Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation

2001 ◽  
Author(s):  
Randolph S. Zounes ◽  
Richard H. Rand
Keyword(s):  
Perturbation Theory ◽  
Nonlinear Term ◽  
Mathieu Equation ◽  
Elliptic Functions ◽  
Action Space ◽  
Global Behavior ◽  
Trigonometric Functions ◽  
Lie Transform ◽  
Analytic Expressions ◽  
Subharmonic Resonances

Abstract We investigate the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation,(1)x..+[δ+ϵ(cos⁡ω1t+cos⁡ω2t)]x+αx3=0. We assume that ϵ ≪ 1 and that the coefficient of the nonlinear term, α, is positive but not necessarily small. We utilize Lie transform perturbation theory with elliptic functions — rather than the usual trigonometric functions — to study subharmonic resonances associated with orbits in 2m : 1 resonance with a respective driver. In particular, we derive analytic expressions that place conditions on (δ, ϵ, ω1, ω2) at which subharmonic resonance bands in a Poincaré section of action space begin to overlap. These results are used in combination with Chirikov’s overlap criterion to obtain an overview of the O(ϵ) global behavior of equation (1) as a function of δ and ω2 with ω1, α, and ϵ fixed.

scholarly journals The chord-length distribution of a polyhedron

2020 ◽  
Vol 76 (4) ◽  
pp. 474-488
Author(s):  
Salvino Ciccariello
Keyword(s):  
Distribution Function ◽  
Length Distribution ◽  
Chord Length ◽  
Square Root ◽  
Trigonometric Functions ◽  
Chord Length Distribution ◽  
Analytic Expressions ◽  
Analytic Expression ◽  
Positive Integers ◽  
Inverse Trigonometric Functions

The chord-length distribution function [γ′′(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ′′(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ1], Δ1 being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ′′(r) can only show singularities of the forms |r − δ|−n and |r − δ|−m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

scholarly journals On solutions of the combined KdV-nKdV equation

2019 ◽  
Vol 30 (2) ◽  
pp. 33
Author(s):  
Mohammed Allami ◽  
A. K. Mutashar ◽  
A. S. Rashid
Keyword(s):  
Vries Equation ◽  
Special Functions ◽  
Elliptic Functions ◽  
Rational Functions ◽  
Trigonometric Functions ◽  
Exponential Functions ◽  
Jacobian Elliptic Functions ◽  
Negative Order ◽  
De Vries ◽  
Wave Reduction

The aim of this work is to deal with a new integrable nonlinear equation of wave propagation, the combined of the Korteweg-de vries equation and the negative order Korteweg-de vries equation (combined KdV-nKdV) equation, which was more recently proposed by Wazwaz. Upon using wave reduction variable, it turns out that the reduced combined KdV-nKdV equation is alike the reduced (3+1)-dimensional Jimbo Miwa (JM) equation, the reduced (3+1)-dimensional Potential Yu-Toda-Sasa-Fukuyama (PYTSF) equation and the reduced (3 + 1)¬dimensional generalized shallow water (GSW) equation in the trav¬elling wave. In fact, the four transformed equations belong to the same class of ordinary differential equation. With the benefit of a well known general solutions for the reduced equation, we show that sub¬jects to some scaling and change of parameters, a variety of families of solutions are constructed for the combined KdV-nKdV equation which can be expressed in terms of rational functions, exponential functions and periodic solutions of trigonometric functions and hyperbolic func¬tions. In addition to that the equation admits solitary waves, and double periodic waves in terms of special functions such as Jacobian elliptic functions and Weierstrass elliptic functions.

scholarly journals A PERTURBATIVE APPROACH TO NONLINEARITIES IN THE INFORMATION CARRIED BY A TWO LAYER NEURAL NETWORK

2001 ◽  
Vol 15 (03) ◽  
pp. 281-295 ◽  
Author(s):  
E. KORUTCHEVA ◽  
V. DEL PRETE ◽  
J.-P. NADAL
Keyword(s):  
Neural Network ◽  
Perturbation Theory ◽  
Mutual Information ◽  
Nonlinear Term ◽  
Exact Expression ◽  
Neural Systems ◽  
Neuronal Dynamics ◽  
Perturbative Approach ◽  
First Order ◽  
Nonlinear Analogue

We evaluate the mutual information between the input and the output of a two layer network in the case of a noisy and nonlinear analogue channel. In the case where the nonlinearity is small with respect to the variability in the noise, we derive an exact expression for the contribution to the mutual information given by the nonlinear term in first order of perturbation theory. Finally we show how the calculation can be simplified by means of a diagrammatic expansion. Our results suggest that the use of perturbation theories applied to neural systems might give an insight on the contribution of nonlinearities to the information transmission and in general to the neuronal dynamics.

