Dynamics of a Nonlinear Parametrically-Excited Partial Differential Equation

1995 ◽  
Author(s):  
Richard H. Rand ◽  
William I. Newman ◽  
Bruce C. Denardo ◽  
Alice L. Newman
Keyword(s):  
Differential Equation ◽  
Perturbation Theory ◽  
Initial Conditions ◽  
Mathieu Equation ◽  
Resonant Mode ◽  
The Other ◽  
State Behavior ◽  
Stable Periodic ◽  
Parameter Values ◽  
Steady State Behavior

Abstract We investigate a nonlinear Mathieu equation with diffusion and damping, using both perturbation theory and numerical integration. The perturbation results predict that for parameters which lie near the 2 : 1 resonance tongue of instability corresponding to a mode shape cos nx the resonant mode achieves a stable periodic motion, while all the other modes are predicted to decay to zero. By numerically integrating the p.d.e. as well as a 3-mode o.d.e. truncation, the predictions of perturbation theory are shown to represent an oversimplified picture of the dynamics. In particular it is shown that steady states exist which involve many modes. The dependence of steady state behavior on parameter values and initial conditions is investigated numerically.

Algebra, analysis and probability for a coupled system of reaction-duffusion equations

1995 ◽  
Vol 350 (1692) ◽  
pp. 69-112 ◽  
Keyword(s):  
Differential Equation ◽  
Large Deviation ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Travelling Wave ◽  
The Other ◽  
Reaction Diffusion Equation ◽  
Local Martingale

This paper is designed to interest analysts and probabilists in the methods of the ‘other’ field applied to a problem important in biology and in other contexts. It does not strive for generality. After § 1 a , it concentrates on the simplest case of a coupled reaction-diffusion equation. It provides a complete treatment of the existence, uniqueness, and asymptotic behaviour of monotone travelling waves to various equilibria, both by differential-equation theory and by probability theory. Each approach raises interesting questions about the other. The differential-equation treatment makes new use of the maximum principle for this type of problem. It suggests a numerical method of solution which yields computer pictures which illustrate the situation very clearly. The probabilistic treatment is careful in its proofs of martingale (as opposed to merely local-martingale) properties. A new change-of-measure technique is used to obtain the best lower bound on the speed of the monotone travelling wave with Heaviside initial conditions. Waves to different equilibria are shown to be related by Doob h -transforms. Large-deviation theory provides heuristic links between alternative descriptions of minimum wave speeds, rigorous algebraic proofs of which are provided. Since the paper was submitted, an alternative method of proving existence of monotone travelling waves has been developed by Karpelevich et al. (1993). We have extended our results in different directions from theirs (one of which is hinted at in § 1 a ), and have found the methods used here well equipped for these generalizations. See the Addendum.

HOPF BIFURCATION FROM LINES OF EQUILIBRIA WITHOUT PARAMETERS IN MEMRISTOR OSCILLATORS

2010 ◽  
Vol 20 (02) ◽  
pp. 437-450 ◽  
Author(s):  
MARCELO MESSIAS ◽  
CRISTIANE NESPOLI ◽  
VANESSA A. BOTTA
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Fixed Set ◽  
Electronic Element ◽  
Stable Periodic ◽  
Parameter Values

The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical existence was postulated in 1971 by L. O. Chua, based on symmetrical and logical properties observed in some electronic circuits. On the other hand its physical realization was announced only recently in a paper published on May 2008 issue of Nature by a research team from Hewlett–Packard Company. In this work, we present the bifurcation analysis of two memristor oscillators mathematical models, given by three-dimensional five-parameter piecewise-linear and cubic systems of ordinary differential equations. We show that depending on the parameter values, the systems may present the coexistence of both infinitely many stable periodic orbits and stable equilibrium points. The periodic orbits arise from the change in local stability of equilibrium points on a line of equilibria, for a fixed set of parameter values. This phenomenon is a kind of Hopf bifurcation without parameters. We have numerical evidences that such stable periodic orbits form an invariant surface, which is an attractor of the systems solutions. The results obtained imply that even for a fixed set of parameters the two systems studied may or may not present oscillations, depending on the initial condition considered in the phase space. Moreover, when they exist, the amplitude of the oscillations also depends on the initial conditions.

