scholarly journals Harmonic and non-periodic solutions of velocity-dependent conservative equations

2021 ◽  
Author(s):  
YEHOSSOU Régis ◽  
ADJAI K. K. Damien ◽  
AKANDE Jean ◽  
MONSIA Marc Delphin
Keyword(s):  
Helmholtz Equation ◽  
Periodic Solutions ◽  
Nonlinear Oscillator ◽  
Quadratic Damping

Abstract We study in this paper a quadratic damping Helmholtz equation presumed to be velocity-dependent conservative nonlinear oscillator. We show that under the usual conditions of existence of particular and exact harmonic solutions, the equation can also exhibit exact and general non-periodic solutions. We show finally the existence of exact and explicit general harmonic and isochronous solutions without requiring that the system Hamiltonian must be identically zero.

TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY

2012 ◽  
Vol 22 (03) ◽  
pp. 1250060 ◽  
Author(s):  
J. C. JI ◽  
X. Y. LI ◽  
Z. LUO ◽  
N. ZHANG
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Solutions ◽  
Nonlinear Oscillator ◽  
Critical Time ◽  
Zero Solution ◽  
Delayed Feedback Control ◽  
Hopf Bifurcations ◽  
Centre Manifold ◽  
Normal Form Theory

The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf–Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present paper studies the dynamics of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf–Hopf bifurcations. The ratio of the frequencies of two Hopf bifurcations is numerically found to be nearly equal to two. The two resonant Hopf bifurcations can generate two respective periodic solutions. Consequently, the centre manifold corresponding to these two solutions is determined by a set of four first-order differential equations under two-to-one internal resonances. It is shown that the amplitudes of the two bifurcating periodic solutions admit the trivial solution and two-mode solutions for the averaged equations on the centre manifolds. Correspondingly, the cumulative behavior of the original nonlinear oscillator exhibits the initial equilibrium and a quasi-periodic motion having two frequencies. Illustrative examples are given to show the unstable zero solution, stable zero solution, and stable two-mode solution of the nonlinear oscillator under the two-to-one resonant Hopf–Hopf interactions.

Calculation of Analytic Approximations to the Periodic Solutions of a “Truly Nonlinear” Oscillator Equation

2005 ◽  
Author(s):  
Ronald E. Mickens
Keyword(s):  
Periodic Solutions ◽  
Homotopy Type ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Fourier Coefficients ◽  
Analytic Approximations ◽  
Method Of Harmonic Balance ◽  
Iteration Technique

We consider a “truly nonlinear oscillator” (TNLO) equation x¨ + x3 = 0 and calculate analytic approximations to its periodic solutions by the method of harmonic balance, a homotopy-type procedure, and an iteration technique. The corresponding angular frequencies and ratio of the first two Fourier coefficients are calculated and compared.

scholarly journals Higher-Order Approximate Periodic Solutions of a Nonlinear Oscillator with Discontinuity by Variational Approach

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
M. Orhan Kaya ◽  
S. Altay Demirbağ ◽  
F. Özen Zengin
Keyword(s):  
Approximate Solution ◽  
Periodic Solutions ◽  
Relative Error ◽  
Variational Approach ◽  
Nonlinear Oscillator ◽  
Higher Order ◽  
Second Order ◽  
Elastic Force ◽  
Force Term ◽  
The Third

He's variational approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). Three levels of approximation have been used. We obtained 1.6% relative error for the first approximate period, 0.3% relative error for the second-order approximate period. The third approximate solution has the accuracy as high as 0.1%.

