Averaging Theorems for Ordinary Differential Equations and Retarded Functional Differential Equations

Versal unfoldings for linear retarded functional differential equations

We consider parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds, and address the question of versality of the resulting parametrized family of linear ordinary differential equations. A sufficient criterion for versality is given in terms of readily computable quantities. In the case where the unfolding is not versal, we show how to construct a perturbation of the original linear RFDE (in terms of delay differential operators) whose finite-dimensional projection generates a versal unfolding. We illustrate the t…

Averaging Principle on Infinite Intervals for Stochastic Ordinary Differential Equations

In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solu…

On a class of differential-algebraic equations with infinite delay

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 equations are equivalent to Retarded Functional (Ordinary) Differential Equations on a manifold. Our study is based on known results about the latter class of equations.

Formulation of the normal forms of Turing-Hopf bifurcation in reaction-diffusion systems with time delay

The normal forms up to the third order for a Hopf-steady state bifurcation of a general system of partial functional differential equations (PFDEs) is derived based on the center manifold and normal form theory of PFDEs. This is a codimension-two degenerate bifurcation with the characteristic equation having a pair of simple purely imaginary roots and a simple zero root, and the corresponding eigenfunctions may be spatially inhomogeneous. The PFDEs are reduced to a three-dimensional system of ordinary differential equations and precise dynamics near bifurcation point can be revealed by two unf…

具有阻尼非共振扰动的振荡系统中的渐近状态

Asymptotic regimes in oscillatory systems with damped non-resonant perturbations

本文研究平面上非线性振荡系统在满足非共振条件且随时间衰减的扰动下的长期渐近行为，结合平均化方法与Lyapunov函数构造，证明受扰系统既可保持极限系统行为，也可能产生新的渐近稳定状态。

An Efficient Numerical Framework for Fiber Spinning Scenarios with Evaporation Effects in Airflows

In many spinning processes, as for example in dry spinning, solvent evaporates out of the spun jets and leads to thinning and solidification of the produced fibers. Such production processes are significantly driven by the interaction of the fibers with the surrounding airflow. Faced with industrial applications producing up to several hundred fibers simultaneously, the direct numerical simulation of the three-dimensional multiphase, multiscale problem is computationally extremely demanding and thus in general not possible. In this paper, we hence propose a dimensionally reduced, efficiently e…