### An inertial manifold and the principle of spatial averaging.

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In this paper, we prove and discuss averaging results for ordinary differential equations perturbed by a small parameter. The conditions we assume on the right-hand sides of the equations under which our averaging results are stated are more general than those considered in the literature. Indeed, often it is assumed that the right-hand sides of the equations are uniformly bounded and a Lipschitz condition is imposed on them. Sometimes this last condition is relaxed to the uniform continuity in...

In this paper, we are interested in the asymptotical behavior of the error between the solution of a differential equation perturbed by a flow (or by a transformation) and the solution of the associated averaged differential equation. The main part of this redaction is devoted to the ascertainment of results of convergence in distribution analogous to those obtained in [10] and [11]. As in [11], we shall use a representation by a suspension flow over a dynamical system. Here, we make an assumption...

The modified generalized Van der Pol-Mathieu equation is generalization of the equation that is investigated by authors Momeni et al. (2007), Veerman and Verhulst (2009) and Kadeřábek (2012). In this article the Bautin bifurcation of the autonomous system associated with the modified generalized Van der Pol-Mathieu equation has been proved. The existence of limit cycles is studied and the Lyapunov quantities of the autonomous system associated with the modified Van der Pol-Mathieu equation are computed....

New oscillation criteria are given for the second order sublinear differential equation $${\left[a\left(t\right)\psi \left(x\left(t\right)\right){x}^{\text{'}}\left(t\right)\right]}^{\text{'}}+q\left(t\right)f\left(x\left(t\right)\right)=0,\phantom{\rule{1.0em}{0ex}}t\ge {t}_{0}>0,$$ where $a\in {C}^{1}\left([{t}_{0},\infty )\right)$ is a nonnegative function, $\psi ,f\in C\left(\mathbb{R}\right)$ 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 $\mathbb{R}\setminus \left\{0\right\}$ 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 ${x}^{\text{'}\text{'}}\left(t\right)+q\left(t\right)x\left(t\right)=0$.

Digestion in the small intestine is the result of complex mechanical and biological phenomena which can be modelled at different scales. In a previous article, we introduced a system of ordinary differential equations for describing the transport and degradation-absorption processes during the digestion. The present article sustains this simplified model by showing that it can be seen as a macroscopic version of more realistic models including biological phenomena at lower scales. In other words,...

The basic idea of this paper is to give the existence theorem and the method of averaging for the system of functional-differential inclusions of the form ⎧$\u1e8b\left(t\right)\in F(t,{x}_{t},{y}_{t})$ (0) ⎨ ⎩$\u1e8b\left(t\right)\in G(t,{x}_{t},{y}_{t})$ (1)

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 $\frac{\mathrm{d}{X}^{\epsilon}\left(t\right)}{\mathrm{d}t}=\epsilon B({X}^{\epsilon}\left(t\right)$, $\mathit{\Xi}\left({q}_{1}\left(t\right)\right),\mathit{\Xi}\left({q}_{2}\left(t\right)\right),...,\mathit{\Xi}\left({q}_{\ell}\left(t\right)\right))$ where $\mathit{\Xi}\left(t\right)$, $t\ge 0$ 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.