phorbol ester-stimulated, isolated adult rat cardiac myocytes. Concomitant with the translocation of PKC isozymes to particulate fractions during myocardial ischemia, increased protein phosphorylation was observed, which was blocked by pretreatment of hearts with the selective PKC inhibitor bisindolylmaleimide I (50 nM). In particular, ischemia resulted in the phosphorylation of 26-, 20-, and 17-kDa particulate-associated proteins. Taken together, the present findings are the first to demonstrate that specific PKC isozymes are translocated to particulate fractions in the ischemic and the reperfused ischemic rat heart, resulting in the phosphorylation of specific particulate-associated proteins.

Identification of specific nuclear protein kinase C isozymes and accelerated protein kinase C-dependent nuclear protein phosphorylation during myocardial ischemia.

Protein kinase C (PKC) has been suggested to mediate, at least in part, multiple processes in the pathophysiological sequelae of myocardial ischemia. The present study demonstrates that the epsilon, eta and iota isozymes of PKC are translocated to nuclei in response to brief intervals of global ischemia as well as reperfusion of ischemic rat myocardium. Concomitant with the translocation of PKC isozymes to nuclei during ischemia, increased PKC-mediated nuclear protein phosphorylation was observed. Taken together, the present results demonstrate that nuclear signaling mechanisms are activated during myocardial ischemia that include PKC translocation and PKC-mediated nuclear protein phosphorylation.