Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities

We analyze the dynamics of the forced singularly perturbed differential equation of Duffing's type. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small parameter at the highest derivative. We give some generalizations of results obtained recently by B.S. Wu, W.P. Sun and C.W. Lim, Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations, Jou...

In this paper we first prove the so-called large twist theorem, then using it to prove the boundedness of all solutions and the existence of quasi-periodic solutions for Duffing's equation $$ \ddot{x}+x^{2n+1}+\dsum_{i=0}^{2n}p_i(t)x^i=0, $$ where $p_i(t)\in C^1(\mathbb{S}) (n+1\leq i\leq 2n)$ and $p_i(t)\in C^0(\mathbb{S}) (0\leq i\leq n)$ with $\mathbb{S}=\mathbb{R}/\mathbb{Z}$.

In the work a nonlinear Duffing oscillator is considered under impulse excitation with two ways of introduction of the random additive term simulating noise, - with help of amplitude modulation and modulation of period of impulses sequence. The scaling properties both in the Feigenbaum scenario and in the tricritical case are shown.

We present analytical and numerical results on the joint dynamics of two coupled Duffing oscillators with nonlinearity of opposite signs (hardening and softening). In particular, we focus on the existence and stability of synchronized oscillations where the frequency is independent of the amplitude. In this regime, the amplitude--frequency interdependence (a--f effect) ---a noxious consequence of nonlinearity, which jeopardizes the use of micromechanical oscillators in the de...

We analyse different types of nonlinear resonances in a weakly damped Duffing oscillator using bifurcation theory techniques. In addition to (i) odd subharmonic resonances found on the primary branch of symmetric periodic solutions with the forcing frequency and (ii) even subharmonic resonances due to symmetry-broken periodic solutions that bifurcate off the primary branch and also oscillate at the forcing frequency, we uncover (iii) novel resonance type due to isolas of peri...

In this paper, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method we find explict periodic waves and we also present a characterization for all positive and periodic solutions for the model using the monotonicity of the period map in terms of the energy levels. The monotonicity of the period map is also useful to obtain the quantity and multiplicity of non-positive eigenvalues for t...

In this paper we have investigated the driven cubic-quintic Duffing equation. Using the Melnikov analysis technique, we are able to predict the number of limit cycles around the equilibrium and to develop a theoretical approach to chaos suppression in damped driven systems. The analytical solution and the bifurcation of limit cycles of the investigated system have been studied., We discussed our nonlinear dynamics, namely, the cubic quintic Duffing equation in the absence of ...

A simple non-autonomous scalar differential equation with delay, exponential decay, nonlinear negative feedback and a periodic multiplicative coefficient is considered. It is shown that stable slowly oscillating periodic solutions with the period of the feedback coefficient, and also with the double period of the feedback coefficient exist. The periodic solutions are built explicitly in the case of piecewise constant feedback function and the periodic coefficient. The periodi...

In this paper we are concerned with the existence of invariant curves of planar twist mappings which are almost periodic in a spatial variable. As an application of this result to differential equations we will discuss the existence of almost periodic solutions and the boundedness of all solutions for superlinear Duffing's equation with an almost periodic external force.

We consider arbitrary one-parameter cubic deformations of the Duffing oscillator $x"=x-x^3$. In the case when the first Melnikov function $M_1$ vanishes, but $M_2\neq 0$ we compute the general form of $M_2$ and study its zeros in a suitable complex domain.