scholarly journals Floquet Theory for Discontinuously Supported Waveguides

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
A. Sorzia
Keyword(s):  
Floquet Theory ◽  
Periodic Coefficient ◽  
Fundamental Solutions ◽  
Hill Equation ◽  
Elastic Waveguide ◽  
Elastic String ◽  
Winkler Elastic Foundation ◽  
The Stability ◽  
The Hill ◽  
Band Gap Structure

We apply Floquet theory of periodic coefficient second-order ODEs to an elastic waveguide. The waveguide is modeled as a uniform elastic string periodically supported by a discontinuous Winkler elastic foundation and, as a result, a Hill equation is found. The fundamental solutions, the stability regions, and the dispersion curves are determined and then plotted. An asymptotic approximation to the dispersion curve is also given. It is further shown that the end points of the band gap structure correspond to periodic and semiperiodic solutions of the Hill equation.

scholarly journals Resonance tongues for the Hill equation with Duffing coefficients and instabilities in a nonlinear beam equation

2017 ◽  
Vol 20 (01) ◽  
pp. 1750022 ◽  
Author(s):  
Carlo Gasparetto ◽  
Filippo Gazzola
Keyword(s):  
Nauk Sssr ◽  
Elliptic Functions ◽  
Periodic Coefficient ◽  
Hill Equation ◽  
Beam Equation ◽  
Stability And Instability ◽  
Beam Equations ◽  
Akad Nauk Sssr ◽  
The Stability ◽  
The Hill

We consider a class of Hill equations where the periodic coefficient is the squared solution of some Duffing equation plus a constant. We study the stability of the trivial solution of this Hill equation and we show that a criterion due to Burdina [Boundedness of solutions of a system of differential equations, Dokl. Akad. Nauk. SSSR 92 (1953) 603–606] is very helpful for this analysis. In some cases, we are also able to determine exact solutions in terms of Jacobi elliptic functions. Overall, we obtain a fairly complete picture of the stability and instability regions. These results are then used to study the stability of nonlinear modes in some beam equations.

STABILITY BOUNDARY AND DURATION TIME OF SYNCHRONIZATION FOR COUPLED CHAOTIC MATHIEU–DUFFING OSCILLATORS

2008 ◽  
Vol 22 (27) ◽  
pp. 4817-4831
Author(s):  
JIANHE SHEN ◽  
JIANPING CAI ◽  
SHUHUI CHEN ◽  
KECHANG LIN
Keyword(s):  
Coupling Strength ◽  
Stability Boundary ◽  
Hill Equation ◽  
Critical Values ◽  
Stability Boundaries ◽  
Synchronization Time ◽  
Duration Time ◽  
Error System ◽  
The Stability ◽  
The Hill

The stability boundaries and behaviors of the duration time of synchronization for chaotic Mathieu–Duffing oscillators are investigated. Based on the unidirectional or bidirectional linear state error feedback coupled scheme, the error system is derived. After replacing the chaotic orbit by a regular orbit containing multi-harmonics, we analyze the asymptotic stability of the error system, which leads to a Hill equation. According to Floquet theory and the properties of the Hill equation, the evolution of the discriminant of the Hill equation with respect to the coupling strength is traced to determine the stability boundaries between the synchronization and desynchronization domains. Thus, the critical values of coupling strength are obtained. These critical values are in good agreement with those from numerical simulations. The behaviors of the synchronization time are numerically investigated in the synchronization domain. It is found that the synchronization time reaches an asymptotic minimal value when the oscillators are unidirectionally or bidirectionally coupled, and the two asymptotic minimal values are almost the same. It is also noted that the slowing down behavior of the synchronization time can occur inside the synchronization domain when the coupling is bidirectional.

Time Domain Approaches to the Stability Analysis of Flexible Dynamical Systems

2015 ◽  
Vol 11 (4) ◽  
Author(s):  
Jielong Wang ◽  
Xiaowen Shan ◽  
Bin Wu ◽  
Olivier A. Bauchau
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Time Domain ◽  
Floquet Theory ◽  
Characteristic Exponent ◽  
Periodic Coefficient ◽  
Theoretical Background ◽  
System Stability ◽  
Lyapunov Characteristic Exponent ◽  
The Stability

