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Lirui Feng

Publications and source records attributed to Lirui Feng.

7 recordsLinked to original sources
arXiv2026

Decomposition of quasi-all invariant bundles for the cocycle with a set of strongly invariant cones

In this paper, we prove that a linear cocycle with a set of strongly invariant cones admits a decomposition of quasi-all invariant bundles. We further obtain the certain location-relations between the fibres of positive invariant bundles and these cones, as well as we get the order-relations among characteristic exponents of the linear cocycle on the positive invariant bundles. The key of proofs is to analyze relations among positive invariant bundles with respect to a linear cocycle that admits both of a $k$-exponential separation and a $k^{\prime}$-exponential separation. These results are applied to a class of semilinear equations on a Hilbert space.

math.DS↗
arXiv2026

On the persistence of $k$-exponential separation of linear cocycles under a small perturbation

In this paper, the concept about a $k$-exponential separation of a linear cocycle $(\tilde{F},\mathcal{G})$ on $\tilde{K}\times X$ is extended for a general linear coclye whose base space $\tilde{K}$ and fibre-map-value map $\mathcal{G}$ become a nonempty set and a continuous map from $\tilde{K}$ to $L(X)$ respectvely, by removing the prior assumptions in the classical sense that $\tilde{K}$ is a compact set contained in the Banach space $X$, and $\mathcal{G}(x)$ is compact for any $x\in K$. We prove that $(\tilde{F},\mathcal{G})$ on $\tilde{K}\times X$ admits a $k$-exponential separation if the cocycle $(\tilde{F},\mathcal{G})$ is generated from a linear cocycle $(F,\mathcal{T})$ on $K\times X$ adimtting a $k$-exponential separation with $K$ being compact in the classical sense via a small perturbation. We also obtain some consequent results with their needed concepts spinning off from the one of a $k$-exponential separation of $(\tilde{F},\mathcal{G})$ on $\tilde{K}\times X$, as well as the unified terminology system around $k$-exponential separation is normalized. We apply our results to analyze the linearized structure near by an invariant set of a system generated from a dissipative system via a small perturbation, where the small perturbation is without the restriction of compactness.

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arXiv2024

Semiflows strongly focusing monotone with respect to high-rank cones: II. Pseudo-ordered principle

We consider a semiflow strongly focusing monotone with respect to a cone of rank k on a Banach space. We prove that the omega-limit set of a pseudo-ordered semiorbit is ordered, which is called as pseudo-ordered principle. Based on this principle, we obtain the solid Poincar\{'}e-Bendixson theorem with the rank k=2, that is, the omega-limit set of a pseudo-ordered semiorbit is either a nontrivial periodic orbit or a set consisting of equilibria with their potential connected orbits. The generic solid dynamics theorem with a general rank k and the generic solid Poincar\{'}e-Bendixson theorem with the rank k=2 are also obtained.

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arXiv2023

Semiflows strongly focusing monotone with respect to high-rank cones. I: Generic dynamics

We consider a smooth semiflow strongly focusing monotone with respect to a cone of rank k on a Banach space. We obtain its generic dynamics, that is, semiorbits with initial data from an open and dense subset of any bounded open set are either pseudo-ordered or convergent to an equilibrium. For the case k=1, it is the celebrated Hirsch's Generic Convergence Theorem. For the case k=2, we obtain the generic Poincare-Bendixson Theorem.

math.DS↗
arXiv2019

Generical behavior of flows strongly monotone with respect to high-rank cones

We consider a $C^{1,α}$ smooth flow in $\mathbb{R}^n$ which is "strongly monotone" with respect to a cone $C$ of rank $k$, a closed set that contains a linear subspace of dimension $k$ and no linear subspaces of higher dimension. We prove that orbits with initial data from an open and dense subset of the phase space are either pseudo-ordered or convergent to equilibria. This covers the celebrated Hirsch's Generic Convergence Theorem in the case $k=1$, yields a generic Poincaré-Bendixson Theorem for the case $k=2$, and holds true with arbitrary dimension $k$. Our approach involves the ergodic argument using the $k$-exponential separation and the associated $k$-Lyapunov exponent (that reduces to the first Lyapunov exponent if $k=1$).

math.DS↗
arXiv2016

Monotone semiflows with respect to high-rank cones on a Banach space

We consider semiflows in general Banach spaces motivated by monotone cyclic feedback systems or differential equations with integer-valued Lyapunov functionals. These semiflows enjoy strong monotonicity properties with respect to cones of high ranks, which imply order-related structures on the $ω$-limit sets of precompact semi-orbits. We show that for a pseudo-ordered precompact semi-orbit the $ω$-limit set $Ω$ is either ordered, or is contained in the set of equilibria, or possesses a certain ordered homoclinic property. In particular, we show that if $Ω$ contains no equilibrium, then $Ω$ itself is ordered and hence the dynamics of the semiflow on $Ω$ is topologically conjugate to a compact flow on $\mathbb{R}^k$ with $k$ being the rank. We also establish a Poincaré-Bendixson type Theorem in the case where $k=2$. All our results are established without the smoothness condition on the semiflow, allowing applications to such cellular or physiological feedback systems with piecewise linear vector fields and to such infinite dimensional systems where the $C^1$-Closing Lemma or smooth manifold theory has not been developed.

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