We formulate nonuniform nonresonance criteria for certain third order differential systems of the form $X^{{}^{\text{'}\text{'}\text{'}}}+A{X}^{{}^{\text{'}\text{'}}}+G(t,X^{{}^{\text{'}}})+CX=P\left(t\right)$, which further improves upon our recent results in [12], given under sharp nonresonance considerations. The work also provides extensions and generalisations to the results of Ezeilo and Omari [5], and Minhós [9] from the scalar to the vector situations.

An abstract theory on general synchronization of a system of several oscillators coupled by a medium is given. By generalized synchronization we mean the existence of an invariant manifold that allows a reduction in dimension. The case of a concrete system modeling the dynamics of a chemical solution on two containers connected to a third container is studied from the basics to arbitrary perturbations. Conditions under which synchronization occurs are given. Our theoretical results are complemented...

Symmetric piecewise linear bi-dimensional systems are very common in control engineering. They constitute a class of non-differentiable vector fields for which classical Hopf bifurcation theorems are not applicable. For such systems, sufficient and necessary conditions for bifurcation of a limit cycle from the periodic orbit at infinity are given.

In this paper the issue of impulsive stabilization and synchronization of uncertain financial hyperchaotic systems with parameters perturbation is investigated. Applying the impulsive control theory, some less conservative and easily verified criteria for the stabilization and synchronization of financial hyperchaotic systems are derived. The control gains and impulsive intervals can be variable. Moreover, the boundaries of the stable region are also estimated according to the equidistant impulse...

New oscillation criteria are given for the second order sublinear differential equation $${\left[a\left(t\right)\psi \left(x\left(t\right)\right)x^{\text{'}}\left(t\right)\right]}^{\text{'}}+q\left(t\right)f\left(x\left(t\right)\right)=0,t\ge t_0>0,$$ where $a\in C^1\left([t_0,\infty )\right)$ is a nonnegative function, $\psi ,f\in C\left(ℝ\right)$ with $\psi \left(x\right)\ne 0$, $xf\left(x\right)/\psi \left(x\right)>0$ for $x\ne 0$, $\psi$, $f$ have continuous derivative on $ℝ\setminus \left\{0\right\}$ with ${[f\left(x\right)/\psi \left(x\right)]}^{\text{'}}\ge 0$ for $x\ne 0$ and $q\in C\left([t_0,\infty )\right)$ has no restriction on its sign. This oscillation criteria involve integral averages of the coefficients $q$ and $a$ and extend known oscillation criteria for the equation $x^{\text{'}\text{'}}\left(t\right)+q\left(t\right)x\left(t\right)=0$.

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ${\left(a\left(t\right){\phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+b\left(t\right){\phi }_p\left(x^{\text{'}}\right)+c\left(t\right){\phi }_p\left(x\right)=0$, where ${\phi }_p\left(x\right)={\left|x\right|}^{p-1}\mathrm{sgn}x$ for $x\in ℝ$ and $p>1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are...

The response of an oscillator to a small amplitude periodic excitation is discussed. In particular, sufficient conditions are formulated for the perturbed oscillator to have an invariant torus in the phase cylinder. When such an invariant torus exists, some perturbed solutions are in the basin of attraction of this torus and are thus entrained to the dynamical behavior of the perturbed system on the torus. In particular, the perturbed solutions in the basin of attraction of the invariant torus are...