This paper presents two approaches to the stability analysis of flexible dynamical systems in the time domain. The first is based on the partial Floquet theory and proceeds in three steps. A preprocessing step evaluates optimized signals based on the proper orthogonal decomposition (POD) method. Next, the system stability characteristics are obtained from partial Floquet theory through singular value decomposition (SVD). Finally, a postprocessing step assesses the accuracy of the identified stability characteristics. The Lyapunov characteristic exponent (LCE) theory provides the theoretical background for the second approach. It is shown that the system stability characteristics are related to the LCE closely, for both constant and periodic coefficient systems. For the latter systems, an exponential approximation is proposed to evaluate the transition matrix. Numerical simulations show that the proposed approaches are robust enough to deal with the stability analysis of flexible dynamical systems and the predictions of the two approaches are found to be in close agreement.

scholarly journals Analitical theory Flouqet-Bloch waves for gyrotropic magnetophotonic krystals

2019 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Hill Equation ◽  
Dispersion Characteristics ◽  
Bloch Waves ◽  
The Third ◽  
Magnetophotonic Crystals ◽  
The Hill ◽  
Third Boundary Value Problem

The relevance of the problem considered in the work is determined by the widespread use of magnetophotonic crystals in various devices of the terahertz microwave and optical ranges. The key is the analytical solution of the third boundary value problem for the Hill equation with mixed Cauchy boundary conditions. This approach made it possible to explicitly find new solutions for electromagnetic fields in the crystal layers and dispersion characteristics for TE and TM waves, which is important for the development of new multifunction devices in the terahertz range.. The purpose of the work is to develop an analytical theory of Floquet-Bloch waves for gyrotropic magnetophotonic crystals with a transverse magnetic field. Materials and methods. Magnetophotonic crystals consist of gyrotropic (gyroelectric or gyromagnetic materials) two-layer structures over a period, the parameters of which vary from the magnitude of the applied magnetic field. Analytical methods for solving the Hill equation through fundamental solutions of the third boundary value problem. Results. The fundamental solutions of the Hill equation are determined in an analytical form. Analytical expressions for the dispersion characteristics of TE and TM Floquet-Bloch waves are found. The existence of bulk and surface waves in the transmission zones of a magnetophotonic crystal is established. The existence of an extraordinary surface wave with an atypical field distribution in the crystal layers for positive effective electric or magnetic permeability is shown. Conclusions. The proposed new approach for determining the solutions of the Hill equation based on the fundamental solutions of the third boundary-value problem made it possible to obtain in an analytical form the dispersion characteristics and fields of controlled gyromagnetic magnetophotonic crystals for TE and TM Floquet-Bloch waves. This will make it relatively easy to calculate various devices based on controlled Bragg structures.

scholarly journals Pharmacodynamics of Daptomycin against Enterococcus faecium and Enterococcus faecalis in the Murine Thigh Infection Model

2018 ◽  
Vol 62 (10) ◽  
Author(s):  
James M. Kidd ◽  
Kamilia Abdelraouf ◽  
Tomefa E. Asempa ◽  
Romney M. Humphries ◽  
David P. Nicolau
Keyword(s):  
Enterococcus Faecalis ◽  
Enterococcus Faecium ◽  
Hill Equation ◽  
Infection Model ◽  
Free Drug Concentration ◽  
Time Curve ◽  
Content Type ◽  
Concentration Time ◽  
The Hill ◽  
Susceptibility Breakpoint

ABSTRACT The Clinical and Laboratory Standards Institute (CLSI) daptomycin MIC susceptibility breakpoint for the treatment of enterococcal infections is ≤4 μg/ml. However, patients receiving daptomycin for the treatment of infections caused by enterococci with MICs of ≤4 μg/ml may experience treatment failures. We assessed the pharmacodynamics of daptomycin against enterococci in a neutropenic murine thigh infection model and determined the exposures necessary for bacteriostasis and a 1-log10-CFU reduction of Enterococcus faecalis and Enterococcus faecium. We further characterized daptomycin efficacy at clinically achievable exposures. Six E. faecium and 6 E. faecalis isolates (daptomycin MICs, 0.5 to 32 μg/ml) were studied. Daptomycin was administered at various doses over 24 h to achieve area under the free drug concentration-time curve-to-MIC ratios (fAUC0–24/MIC) ranging from 1 to 148. Daptomycin regimens that simulate mean human exposures following doses of 6, 8, and 10 mg/kg of body weight/day were also studied. Efficacy was assessed by the differences in the number of log10 CFU per thigh at 24 h. The Hill equation was used to estimate the fAUC0–24/MIC required to achieve bacteriostasis and a 1-log10-CFU reduction. For E. faecium, a 1-log10-CFU reduction required an fAUC0–24/MIC of 12.9 (R2 = 0.71). For E. faecalis, a 1-log10-CFU reduction was not achieved, while the fAUC0–24/MIC required for stasis was 7.2 (R2 = 0.8). With a human-simulated regimen of 6 mg/kg/day, a 1-log10-CFU reduction was observed in 3/3 E. faecium isolates with MICs of <4 μg/ml and 0/3 E. faecium isolates with MICs of ≥4 μg/ml; however, a 1-log10-CFU reduction was not achieved for any of the 6 E. faecalis isolates. These results, alongside clinical data, prompt a reevaluation of the current breakpoint.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