scholarly journals An Efficient Fast and Convergence-Controlled Algorithm for Sidelobes Simultaneous Reduction (SSR) and Spatial Filtering

Electronics ◽  
2021 ◽  
Vol 10 (9) ◽  
pp. 1071
Author(s):  
Yasser Albagory
Keyword(s):  
Processing Time ◽  
Spatial Filtering ◽  
The Other ◽  
Angular Range ◽  
State Behavior ◽  
Simultaneous Reduction ◽  
One Dimensional ◽  
Filtering Algorithm ◽  
Spacing Distance ◽  
Steady State Behavior

In this paper, an efficient sidelobe levels (SLL) reduction and spatial filtering algorithm is proposed for linear one-dimensional arrays. In this algorithm, the sidelobes are beamspace processed simultaneously based on its orientation symmetry to achieve very deep SLL at much lower processing time compared with recent techniques and is denoted by the sidelobes simultaneous reduction (SSR) algorithm. The beamwidth increase due to SLL reduction is found to be the same as that resulting from the Dolph-Chebyshev window but at considerably lower average SLL at the same interelement spacing distance. The convergence of the proposed SSR algorithm can be controlled to guarantee the achievement of the required SLL with almost steady state behavior. On the other hand, the proposed SSR algorithm has been examined for spatial selective sidelobe filtering and has shown the capability to effectively reduce any angular range of the radiation pattern effectively. In addition, the controlled convergence capability of the proposed SSR algorithm allows it to work at any interelement spacing distance, which ranges from tenths to a few wavelength distances, and still provide very low SLL.

scholarly journals Commentary on the mutual interaction model of McCarley and Massaquoi for REM-NREM cycle

1986 ◽  
Vol 251 (6) ◽  
pp. R1030-R1032
Author(s):  
S. Daan ◽  
D. G. Beersma
Keyword(s):  
Limit Cycle ◽  
Initial Conditions ◽  
Circadian Variation ◽  
Sleep Onset ◽  
Interaction Model ◽  
The Other ◽  
Circadian Cycle ◽  
Rem Latency ◽  
Parameter Values ◽  
Cycle Dependence

McCarley and Massaquoi successfully simulated human REM-NREM cycle characteristics by extending the McCarley-Hobson model with two sets of assumptions, one creating limit cycle behavior, the other introducing two sources of circadian variation. We argue that the limit cycle assumptions, due to freedom in choosing parameter values, suffice to explain variation in REM across the night. Nonmonotonic circadian variation in REM latency requires a circadian cycle dependence only of initial conditions at sleep onset.

The Effects of Strong Cubic Nonlinearity on the Existence of Periodic Solutions of the Mathieu–Duffing Equation

2009 ◽  
Vol 76 (5) ◽  
Author(s):  
Ivana Kovacic ◽  
Livija Cveticanin
Keyword(s):  
Periodic Solutions ◽  
Initial Conditions ◽  
Mathieu Equation ◽  
Duffing Equation ◽  
Cubic Nonlinearity ◽  
Strongly Nonlinear ◽  
Numerical Integrations ◽  
Strongly Nonlinear System ◽  
Existence Of Periodic Solutions ◽  
Parameter Values

This paper deals with systems governed by the Mathieu–Duffing equation, with a time-dependent coefficient of the linear term and a constant, not necessarily small coefficient of the cubic term. This coefficient can be positive or negative. The method of strained parameters applied to a linear system governed by the Mathieu equation is extended to a strongly nonlinear system. As a result, the curves corresponding to the parameter values at which periodic solutions exist are obtained. It is shown that they strongly depend on the value of the coefficient of nonlinearity and the initial conditions. The corresponding parameter planes are plotted. Numerical integrations are carried out to confirm the analytical results.

