U. A. Rozikov, A. Zada
We consider $\ell$-Volterra quadratic stochastic operators defined on $(m-1)$-dimensional simplex, where $\ell\in\{0,1,...,m\}$. Under some conditions on coefficients of such operators we describe Lyapunov functions and apply them to obtain upper estimates for the set of $ω$- limit points of trajectories. We describe a set of fixed points of $\ell$-Volterra operators.
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Global dynamical behaviors of the competitive Lotka-Volterra system even in 3-dimension are not fully understood. The Lyapunov function can provide us such knowledge once it is constructed. In this paper, we construct explicitly the Lyapunov function in three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-dimensional case; (2) a 3-dimensional model; (3) the model of May-Leonard. The dynamics of these examples include bistable case and cyclical behavior. The first two examples are the generalized gradient system defined in the Appendixes, while th…
A Perron-Volterra Lyapunov function for mathematical epidemiology reaction networks (MERN) with non-interacting rank-one strains
针对具有递增凹发病率和标量、不相关菌株块的多菌株模型，通过构造边界和共存平衡点的显式全局Lyapunov函数，证明了参数空间的竞争排斥划分，并将构造扩展到包含一个不可约秩一感染块和多个不相关标量菌株的模型。
We introduce a notion of $\ell$-Volterra quadratic stochastic operator defined on $(m-1)$-dimensional simplex, where $\ell\in\{0,1,...,m\}$. The $\ell$-Volterra operator is a Volterra operator iff $\ell=m$. We study structure of the set of all $\ell$-Volterra operators and describe their several fixed and periodic points. For $m=2$ and 3 we describe behavior of trajectories of $(m-1)$-Volterra operators. The paper also contains many remarks with comparisons of $\ell$-Volterra operators and Volterra ones.
In the present paper, we are aiming to study limiting behavior of infinite dimensional Volterra operators. We introduce two classes $\tilde {\mathcal{V}}^+$ and $\tilde{\mathcal{V}}^-$of infinite dimensional Volterra operators. For operators taken from the introduced classes we study their omega limiting sets $ω_V$ and $ω_V^{(w)}$ with respect to $\ell^1$-norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to $\tilde {\mathca…
We present a method for determining the local stability of equilibrium points of conservative generalizations of the Lotka-Volterra equations. These generalizations incorporate both an arbitrary number of species -including odd-dimensional systems- and nonlinearities of arbitrarily high order in the interspecific interaction terms. The method combines a reformulation of the equations in terms of a Poisson structure and the construction of their Lyapunov functionals via the energy-Casimir method. These Lyapunov functionals are a generalization of those traditionally known for Lotka-Volterra sys…
A family of non-Volterra quadratic operators corresponding to permutations
本文研究一族参数化的非Volterra二次随机算子，确定其所有不动点，构造Lyapunov函数，完整描述极限点集并证明遍历性。
This paper examines the cycling behavior of a deterministic and a stochastic version of the economic interpretation of the Lotka-Volterra model, the Goodwin model. We provide a characterization of orbits in the deterministic highly non-linear model. We then study the cycling behavior for a stochastic version, where a Brownian noise is introduced via an heterogeneous productivity factor. Sufficient conditions for existence of the system are provided. We prove that the system produces cycles around an equilibrium point in finite time for general volatility levels, using stochastic Lyapunov techn…
We consider a discrete-time dynamical system generated by a nonlinear operator (with four real parameters $a,b,c,d$) of ocean ecosystem. We find conditions on the parameters under which the operator is reduced to a $\ell$-Volterra quadratic stochastic operator mapping two-dimensional simplex to itself. We show that if $cd(c+d)=0$ then (under some conditions on $a,b$) this $\ell$-Volterra operator may have up to three or a countable set of fixed points; if $cd(c+d)\ne 0$ then the operator has up to three fixed points. Depending on the parameters the fixed points may be attracting, repelling or …
