Oscillation of Second-Order Nonlinear Delay Dynamic Equations on Time Scales

In this work, we use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation (p(t)(xΔ(t))γ)Δ+q(t)f(x(τ(t)))=0, on a time scale , where γ is the quotient of odd positive integers and p(t) and q(t) are positive right-dense continuous (rd-continuous) functions on 𝕋. Our results improve and extend some results established by Sun et al. 2009. Also our results unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our main results

Oscillation of Second-Order Nonlinear Delay Dynamic Equations with Damping on Time Scales

We use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scale T(r(t)g(x(t), xΔ(t)))Δ+p(t)g(x(t), xΔ(t))  + q(t)f(x(τ(t)))=0, where r(t), p(t), and q(t) are positive right dense continuous (rd-continuous) functions on T. Our results improve and extend some results established by Zhang et al., 2011. Also, our results unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping. Finally, we give some examples to illustrate our main results

Interval Oscillation Criteria for Forced Second-Order Nonlinear Delay Dynamic Equations with Damping and Oscillatory Potential on Time Scales

We are concerned with the interval oscillation of general type of forced second-order nonlinear dynamic equation with oscillatory potential of the form rtg1xt,xΔtΔ+p(t)g2(x(t),xΔ(t))xΔ(t)+q(t)f(x(τ(t)))=e(t), on a time scale T. We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced second-order nonlinear delay differential equation and the forced second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our results

Qualitative analysis of dynamic equations on time scales

In this article, we establish the Picard-Lindelof theorem and approximating results for dynamic equations on time scale. We present a simple proof for the existence and uniqueness of the solution. The proof is produced by using convergence and Weierstrass M-test. Furthermore, we show that the Lispchitz condition is not necessary for uniqueness. The existence of epsilon-approximate solution is established under suitable assumptions. Moreover, we study the approximate solution of the dynamic equation with delay by studying the solution of the corresponding dynamic equation with piecewise constant argument. We show that the exponential stability is preserved in such approximations.Comment: 13 page

Effects of noise on the internal resonance of a nonlinear oscillator

We numerically analyze the response to noise of a system formed by two coupled mechanical oscillators, one of them having Duffing and van der Pol nonlinearities, and being excited by a self-sustaining force proportional to its own velocity. This system models the internal resonance of two oscillation modes in a vibrating solid beam clamped at both ends. In applications to nano- and micromechanical devices, clamped-clamped beams are subjected to relatively large thermal and electronic noise, so that characterizing the fluctuations induced by these effects is an issue of both scientific and technological interest. We pay particular attention to the action of stochastic forces on the stability of internal-resonance motion, showing that resonant oscillations become more robust than other forms of periodic motion as the quality factor of the resonant mode increases. The dependence on other model parameters - in particular, on the coupling strength between the two oscillators - is also assessed.Fil: Zanette, Damian Horacio. Comisión Nacional de Energía Atómica. Gerencia del Area de Investigación y Aplicaciones No Nucleares. Gerencia de Física (Centro Atómico Bariloche); Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Halanay type inequalities on time scales with applications

This paper aims to introduce Halanay type inequalities on time scales. By means of these inequalities we derive new global stability conditions for nonlinear dynamic equations on time scales. Giving several examples we show that beside generalization and extension to q-difference case, our results also provide improvements for the existing theory regarding differential and difference inequalites, which are the most important particular cases of dynamic inequalities on time scales

Lyapunov functions for linear nonautonomous dynamical equations on time scales

The existence of a Lyapunov function is established following a method of Yoshizawa for the uniform exponential asymptotic stability of the zero solution of a nonautonomous linear dynamical equation on a time scale with uniformly bounded graininess

Oscillation theorems for certain third order nonlinear delay dynamic equations on time scales

In this paper, we establish some new oscillation criteria for the third order nonlinear delay dynamic equations $$\left(b(t)\left([a(t)(x^\Delta(t))^{\alpha_1}]^\Delta\right)^{\alpha_2}\right)^\Delta+q(t)x^{\alpha_3}(\tau(t))=0$$ on a time scale $\mathbb{T}$, where $\alpha_i$ are ratios of positive odd integers, $i=1,\ 2,\ 3,$ $b,\ a$ and $q$ are positive real-valued rd-continuous functions defined on $\mathbb{T}$, and the so-called delay function $\tau:\mathbb{T}\rightarrow \mathbb{T}$ is a strictly increasing function such that $\tau(t)\leq t$ for $t\in \mathbb{T}$ and $\tau(t)\rightarrow\infty$ as $t\rightarrow\infty.$ By using the Riccati transformation technique and integral averaging technique, some new sufficient conditions which insure that every solution oscillates or tends to zero are established. Our results are new for third order nonlinear delay dynamic equations and complement the results established by Yu and Wang in J. Comput. Appl. Math., 2009, and Erbe, Peterson and Saker in J. Comput. Appl. Math., 2005. Some examples are given here to illustrate our main results