Signature change in p-adic and noncommutative FRW cosmology

2014 ◽  
Vol 29 (27) ◽  
pp. 1450155 ◽  
Author(s):  
Goran S. Djordjevic ◽  
Ljubisa Nesic ◽  
Darko Radovancevic
Keyword(s):  
Loop Quantum Gravity ◽  
Initial Conditions ◽  
Planck Scale ◽  
The Other ◽  
Frw Cosmology ◽  
Signature Change ◽  
Discrete Nature ◽  
Frw Metric ◽  
The Universe ◽  
Friedmann Robertson Walker

The significant matter for the construction of the so-called no-boundary proposal is the assumption of signature transition, which has been a way to deal with the problem of initial conditions of the universe. On the other hand, results of Loop Quantum Gravity indicate that the signature change is related to the discrete nature of space at the Planck scale. Motivated by possibility of non-Archimedean and/or noncommutative structure of space–time at the Planck scale, in this work we consider the classical, p-adic and (spatial) noncommutative form of a cosmological model with Friedmann–Robertson–Walker (FRW) metric coupled with a self-interacting scalar field.

scholarly journals Interaction of buoyant plumes in open-channel flow

1992 ◽  
Vol 33 (4) ◽  
pp. 451-473
Author(s):  
P. J. Wicks
Keyword(s):  
Diffusion Equation ◽  
Channel Flow ◽  
Open Channel ◽  
Nonlinear Term ◽  
Initial Conditions ◽  
Hermite Polynomials ◽  
Open Channel Flow ◽  
The Other ◽  
Buoyant Plumes ◽  
Rich Variety

AbstractIn this paper, a model for lateral dispersion in open-channel flow is studied involving a diffusion equation which has a nonlinear term describing the effect of buoyancy. The model is used to investigate the interaction of two buoyant pollutant plumes. An approximate analytic technique involving Hermite polynomials is applied to the resulting PDEs to reduce them to a system of ODEs for the centroids and widths of the two plumes. The ODEs are then solved numerically. A rich variety of behaviour occurs depending on the relative positions, widths and strengths of the initial discharges. It is found that for two plumes of equal strength and width discharged side-by-side, the plumes move apart and the rate of spreading is inhibited by their interaction, whereas when one plume is initially much wider than the other, both plumes tend to drift to the side of the narrower plume. Finally, the PDEs are solved numerically for two sets of initial conditions and a comparison is made with the ODE solutions. Agreement is found to be good.

Observations Regarding Numerical Results Obtained by the Floquet-Method

2013 ◽  
Author(s):  
Till J. Kniffka ◽  
Horst Ecker
Keyword(s):  
Numerical Methods ◽  
Monodromy Matrix ◽  
Mathieu Equation ◽  
Numerical Results ◽  
The Other ◽  
Benchmark Problem ◽  
Stability Studies ◽  
Floquet Method ◽  
System Equations ◽  
Time Periodic

Stability studies of parametrically excited systems are frequently carried out by numerical methods. Especially for LTP-systems, several such methods are known and in practical use. This study investigates and compares two methods that are both based on Floquet’s theorem. As an introductary benchmark problem a 1-dof system is employed, which is basically a mechanical representation of the damped Mathieu-equation. The second problem to be studied in this contribution is a time-periodic 2-dof vibrational system. The system equations are transformed into a modal representation to facilitate the application and interpretation of the results obtained by different methods. Both numerical methods are similar in the sense that a monodromy matrix for the LTP-system is calculated numerically. However, one method uses the period of the parametric excitation as the interval for establishing that matrix. The other method is based on the period of the solution, which is not known exactly. Numerical results are computed by both methods and compared in order to work out how they can be applied efficiently.

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