arXiv subjects

Jinxiang Yao

Publications and source records attributed to Jinxiang Yao.

7 recordsLinked to original sources

arXiv2024

Global Dynamics of Granular Media Equations via Stochastic Order

This paper studies the rich dynamics of one-dimensional granular media equations with attractive quadratic interactions. Building on the monotone dynamical systems framework developed in an earlier work, we allow for multiplicative noise, in contrast to most existing results restricted to additive noise. Within this framework, we show that, in the one-dimensional setting, invariant measures are totally ordered with respect to the stochastic order. The basins of attraction of the minimal and maximal invariant measures contain unbounded open sets in the 2-Wasserstein space, which is vacant in previous research even for additive noises. Also, our main results address the global convergence to the order interval enclosed by the minimal and maximal invariant measures, and an alternating arrangement of invariant measures in terms of stability (locally attracting) and instability (as the backward limit of a connecting orbit). Our theorems cover a wide range of classical granular media equations, such as double-well and multi-well landscapes. Specific values for the parameter ranges, explicit descriptions of attracting sets and phase diagrams are provided.

math.PR↗

arXiv2024

Connecting Orbits in Cooperative McKean-Vlasov SDEs

In this work we extend the framework of monotone dynamical systems to a broad and important class of stochastic equations, namely cooperative McKean-Vlasov SDEs with multiplicative noise. Under a locally dissipative assumption, our main theorem establishes the existence of multiple order-related invariant measures in the the Wasserstein space together with monotone connecting orbits (heteroclinic orbits) between them, with respect to the stochastic order. The presence of such connecting orbits also reveals the unstable nature of those invariant measures appearing as their backward limits, a dynamical feature that has remained largely unexplored in stochastic equations. The framework applies to a wide range of classical models, including granular media equations in double-well and multi-well confining potentials with quadratic interaction, perturbed double-well landscapes, and interacting multi-species population models. Our method is based on building a monotone dynamical system that preserves the stochastic order, achieved through a cone compatible with this order and an extension of the classical Dancer-Hess connecting orbit theorem.

math.PR↗

arXiv2022

Nonexistence of observable chaos and its robustness in strongly monotone dynamical systems

For strongly monotone dynamical systems on a Banach space, we show that the largest Lyapunov exponent $λ_{\max}>0$ holds on a shy set in the measure-theoretic sense. This exhibits that strongly monotone dynamical systems admit no observable chaos, the notion of which was formulated by L.S. Young. We further show that such phenomenon of no observable chaos is robust under the $C^1$-perturbation of the systems.

math.DS↗

arXiv2022

Prevalent behavior and almost sure Poincare-Bendixson Theorem for smooth flows with invariant k-cones

We investigate the global dynamics from a measure-theoretic perspective for smooth flows with invariant cones of rank k. For such systems, it is shown that prevalent (or equivalently, almost all) orbits will be pseudo-ordered or convergent to equilibria. This reduces to Hirsch's prevalent convergence Theorem if the rank k=1; and implies an almost-sure Poincare-Bendixson Theorem for the case k=2. These results are then applied to obtain an almost sure Poincare-Bendixson theorem for high-dimensional differential equations.

math.DS↗

arXiv2021

Sharpened dynamics alternative and its $C^1$-robustness for strongly monotone discrete dynamical systems

For strongly monotone dynamical systems, the dynamics alternative for smooth discrete-time systems turns out to be a perfect analogy of the celebrated Hirsch's limit-set dichotomy for continuous-time semiflows. In this paper, we first present a sharpened dynamics alternative for $C^1$-smooth strongly monotone discrete-time dissipative system $\{F_0^n\}_{n\in \mathbb{N}}$ (with an attractor $A$), which concludes that there is a positive integer $m$ such that any orbit is either manifestly unstable; or asymptotic to a linearly stable cycle whose minimal period is bounded by $m$. Furthermore, we show the $C^1$-robustness of the sharpened dynamics alternative, that is, for any $C^1$-perturbed system $\{F_ε^n\}_{n\in \mathbb{N}}$ ($F_ε$ not necessarily monotone), any orbit initiated nearby $A$ will admit the sharpened dynamics alternative with the same $m$. The improved generic convergence to cycles for the $C^1$-system $\{F_0^n\}_{n\in \mathbb{N}}$, as well as for the perturbed system $\{F_ε^n\}_{n\in \mathbb{N}}$, is thus obtained as by-products of the sharpened dynamics alternative and its $C^1$-robustness. The results are applied to nonlocal $C^1$-perturbations of a time-periodic parabolic equations and give typical convergence to periodic solutions whose minimal periods are uniformly bounded.

math.DS↗

arXiv2021

Prevalent Behavior of Smooth Strongly Monotone Discrete-Time Dynamical Systems

For C1-smooth strongly monotone discrete-time dynamical systems, it is shown that ``convergence to linearly stable cycles" is a prevalent asymptotic behavior in the measuretheoretic sense. The results are then applied to classes of time-periodic parabolic equations and give new results on prevalence of convergence to periodic solutions. In particular, for equations with Neumann boundary conditions on convex domains, we show the prevalence of the set of initial conditions corresponding to the solutions that converge to spatiallyhomogeneous periodic solutions. While, for equations on radially symmetric domains, we obtain the prevalence of the set of initial values corresponding to solutions that are asymptotic to radially symmetric periodic solutions.

math.DS↗

arXiv2021

Almost automorphy of minimal sets for $C^1$-smooth strongly monotone skew-product semiflows on Banach spaces

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.

math.DS↗