Isochronous and Unexpected Behavior for Complex-Valued Nonlinear Oscillators with Parametric Excitation ​

Usually oscillators with periodic excitations show a periodic motion with frequency equal to the forcing one. A complex-valued nonlinear oscillator under parametric excitation is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four differential equations for two nonlinearly coupled oscillators are derived. Approximate solutions are obtained and their stability is discussed. We found that the resulting motion is periodic with a frequency equal to the forcing one, if appropriate inequalities are satisfiedand then for a large parameter range. The system is isochronous because periodic solutions are possible in a well defined phase region and not only for certain discrete values. Moreover we demonstrate that if we insert a gyroscopic term the motion can be always periodic for a well defined parameter range but with a frequency different from the forcing frequency.Analytic approximate solutions are checked by numerical integration.

On a Peculiar Attractor for Weakly Nonlinear Oscillators with a Two Period Quasiperiodic Forcing

We study a very peculiar nonlinear oscillator with an external two period quasiperiodic excitation, being the golden mean the ratio between the two frequencies. The two period quasi periodic forcing configuration gets an infinite frequencies number. As a consequence, we find the motion settles down in a two period quasi periodic atttractor for a wide excitation amplitude range. The competition between the two frequencies does not produce a closed curve but fills a well defined phase space region in the Poincarè section. This attractor somehow resembles strange nonchaotic attractors because both are characterized by quasiperiodic forcing. Using a suitable perturbation method, we can understand the new attractor most important characteristics and find an approximate solution for its dynamical behavior. Numerical simulations are used to check out the analytical investigation.

Fractal Oscillators

We consider a weakly nonlinear oscillator with a fractal forcing, given by the Weierstrass function, and use the asymptotic perturnation (AP) method to study its behavior. Being this function nowhere differentiable we can only use adequate approximations. We find that while in the linear case the resulting motion is a simple superposition between the fractal forcing and the standard oscillation, on the contrary in the nonlinear case the oscillator phase and its frequency also become fractal. We obtain the Poincarè sections in various cases and all theoretical findings are corroborated with numerical simulation

Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations

A new algorithm is presented to find exact traveling wave solutions of differential-difference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the equations might admit polynomial solutions in tanh. Examples illustrate the key steps of the algorithm. Parallels are drawn through discussion and example to the tanh-method for partial differential equations. The new algorithm is implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute traveling wave solutions of nonlinear polynomial differential-difference equations. Use of the package, implementation issues, scope, and limitations of the software are addressed.Comment: 19 pages submitted to Computer Physics Communications. The software can be downloaded at http://www.mines.edu/fs_home/wherema

Structure of New Solitary Solutions for The Schwarzian Korteweg De Vries Equation And (2+1)-Ablowitz-Kaup-Newell-Segur Equation

In this research, we introduce and represent the modified Khater method on two basic models in the optical fiber. These two models describe the dynamics of the wave movement in the optical fiber.  It is a new modification of new recent method which developed by Mostafa M. A. Khater in 2017. We implement this new modified technique on Schwarzian Korteweg de Vries equation and (2+1)-Ablowitz-Kaup-Newell-Segur equation. This modification of Khater method produces more closed solutions than many other methods. Schwarzian Korteweg de Vries (SKdV) equation has a closed relationship with (2+1)-Ablowitz- Kaup-Newell-Segur equation. Schwarzian Korteweg de Vries equation prescribes the location in a micro-segment of space and motion of the isolated waves in varied fields which localized in a tiny portion of space. It is a great and basic system in fluid mechanics, nonlinear optics, plasma physics, and quantum field theory

Additive resonances of a controlled van der Pol-Duffing oscillator

The trivial equilibrium of a controlled van der Pol-Duffing oscillator with nonlinear feedback control may lose its stability via a non-resonant interaction of two Hopf bifurcations when two critical time delays corresponding to two Hopf bifurcations have the same value. Such an interaction results in a non-resonant bifurcation of co-dimension two. In the vicinity of the non-resonant Hopf bifurcations, the presence of a periodic excitation in the controlled oscillator can induce three types of additive resonances in the forced response, when the frequency of the external excitation and the frequencies of the two Hopf bifurcations satisfy a certain relationship. With the aid of centre manifold theorem and the method of multiple scales, three types of additive resonance responses of the controlled system are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. The amplitudes of the free-oscillation terms are found to admit three solutions; two non-trivial solutions and the trivial solution. Of two non-trivial solutions, one is stable and the trivial solution is unstable. A stable non-trivial solution corresponds to a quasi-periodic motion of the original system. It is also found that the frequency-response curves for three cases of additive resonances are an isolated closed curve. It is shown that the forced response of the oscillator may exhibit quasi-periodic motions on a three-dimensional torus consisting of three frequencies; the frequencies of two bifurcating solutions and the frequency of the excitation. Illustrative examples are given to show the quasi-periodic motions. © 2008 Elsevier Ltd. All rights reserved

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation. © 2009 Elsevier Ltd. All rights reserved