In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model is obtained. Furthermore, by means of the almost periodic functional hull theory on time scales and Lyapunov functional, some criteria are obtained for the existence, uniqueness and global attractivity of almost periodic solutions of the model. Our results complement and extend some scientific work in recent years. Finally, an example is given to illustrate t…
Optimal Harvesting of a Stochastic Lotka-Volterra Competition Model with Periodic Coefficients
本文研究了具有周期系数的随机Lotka-Volterra竞争模型的最优捕获问题，通过建立充分条件证明了模型的稳定性，并给出了最优捕获策略和最大可持续产量的显式表达式。
Results for Global Attractivity of Interior Equilibrium Points for Lotka-Volterra Systems
本文研究了Lotka-Volterra系统内部平衡点的全局吸引性，提出了一般性结果并验证了三维案例的吸引性。
A practical approach to computing Lyapunov exponents of renewal and delay equations
提出一种实用方法，通过将延迟方程转化为抽象微分方程并约简为常微分方程组，结合离散QR方法计算更新方程及耦合系统的Lyapunov指数，实验验证有效并提供MATLAB实现。
Solution space characterisation of perturbed linear discrete and continuous stochastic Volterra convolution equations: the $\ell^p$ and $L^p$ cases
本文研究扰动线性随机Volterra方程的解空间特征，探讨$p$-可和性与可积性条件，并在对角噪声下分析解的收敛性。
Lyapunov exponents of renewal equations: numerical approximation and convergence analysis
针对更新方程的李雅普诺夫指数计算问题，提出基于离散QR技术、傅里叶投影与伪谱配点的数值方法，证明其收敛性并提供MATLAB实现，解决了Volterra型延迟方程相关指数的高效计算难题。
Existence and asymptotic stability of a generic Lotka-Volterra system with nonlinear spatially heterogeneous cross-diffusion
本文针对具有空间异质捕食函数和趋化系数的Lotka-Volterra系统，引入加权范数并推广不等式，结合能量估计和bootstrap方法证明了解的存在性，并利用推广的LaSalle原理分析了常数平衡点的渐近稳定性。
Consider a $C^1$ vector field together with an ergodic invariant probability that has $\ell$ nonzero Lyapunov exponents. Using orthonormal moving frames along certain transitive orbits we construct a linear system of $\ell$ differential equations which is a reduced form of Liao's "standard system". We show that the Lyapunov exponents of this linear system coincide with all the nonzero exponents of the given vector field with respect to the given probability. Moreover, we prove that these Lyapunov exponents have a persistence property that implies that a "Liao perturbation" preserves both sign …
Analytical solutions of one class of Volterra quadratic stochastic operators of a two-sex population with continuous time
本文用同伦分析法求解连续时间两性群体Volterra二次随机算子，证明状态空间正不变性，给出ω-极限集条件、周期性与级数收敛性，并获显式解与高精度近似。
In this work, we give a new proof of the classification of the Lotka-Volterra and Reversible foliations, originally given by Gautier. This new proof, involves an unified technique for both cases, using the theory of foliations. In addition, we obtain a linear family of elliptical foliations with a non-invariant tangency set.
Synchronisation in two-dimensional damped-driven Navier-Stokes turbulence: insights from data assimilation and Lyapunov analysis
研究二维阻尼驱动NS湍流中通过数据同化和李雅普诺夫分析确定的等效长度尺度，发现其接近强迫尺度而非耗散尺度，并探讨其与三维情况的差异来源。
Lyapunov instability in KAM stable Hamiltonians with two degrees of freedom
本文证明存在任意接近可积KAM非退化解析哈密顿函数的Gevrey光滑哈密顿函数，其具有频率向量满足$\omega_1\omega_2<0$的李雅普诺夫不稳定但KAM稳定的椭圆平衡点，并给出近可积情形下的类似例子。
Extinction of multiple populations and a team of Die-out Lyapunov functions
本研究通过构造“消亡”李雅普诺夫函数组，判定微分方程模型中多个物种必然指数灭绝的条件，并给出广义Lotka-Volterra系统的营养级条件以确保解有界。
Consider a family of K3 surfaces over a hyperbolic curve (i.e. Riemann surface). Their second cohomology groups form a local system, and we show that its top Lyapunov exponent is a rational number. One proof uses the Kuga-Satake construction, which reduces the question to Hodge structures of weight 1. A second proof uses integration by parts. The case of maximal Lyapunov exponent corresponds to modular families, given by the Kummer construction on a product of isogenous elliptic curves.
Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels
本文研究了广义朗之万方程中记忆核近似对轨迹预测误差的影响，通过理论分析与数值验证，揭示了核估计精度与轨迹预测精度之间的定量关系。
Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids
本文通过结合概率论、动力系统、数值分析、优化和控制理论的方法，将确定性延拓算法扩展到随机微分方程的亚稳态平衡点跟踪，利用椭球定义样本路径的高集中区域，并通过迭代方法高效计算椭球及其距离。
The purpose of this paper is to present an example of an Ordinary Differential Equation $x'=F(x)$ in the infinite-dimensional Hilbert space $\ell^2$ with $F$ being of class $\mathcal{C}^1$ in the Fréchet sense, such that the origin is an asymptotically stable equilibrium point but the spectrum of the linearized operator $DF(0)$ intersects the half-plane $\Re(z)>0$. The possible existence or not of an example of this kind has been an open question until now, to our knowledge. An analogous example, but of a non-invertible map instead of a flow defined by an ODE was recently constructed by the au…
We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov- Schmidt reduction procedure. This leads to the study of a singularity which inherits certain structure from the Hamiltonian nature of the system. Under non-degeneracy assumptions, we classify the possible Morse indices of this singularity, permitting a local description of the set of homoclinic orbits. We also consider the case of time-reversible Hamiltonian systems.
Dense Phenomena for Ergodic Schrödinger Operators: I. Spectrum, Integrated Density of States, and Lyapunov Exponent
本文证明对紧致度量空间上同胚的遍历轨道采样得到的薛定谔算子，存在稠密采样函数集使谱间隙有限、态密度积分光滑且李雅普诺夫指数光滑为正，并肯定回答Walters的非一致上循环存在性问题。
Moment stability and large deviations for random dynamical systems on non-compact manifolds
本文研究非紧流形上随机动力系统的矩Lyapunov指数与大偏差，通过增长条件将紧情形的特征值刻画推广到非紧情形，并给出中心极限定理和中偏差估计。
This paper provides a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed of infinitely many finite-dimensional systems. We model these networks on infinite-dimensional $\ell_{\infty}$-type spaces. A crucial assumption in our results is that the internal Lyapunov gains, modeling the influence of the subsystems on each other, are linear functions. Moreover, the gain operator built from the internal gains is assumed to be subadditive and homogeneous, which covers both max-type and sum-type formulations for the ISS Lyapunov functions of the subsystems. As a c…
Data-driven discovery of polynomial ODEs with provably bounded solutions
本文提出SILAS框架，通过非凸优化发现多项式微分方程并证明其解有界，利用多项式Lyapunov函数定义紧致吸收集，验证方法适用于更广泛的微分方程类。
Contraction theory: Hausdorff--Riemann Measures as Set-Based Lyapunov Functions
本文提出基于Hausdorff-Riemann测度的集基Lyapunov函数理论，用于自主系统稳定性分析和反馈设计。
We prove the Hölder continuity of the integrated density of states for a class of quasi-periodic long-range operators on $\ell^2(\Z^d)$ with large trigonometric polynomial potentials and Diophantine frequencies. Moreover, we give the Hölder exponent in terms of the cardinality of the level sets of the potentials, which improves, in the perturbative regime, the result obtained by Goldstein and Schlag \cite{gs2}. Our approach is a combination of Aubry duality, generalized Thouless formula and the regularity of the Lyapunov exponents of analytic quasi-periodic $GL(m,\C)$ cocycles which is proved …
We show that the complex-valued ODE \begin{equation*} \dot z_t = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0, \end{equation*} which necessarily has trajectories along which the dynamics blows up in finite time, can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term. We also show that the stochastic perturbation has a unique invariant measure which is heavy-tailed yet is uniformly, exponentially attracting. The methods turn on the construction of Lyapunov functions. The techniques used in the construction are general and can likely be used in other settin…
In this paper we give a characterization of Avila's quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove $β$-Hölder continuity of the integrated density of states for supercritical quasi-periodic Schrödinger operators restricted to the $\ell$-th stratum, for any $β<(2(\ell-1))^{-1}$ and $\ell\ge2$. We establish Anderson localization for all Diophantine frequencies for the operator with even analytic potential function on the first supercritical stratum, which has positive measure if it is nonempty.
Positivity and Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators
针对多频斜移势的一维离散薛定谔算子，证明了在Diophantine频率和大耦合下李雅普诺夫指数一致正且关于能量log-Hölder连续，将结果从单频或简单拟周期模型推广到多频斜移情形。
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