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 Lyapunov function and a first approximation of the DA. The "embryo" is used for the determination of a new "embryo" and a new part of the DA. In this way, computing new "embryos" and new domains, the DA is gradually approximated. Numerical examples are given for polynomial systems.
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.
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, 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.
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 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…
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…
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 $N_{p^{x}}$ …
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…
Calculating Domain of Attraction Boundary of Power Systems Based on the Gentlest Ascent Dynamics
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Electric Power Engineering, South China University of Technology（华南理工大学电力学院）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）；Energy and Power Planning Laboratory of Energy Development Research Institute, China Southern Power Grid（中国南方电网能源发展研究院能源与电力规划实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Separatrices divide the phase space of some holomorphic dynamical systems into separate basins of attraction or 'stability regions' for distinct fixed points. 'Bundling' (high density) and mutual 'repulsion' of trajectories are often observed at separatrices in phase portraits, but their global mathematical characterisation is a difficult problem. For 1-D complex polynomial dynamical systems we prove the existence of a separatrix for each critical point at infinity via transformation to the Poincaré sphere. We show that introduction of complex time allows a significantly extended view with the…
University of Cambridge（剑桥大学）；University of California, Los Angeles（加州大学洛杉矶分校）；University of Utah（犹他大学）
Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning models efficiently. First-order methods such as gradient descent are usually the methods of choice for training machine learning models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent [Osher et al., arXiv:1806.06317], called modified Laplacian smoothing gradient descent (mLSGD), and demonstrate its potential to avoid saddle points wi…