Oscillation centres and mode coupling in non-uniform Vlasov plasma

1979 ◽  
Vol 22 (1) ◽  
pp. 105-119 ◽  
Author(s):  
Shayne Johnston ◽  
Allan N. Kaufman
Keyword(s):  
Perturbation Theory ◽  
Single Particle ◽  
Mode Coupling ◽  
Poisson Brackets ◽  
Lie Transform ◽  
Time Integral ◽  
Compact Expression ◽  
Hamiltonian Perturbation Theory ◽  
Insight Into ◽  
Vlasov Plasma

The general coupling coefficient for three electromagnetic linear modes of an inhomogeneous and relativistic plasma is derived from the oscillation-centre viewpoint. A concise and manifestly symmetric formula is obtained; it is cast in terms of Poisson brackets of the single-particle perturbation Hamiltonian and its convective time-integral along unperturbed orbits. The simplicity of the compact expression obtained is shown to lead to a new insight into the essence of three-wave coupling and of the Manley–Rowe relations governing such interactions. Thus, the interaction Hamiltonian of the three waves is identified as simply the trilinear contribution to the single-particle (new) Hamiltonian, summed over all non-resonant particles. The relation between this work and the Lie-transform approach to Hamiltonian perturbation theory is discussed.

Refined Dynamic Stability Theory of Laminated Isotropic Circular Plates

1998 ◽  
Vol 65 (2) ◽  
pp. 334-340 ◽  
Author(s):  
G. Krizhevsky ◽  
Y. Stavsky
Keyword(s):  
Variational Principle ◽  
Dynamic Stability ◽  
Transverse Shear ◽  
Rotational Inertia ◽  
Trigonometric Functions ◽  
Circular Plates ◽  
Analytic Expressions ◽  
Transversally Isotropic ◽  
Instability Regions ◽  
The Galerkin Method

Hamilton’s variational principle is used for the derivation of transversally isotropic laminated circular plates motion. Nonlinear strain-displacements relations are considered. Linearized dynamic stability equations are obtained for circular plates subjected to the same uniformly distributed periodic radial loads. The effects of transverse shear and rotational inertia are included. The exact solutions of vibrations and buckling problems are given initially in the terms of Bessel, power, and trigonometric functions. The vibrational modal functions are used then as a basis in the Galerkin method that reduced the study of dynamic stability to investigation of bounds of instability of Mathieu’s equations. Analytic expressions for the bounds of both principal and combination-type instability regions are obtained using the methods of Bolotin and Tamir. A new effect—sensitivity of certain instability regions to slight imperfections in the symmetry of lamination—is found out and discussed here.

scholarly journals Approximation of Elliptic Functions by Means of Trigonometric Functions with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Alvaro H. Salas ◽  
Lorenzo J. H. Martinez ◽  
David L. R. Ocampo R.
Keyword(s):  
Elliptic Function ◽  
General Principle ◽  
Nonlinear Problems ◽  
Elliptic Functions ◽  
Trigonometric Functions ◽  
Elementary Functions ◽  
Nonlinear Odes ◽  
Weierstrass Elliptic Function ◽  
Weierstrass Functions ◽  
Approximate Expressions

In this work, we give approximate expressions for Jacobian and elliptic Weierstrass functions and their inverses by means of the elementary trigonometric functions, sine and cosine. Results are reasonably accurate. We show the way the obtained results may be applied to solve nonlinear ODEs and other problems arising in nonlinear physics. The importance of the results in this work consists on giving easy and accurate way to evaluate the main elliptic functions cn, sn, and dn, as well as the Weierstrass elliptic function and their inverses. A general principle for solving some nonlinear problems through elementary functions is stated. No similar approach has been found in the existing literature.

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.

scholarly journals A PAIR OF PERFECTLY CONDUCTING DISKS IN AN EXTERNAL FIELD

2015 ◽  
Vol 20 (2) ◽  
pp. 273-288 ◽  
Author(s):  
Natalia Rylko
Keyword(s):  
Closed Form Solution ◽  
Elliptic Functions ◽  
Convergent Series ◽  
Fiber Composites ◽  
Form Solution ◽  
Local Fields ◽  
Trigonometric Functions ◽  
Method Of Images ◽  
Asymptotic Investigation ◽  
Asymptotic Formulae

A pair of non-overlapping perfectly conducting equal disks embedded in a two-dimensional background was investigated by the classic method of images, by Poincar´e series, by use of the bipolar coordinates and by the elliptic functions in the previous works. In particular, successive application of the inversions with respect to circles were applied to obtain the field in the form of a series. For closely placed disks, the previous methods yield slowly convergent series. In this paper, we study the local fields around closely placed disks by the elliptic functions. The problem of small gap is completely investigated since the obtained closed form solution admits a precise asymptotic investigation in terms of the trigonometric functions when the gap between the disks tends to zero. The exact and asymptotic formulae are extended to the case when a prescribed singularity is located in the gap. This extends applications of structural approximations to estimations of the local fields in densely packed fiber composites in various external fields.

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