Effect of osmotic swelling and shrinkage on Na(+)-K+ pump activity in mammalian cardiac myocytes

1993 ◽  
Vol 265 (5) ◽  
pp. C1201-C1210 ◽  
Author(s):  
D. W. Whalley ◽  
L. C. Hool ◽  
R. E. Ten Eick ◽  
H. H. Rasmussen
Keyword(s):  
Ventricular Myocytes ◽  
Pump Current ◽  
Hill Equation ◽  
Cell Swelling ◽  
Intracellular Na Activity ◽  
Single Ventricular Myocytes ◽  
Hyperosmolar Solutions ◽  
Hill Coefficients ◽  
The Hill ◽  
The Relationship

The effect on the sarcolemmal Na(+)-K+ pump of exposure to anisosmolar solutions was examined using whole cell patch clamping and ion-selective microelectrodes. Na(+)-K+ pump currents were measured in single ventricular myocytes by using pipette Na+ concentrations ([Na]pip) of 0-70 mM. The relationship between [Na]pip and pump current was well described by the Hill equation. The [Na]pip for half-maximal pump current (K0.5) was 21.4 mM in isosmolar (310 mosM) solution. K0.5 was 12.8 mM during cell swelling in hyposmolar solution (240 mosM) and 39.0 mM during cell shrinkage in hyperosmolar solution (464 mosM). The maximal pump currents, derived from the best fit of the Hill equation, and the Hill coefficients were similar in isosmolar, hyposmolar, and hyperosmolar solutions. A sustained (> 20 min) decrease in the intracellular Na+ activity developed during exposure of intact papillary muscles to hyposmolar solutions, and a sustained increase developed during exposure to hyperosmolar solutions. We conclude that osmotic myocyte swelling stimulates the sarcolemmal Na(+)-K+ pump at near-physiological levels of intracellular Na+, whereas shrinkage inhibits the pump. These changes are due to increases and decreases, respectively, in the apparent affinity of the pump for Na+.

Perspective on the relationship between GABAA receptor activity and the apparent potency of an inhibitor

2021 ◽  
Vol 19 ◽  
Author(s):  
Allison L. Germann ◽  
Spencer R. Pierce ◽  
Alex S. Evers ◽  
Joe Henry Steinbach ◽  
Gustav Akk
Keyword(s):  
Gabaa Receptor ◽  
Hill Equation ◽  
Response Relationship ◽  
Inhibition Concentration ◽  
Level Of Activity ◽  
Control Response ◽  
The Hill ◽  
The Relationship ◽  
Inhibition Curve ◽  
Receptor Channel

Background : In electrophysiological experiments inhibition of a receptor-channel, such as the GABAA receptor, is measured by co-applying an agonist producing a predefined control response with an inhibitor to calculate the fraction of the control response remaining in the presence of the inhibitor. The properties of the inhibitor are determined by fitting the inhibition concentration-response relationship to the Hill equation to estimate the midpoint (IC50) of the inhibition curve. Objective: We sought to estimate here the sensitivity of the fitted IC50 to the level of activity of the control response. Methods: The inhibition concentration-response relationships were calculated for models with distinct mechanisms of inhibition. In Model I, the inhibitor acts allosterically to stabilize the resting state of the receptor. In Model II, the inhibitor competes with the agonist for a shared binding site. In Model III, the inhibitor stabilizes the desensitized state. Results: The simulations indicate that the fitted IC50 of the inhibition curve is sensitive to the degree of activity of the control response. In Models I and II, the IC50 of inhibition was increased as the probability of being in the active state (PA) of the control response increased. In Model III, the IC50 of inhibition was reduced at higher PA. Conclusions: We infer that the apparent potency of an inhibitor depends on the PA of the control response. While the calculations were carried out using the activation and inhibition properties that are representative of the GABAA receptor, the principles and conclusions apply to a wide variety of receptor-channels.

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