Liming Wang, Eduardo D. Sontag
This paper deals with global convergence to equilibria, and in particular Hirsch's generic convergence theorem for strongly monotone systems, for singular perturbations of monotone systems.
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We develop a theory which allows making qualitative conclusions about the dynamics of both monotone and non-monotone Moreau sweeping processes. Specifically, we first prove that any sweeping processes with almost periodic monotone right-hand-sides admits a globally exponentially stable almost periodic solution. And then we describe the extent to which such a globally stable solution persists under non-monotone perturbations.
We focus on the presence of almost automorphy in strongly monotone skew-product semiflows on Banach spaces. Under the $C^1$-smoothness assumption, it is shown that any linearly stable minimal set must be almost automorphic. This extends the celebrated result of Shen and Yi [Mem. Amer. Math. Soc. 136(1998), No. 647] for the classical $C^{1,α}$-smooth systems. Based on this, one can reduce the regularity of the almost periodically forced differential equations and obtain the almost automorphic phenomena in a wider range.
Almost Periodic Solutions of Lattice Dynamical Systems with Monotone Nonlinearity
本文研究了具有单调非线性的格点动力系统的几乎周期解存在性，证明在单调非线性条件下系统存在唯一几乎周期解。
In this paper, we study the Poisson stability (in particular, stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations (ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above …
We investigate the singularly perturbed monotone systems with respect to cones of rank $2$ and obtain the so called Generic Poincaré-Bendixson theorem for such perturbed systems, that is, for a bounded positively invariant set, there exists an open and dense subset $\mathcal{P}$ such that for each $z\in \mathcal{P}$, the $ω$-limit set $ω(z)$ that contains no equilibrium points is a closed orbit.
In this paper we study the problem of almost periodicity of solutions for dissipative differential equations (Bronshtein's conjecture). We give a positive answer to this conjecture for monotone almost periodic systems of differential/difference equations.
In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense…
Intraspecific and monotone enzyme catalysis with oscillatory substrate and inhibitor supplies
本文研究底物与抑制剂供给呈概周期变化的开放竞争性抑制酶催化系统，证明了相应非自治动力系统概周期解的全局稳定性，为种内单调开放反应网络的全局稳定性研究提供首个案例。
Strong averaging principle for nonautonomous multi-scale SPDEs with fully local monotone and almost periodic coefficients
针对带完全局部单调概周期系数的非自治多尺度SPDE，引入测度演化系统证明慢分量的强平均原理，拓展既有框架并放宽系数条件，可应用于多类非线性随机偏微分方程。
In this paper, we establish the second Bogolyubov theorem and global averaging principle for stochastic partial differential equations (in short, SPDEs) with monotone coefficients. Firstly, we prove that there exists a unique $L^{2}$-bounded solution to SPDEs with monotone coefficients and this bounded solution is globally asymptotically stable in square-mean sense. Then we show that the $L^{2}$-bounded solution possesses the same recurrent properties (e.g. periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, Levitan almost periodic, etc.) in distribution sense as…
The exponential ordering is exploited in the context of non-auto\-no\-mous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost perio…
Stochastic functional partial differential equations with monotone coefficients: Poisson stability measures, exponential mixing and limit theorems
针对单调系数随机泛函偏微分方程，本文利用耗散性与Shcherbakov比较法证明入口测度的存在唯一性及回复性继承，并建立极限点的一致指数混合、极限定理与收敛速率。
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