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We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities. We obtain sharp bounds for h such that the equation has exactly three ordered T-periodic solutions. Moreover, when h is…
We explore stability and instability of rapidly oscillating solutions $x(t)$ for the hard spring delayed Duffing oscillator $$x''(t)+ ax(t)+bx(t-T)+x^3(t)=0.$$ Fix $T>0$. We target periodic solutions $x_n(t)$ of small minimal periods…
In this work we investigate the dynamics of the nonlinear DDE (delay-differential equation) x''(t)+x(t-T)+x(t)^3=0 where T is the delay. For T=0 this system is conservative and exhibits no limit cycles. For T>0, no matter how small, an…
A class of modified Duffing oscillator differential equations, having nonlinear damping forces, are shown to have finite time dynamics, i.e., the solutions oscillate with only a finite number of cycles, and, thereafter, the motion is zero.…
We study the asymptotic behavior of the solutions of the time-delayed higher-order dispersive nonlinear differential equation \begin{equation*} u_t(x,t)+Au(x,t) +\lambda_0(x) u(x,t)+\lambda(x) u(x,t-\tau )=0 \end{equation*} where…
We consider a second-order equation of Duffing type. Bounds for the derivative of the restoring force are given which ensure the existence and uniqueness of a periodic solution. Furthermore, the unique periodic solution is asymptotically…
We study bounded, unbounded and blow-up solutions of a delay logistic equation without assuming the dominance of the instantaneous feedback. It is shown that there can exist an exponential (thus unbounded) solution for the nonlinear…
We investigate the scalar autonomous equation with two discrete delays $$ \dot{x}(t)=f(x(t),x(t-r),x(t-\sigma)), $$ where $f:\mathbb{R}^3\rightarrow \mathbb{R}$ is a continuously differentiable non-linear function such that $f(0,0,0)=0$. It…
We study the set of $T$-periodic solutions of a class of $T$-periodically perturbed Differential-Algebraic Equations, allowing the perturbation to contain a distributed and possibly infinite delay. Under suitable assumptions, the perturbed…
For ordinary differential equations and functional differential equations the following result is well known. Suppose any solution is bounded on the half-line for each bounded on the half-line right-hand side. Then under certain conditions…
For the delay differential equations $$ \ddot{x}(t) +a(t)\dot{x}(g(t))+b(t)x(h(t))=0, g(t)\leq t, h(t)\leq t, $$ and $$ \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)+a_1(t)\dot{x}(g(t))+b_1(t)x(h(t))=0 $$ explicit exponential stability conditions…
In this paper we will construct a continuous positive periodic function $p(t)$ such that the corresponding superlinear Duffing equation $$ x"+a(x)\,x^{2n+1}+p(t)\,x^{2m+1}=0,\ \ \ \ n+2\leq 2m+1<2n+1 $$ possesses a solution which escapes to…
This paper explores the exponential stability of two nonlinear wave equations coupled through their velocities. The analysis is divided into two main cases. First, we consider a system where one equation is damped, while the other…
In this paper, we prove the existence of two positive $T$-periodic solutions of an electrostatic actuator modeled by the time-delayed Duffing equation $$\ddot{x}(t)+f_{D}(x(t),\dot{x}(t))+ x(t)=1- \dfrac{e…
In this paper, we consider the second-order equations of Duffing type. Bounds for the derivative of the restoring force are given that ensure the existence and uniqueness of a periodic solution. Furthermore, the stability of the unique…
In this paper, by using the spectral theory of functions and properties of evolution semigroups, we establish conditions on the existence, and uniqueness of asymptotic 1-periodic solutions to a class of abstract differential equations with…
In this paper we prove that the following delay differential equation \[ \frac{d}{dt}x(t)=rx(t)\left(1-\int_{0}^{1}x(t-s)ds\right), \] has a periodic solution of period two for $r>\frac{\pi^{2}}{2}$ (when the steady state, $x=1$, is…
Suppose any solution of a linear impulsive delay differential equation $$ \dot{x} (t) + \sum_{i=1}^m A_i (t) x[h_i (t)] = 0,~t \geq 0, x(s) = 0, s < 0, $$ $$ x(\tau_j +0) = B_j x(\tau_j -0) + \alpha_j, ~j=1,2, ... ,$$ is bounded for any…
This manuscript deals with the stability and bifurcation analysis of the equation $D^{2\alpha}x(t)+c D^{\alpha}x(t)=a x(t)+b x(t-\tau)$, where $0<\alpha<1$ and $\tau>0$. We sketch the boundaries of various stability regions in the parameter…
For a nonlinear equation with several variable delays $$ \dot{x}(t)=\sum_{k=1}^m f_k(t, x(h_1(t)),\dots,x(h_l(t)))-g(t,x(t)), $$ where the functions $f_k$ increase in some variables and decrease in the others, we obtain conditions when a…