The paper deals with the existence of periodic solutions for a kind of non-autonomous time-delay Rayleigh equation. With the continuation theorem of the coincidence degree and a priori estimates, some new results on the existence of periodic solutions for this kind of Rayleigh equation are established.

We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form ẋ = A(t)x + f(t) (*), with f having precompact range, which is then applied to find new spectral criteria for the existence of almost periodic solutions with specific spectral properties in the resonant case where $\overline{e^{isp\left(f\right)}}$ may intersect the spectrum of the monodromy operator P of (*) (here sp(f) denotes the Carleman spectrum of f). We show that if (*) has a bounded...

In this paper we consider cubic polynomial systems of the form: x' = y + P(x, y), y' = −x + Q(x, y), where P and Q are polynomials of degree 3 without linear part. If M(x, y) is an integrating factor of the system, we propose its reciprocal V (x, y) = 1 / M(x,y) as a linear function of certain coefficients of the system. We find in this way several new sets of sufficient conditions for a center. The resulting integrating factors are of Darboux type and the first integrals are in the Liouville form.By...

We study the existence of nodal solutions of the $m$-point boundary value problem $$u^{\text{'}\text{'}}+f\left(u\right)=0,0<t<1,u^{\text{'}}\left(0\right)=0,u\left(1\right)=\sum _{i=1}^{m-2}{\alpha }_{i}u\left({\eta }_i\right)$$ where ${\eta }_i\in \mathbb{Q}$$(i=1,2,\cdots ...$

Let $\gamma$ be an integral solution of an analytic real vector field $\xi$ defined in a neighbordhood of $0\in {ℝ}^3$. Suppose that $\gamma$ has a single limit point, $\omega \left(\gamma \right)=\left\{0\right\}$. We say that $\gamma$ is non oscillating if, for any analytic surface $H$, either $\gamma$ is contained in $H$ or $\gamma$ cuts $H$ only finitely many times. In this paper we give a sufficient condition for $\gamma$ to be non oscillating. It is established in terms of the existence of “generalized iterated tangents”, i.e. the existence of a single limit point for any transform property for...

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force $g(x,t)$. We assume that $g(x,t)$ is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if $g(x,t)$ is a quasiperiodic function with respect to $t$, then the attractor is a continuous image of a torus. Moreover the...

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force g(x,t). We assume that g(x,t) is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if g(x,t) is a quasiperiodic function with respect to t, then the attractor is a continuous image...

We consider “nonconventional” averaging setup in the form $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B(X^{\epsilon }\left(t\right)$, $𝛯\left(q_1\left(t\right)\right),𝛯\left(q_2\left(t\right)\right),...,𝛯\left(q_{\ell }\left(t\right)\right))$ where $𝛯\left(t\right)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_j\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}lt;{\alpha }_{2}lt;\cdots lt;{\alpha }_k$ and $q_j$, $j=k+1,...,\ell$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

In this paper we consider complex differential systems in the plane, which are linearizable in the neighborhood of a nondegenerate centre. We find necessary and sufficient conditions for linearizability for the class of complex quadratic systems and for the class of complex cubic systems symmetric with respect to a centre. The sufficiency of these conditions is shown by exhibiting explicitly a linearizing change of coordinates, either of Darboux type or a generalization of it.