E. Kaslik, A. M. Balint, A. Grigis, St. Balint
The purpose of this paper is to show that if the spectral radius of the matrix of the linearized system of a nonlinear discrete dynamical system is less then one then the characterization theorem from \cite{KBBB1,KBGB2} holds.
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In this paper, we show that the domain of attraction of a compact asymptotically stable submanifold of a finite-dimensional smooth manifold of an autonomous system is homeomorphic to its tubular neighborhood. The compactness of the attractor is crucial, without which this result is false; two counterexamples are provided to demonstrate this.
We characterize and compute the maximal admissible positively invariant set for asymptotically stable constrained switching linear systems. Motivated by practical problems found, e.g., in obstacle avoidance, power electronics and nonlinear switching systems, in our setting the constraint set is formed by a finite number of polynomial inequalities. First, we observe that the so-called Veronese lifting allows to represent the constraint set as a polyhedral set. Next, by exploiting the fact that the lifted system dynamics remains linear, we establish a method based on reachability computations to…
In this paper a first order analytical system of difference equations is considered. For an asymptotically stable fixed point x0 of the system a gradual approximation of the domain of attraction DA is presented in the case when the matrix of the linearized system in x0 is a contraction. This technique is based on the gradual extension of the "embryo" of an analytic function of several variables. The analytic function is a Lyapunov function whose natural domain of analyticity is the DA and which satisfies an iterative functional equation. The equation permits to establish an "embryo" of the Lya…
Let $G$ be a discrete infinite amenable group, which acts from the left on a compact metric space $X$. In this paper, we study the chaotic dynamics exhibited inside and near a minimal center of attraction of $(G,X)$ relative to any Følner net in $G$.
In this paper we propose a convex programming based method for computing robust regions of attraction for state-constrained perturbed discrete-time polynomial systems. The robust region of attraction of interest is a set of states such that every possible trajectory initialized in it will approach an equilibrium state while never violating the specified state constraint, regardless of the actual perturbation. Based on a Bellman equation which characterizes the interior of the maximal robust region of attraction as the strict one sub-level set of its unique bounded and continuous solution, we c…
A method for determination and two methods for approximation of the domain of attraction $D_{a}(0)$ of an asymptotically stable steady state of an autonomous, $\mathbb{R}$-analytical, discrete system is presented. The method of determination is based on the construction of a Lyapunov function $V$, whose domain of analyticity is $D_{a}(0)$. The first method of approximation uses a sequence of Lyapunov functions $V_{p}$, which converges to the Lyapunov function $V$ on $D_{a}(0)$. Each $V_{p}$ defines an estimate $N_{p}$ of $D_{a}(0)$. For any $x\in D_{a}(0)$ there exists an estimate $N_{p^{x}}$ …
This paper considers the problem of approximating the "maximal" region of attraction (the set that contains all asymptotically stable sets) of any given set of locally exponentially stable nonlinear Ordinary Differential Equations (ODEs) with a sufficiently smooth vector field. Given a locally exponential stable ODE with a differentiable vector field, we show that there exists a globally Lipschitz continuous converse Lyapunov function whose 1-sublevel set is equal to the maximal region of attraction of the ODE. We then propose a sequence of d-degree Sum-of-Squares (SOS) programming problems th…
An exterior space is a topological space provided with a quasi-filter of open subsets (closed by finite intersections). In this work, we analyze some relations between the notion of an exterior space and the notion of a discrete semi-flow. On the one hand, for an exterior space, one can consider limits, bar-limits and different sets of end points (Steenrod, \v Cech, Brown-Grossman). On the other hand, for a discrete semi-flow one can analyze fixed points, periodic points, omega-limits, et cetera. In this paper, we see that a discrete semi-flow can be provided with the canonical structure of an…
On topological properties of compact attractors on Hausdorff spaces
本文研究了在Hausdorff空间中紧吸引子作为其吸引域强变形抵消的条件。
In this paper, an $\mathbb{R}$-analytical function and the sequence of its Taylor polynomials (which are Lyapunov functions different from those of Vanelli & Vidyasagar (1985, Automatica, 21(1):6 9--80)) is presented, in order to determine and approximate the domain of attraction of the exponentially asymptotically stable zero steady state of an autonomous, $\mathbb{R}$-analytical system of differential equations. The analytical function and the sequence of its Taylor polynomials are constructed by recurrence formulae using the coefficients of the power series expansion of $f$ at 0.
Continuous-Time and Discrete-Time Quasilinear Systems with Asymptotically Unpredictable Solutions
该研究提出渐近不可预测序列概念，利用压缩映射原理证明带时滞拟线性系统渐近不可预测解的存在唯一性，同时研究离散时间系统的对应轨道并给出示例。
This paper considers discrete-time switched systems under dwell-time switching and in the presence of saturation nonlinearity. Based on Multiple Lyapunov Functions and using polytopic representation of nested saturation functions, a sufficient condition for asymptotic stability of such systems is derived. It is shown that this condition is equivalent to linear matrix inequalities (LMIs) and as a result, the estimation of domain of attraction is formulated into a convex optimization problem with LMI constraints. Through numerical examples, it is shown that the proposed approach is less conserva…
The main objective of this paper is to show that two asymptotically stable steady states which belong to an analytic path of asymptotically stable steady states can be gradually transferred one to the other by successive changes of the control parameters.
Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimation
本文提出了一种基于集合的特征化和可验证神经网络估计方法，用于准确估计离散时间非线性不确定系统的安全和鲁棒吸引域。
On the basin of attraction of a critical three-cycle of a model for the secant map
该研究针对割线法对应的离散动力系统的临界3周期轨道，构建模型映射刻画其动力学行为，揭示了吸引域边界的几何拓扑性质与参数的关联，补充了全局动力学研究结论。
In this paper an autonomous analytical system of ordinary differential equations is considered. For an asymptotically stable steady state x0 of the system a gradual approximation of the domain of attraction DA is presented in the case when the matrix of the linearized system in x0 is diagonalizable. This technique is based on the gradual extension of the "embryo" of an analytic function of several complex variables. The analytic function is the transformed of a Lyapunov function whose natural domain of analyticity is the DA and which satisfies a linear non-homogeneous partial differential equa…
We study the discrete dynamical system defined on a subset of $R^2$ given by the iterates of the secant method applied to a real polynomial $p$. Each simple real root $α$ of $p$ has associated its basin of attraction $\mathcal A(α)$ formed by the set of points converging towards the fixed point $(α,α)$ of $S$. We denote by $\mathcal A^*(α)$ its immediate basin of attraction, that is, the connected component of $\mathcal A(α)$ which contains $(α,α)$. We focus on some topological properties of $\mathcal A^*(α)$, when $α$ is an internal real root of $p$. More precisely, we show the existence of a…
Lyapunov functions for Morse-Smale synchronisation diffeomorphisms
针对三对称耦合振荡器的相位差系统，构造离散Lyapunov函数证明T^2上两个Huygens同步不动点渐近稳定，且其吸引域并集闭包覆盖全环面。
Global linearization of asymptotically stable systems without hyperbolicity
本文推广Hartman-Grobman定理至非双曲渐近稳定平衡点，证明全局线性化共轭的完整性与光滑性，并揭示5维情形与4维庞加莱猜想的等价性。
We consider a homoclinic orbit to a saddle fixed point of an arbitrary $C^\infty$ map $f$ on $\mathbb{R}^2$ and study the phenomenon that $f$ has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory, this requires $f$ to have a homoclinic tangency. We show it also necessary for $f$ to satisfy a `global resonance' condition and for the eigenvalues associated with the fixed point, $λ$ and $σ$, to satisfy $|λσ| = 1$. The phenomenon is codimension-three in the case $λσ= -1$, but codimension-four in the case $λσ= 1$ because here the coefficients of the…
Learning Regions of Attraction in Unknown Dynamical Systems via Zubov-Koopman Lifting: Regularities and Convergence
本文提出一种基于 Zubov-Koopman 算子提升的方法，通过在无限维函数空间中学习算子来间接逼近 Zubov 方程的解，从而精确估计未知非线性动力系统的吸引区域，并证明了该方法的正则性、强收敛性及数据效率。
We present a fully automated method that identifies attractors and their basins of attraction without approximations of the dynamics. The method works by defining a finite state machine on top of the system flow. The input to the method is a dynamical system evolution rule and a grid that partitions the state space. No prior knowledge of the number, location, or nature of the attractors is required. The method works for arbitrarily-high-dimensional dynamical systems, both discrete and continuous. It also works for stroboscopic maps, Poincaré maps, and projections of high-dimensional dynamics t…
Calculating Domain of Attraction Boundary of Power Systems Based on the Gentlest Ascent Dynamics
本文基于最温和上升动力学方法、伴随算子方法和稳定流形算法，计算电力系统吸引域边界，验证了该方法在暂态稳定分析中的有效性和准确性。
In this article, we study the global dynamics of a discrete two dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except a Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that th…
Global stability of the Lengyel-Epstein systems
针对Lengyel-Epstein微分系统，证明其唯一平衡点无全局渐近稳定性，给出其渐近稳定的显式条件并研究吸引盆，同时分析Hopf分支及存在两个极限环的参数子集。
Asymptotically large free semigroups in Zariski dense discrete subgroups of Lie groups
本文证明 Zariski 稠密离散子群中存在临界指数任意接近原群的自由有限生成子半群，且这些子半群具有 Zariski 稠密性和 P-Anosov 性，从而说明李群离散子半群临界指数无间隙现象，并推出临界指数在 Chabauty 拓扑下下半连续。
We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized by a circle under assumptions involving the topological properties of the asymptotic stable bundles.
A Lyapunov function for a Synchronisation diffeomorphism of three clocks
针对三个直线排列的相同极限环时钟的Huygens同步微分同胚系统，通过构造离散Lyapunov函数重新证明其在环面T2上存在唯一渐近稳定不动点，且吸引域闭包为全环面，验证了该同步行为的普适性与鲁棒性。
A discrete dynamical system in Euclidean m-space generated by the iterates of an asymptotically zero map f, satisfying f(x) goes to zero as x goes to infinity, must have a compact global attracting set $A $. The question of what additional hypotheses are sufficient to guarantee that A has a minimal (invariant) subset A* that is a chaotic strange attractor is answered in detail for a few types of asymptotically zero maps. These special cases happen to have many applications (especially as mathematical models for a variety of processes in ecological and population dynamics), some of which are pr…
We show that nontrivial homoclinic trajectories of a family of discrete, nonautonomous, asymptotically hyperbolic systems parametrized by a circle bifurcate from a stationary solution if the asymptotic stable bundles Es(+{\infty}) and Es(-{\infty}) of the linearization at the stationary branch are twisted in different ways.
We consider a collective version of Parrondo's games with probabilities parametrized by rho in (0,1) in which a fraction phi in (0,1] of an infinite number of players collectively choose and individually play at each turn the game that yields the maximum average profit at that turn. Dinis and Parrondo (2003) and Van den Broeck and Cleuren (2004) studied the asymptotic behavior of this greedy strategy, which corresponds to a piecewise-linear discrete dynamical system in a subset of the plane, for rho=1/3 and three choices of phi. We study its asymptotic behavior for all (rho,phi) in (0,1)x(0,1]…
We develop a $K$-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author.
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