A fast Galerkin method to obtain the periodic solutions of a nonlinear oscillator

1997 ◽  
Vol 86 (2-3) ◽  
pp. 261-282 ◽  
Author(s):  
Juan Peña Miralles ◽  
Pedro José Jiménez Olivo ◽  
Damián Ginestar Peiró ◽  
Gumersindo Verdú Martín ◽  
JoséLuis Muñoz-Cobo González
Keyword(s):  
Galerkin Method ◽  
Periodic Solutions ◽  
Nonlinear Oscillator

On Asymptotic Approximations of First Integrals for Weakly Nonlinear Oscillators

2001 ◽  
Author(s):  
S. B. Waluya ◽  
W. T. van Horssen
Keyword(s):  
Perturbation Method ◽  
Periodic Solutions ◽  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
First Integrals ◽  
Asymptotic Approximations ◽  
Weakly Nonlinear ◽  
Time Periodic Solutions ◽  
Time Periodic

Abstract In this paper a nonlinear oscillator problem will be studied. It will be shown that the recently developed perturbation method based on integrating vectors can be used to approximate first in tegrals and periodic solutions. The existence, uniqueness, and stability of time-periodic solutions are obtained by using the approximations for the first integrals.

On periodic solutions of the equation of a nonlinear oscillator with pulse influence

1999 ◽  
Vol 51 (6) ◽  
pp. 926-933 ◽  
Author(s):  
A. M. Samoilenko ◽  
V. G. Samoilenko ◽  
V. V. Sobchuk
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Oscillator ◽  
Pulse Influence

Free Oscillations of a Nonlinear Oscillator with an Exponential Non-Viscous Damping

2015 ◽  
Vol 801 ◽  
pp. 38-42 ◽  
Author(s):  
Remus Daniel Ene ◽  
Vasile Marinca ◽  
Bogdan Marinca
Keyword(s):  
Nonlinear Equation ◽  
Perturbation Method ◽  
Periodic Solutions ◽  
Nonlinear Oscillator ◽  
Homotopy Perturbation Method ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Viscous Damping ◽  
Free Oscillations ◽  
Homotopy Perturbation

This paper deals with the nonlinear oscillations of an exponential non-viscous damping oscillator. An analytic technique, namely Optimal Homotopy Perturbation Method (OHPM) is employed to propose an analytic approach to solve nonlinear oscillations. Our procedure proved to very effective and accurate and did not require a small or large parameters in the nonlinear equation or in the initial conditions. An excellent agreement of the approximate frequencies and periodic solutions with the numerical ones has been demonstrated.

scholarly journals Nonlinear Dynamics of the Quadratic-Damping Helmholtz Oscillator

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
R. Fangnon ◽  
C. Ainamon ◽  
A. V. Monwanou ◽  
C. H. Miwadinou ◽  
J. B. Chabi Orou
Keyword(s):  
Helmholtz Equation ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Harmonic Balance Method ◽  
Melnikov Method ◽  
Multiple Scale ◽  
Conditions Of Stability ◽  
Complex Phenomena ◽  
Quadratic Damping ◽  
Predator System

In this paper, the Helmholtz equation with quadratic damping themes is used for modeling the dynamics of a simple prey-predator system also called a simple Lotka–Volterra system. From the Helmholtz equation with quadratic damping themes obtained after modeling, the equilibrium points have been found, and their stability has been analyzed. Subsequently, the harmonic oscillations have been studied by the harmonic balance method, and the phenomena of resonance and hysteresis are observed. The primary and secondary resonances have been researched by the multiple-scale method, and the conditions of stability of the amplitudes of oscillations are determined. Chaos is detected analytically by the Melnikov method and numerically using the basin of attraction, the bifurcation diagram, the Lyapunov exponent, the phase portrait, and the Poincaré section. The effects of all the parameters of the system are analyzed in detail, and special emphasis is placed on the new parameters. Through this analysis, the complex phenomena such as hysteresis, bistability, amplitude jump, resonances, and chaos have been obtained. The control of the parameters and the necessary conditions to control the aforementioned phenomena have been found.

A Modified Lindstedt–Poincaré Method for Strongly Mixed-Parity Nonlinear Oscillators

2006 ◽  
Vol 2 (2) ◽  
pp. 141-145 ◽  
Author(s):  
W. P. Sun ◽  
B. S. Wu ◽  
C. W. Lim
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
Optimal Value

By introducing linear and constant terms with an undetermined parameter and subsequently using certain rules to determine the optimal value of the parameter, we establish analytical approximate frequencies and the corresponding periodic solutions for strongly mixed-parity nonlinear oscillators. A quadratic–cubic nonlinear oscillator is used to verify and illustrate the usefulness and effectiveness of the proposed method.

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