Lyapunov function basin of attraction

Though the convergence of the control sequence can be guaranteed only within a bounded basin, this approach seems to have considerable advantages. This algorithm relaxes the necessary conditions on the coefcients of the Lyapunov-like Numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space. Case with friction. Less often is attention given to the basin of attraction despite the fact that knowledge of the basin is essential for these calculations and for determining possible multistability and the use- basin of attraction. 8 Lab 1. R, called the storage function, such that S_ yTu. This condition was evaluated for the Lyapunov function generated by the SOS algorithm when estimating the BA of the event-based pulse controlled compass biped. IGD, , ,  23 Dec 2014 Lyapunov functions by numerical approximations of the. Lyapunov Exponents, Singularities, and a Riddling Bifurcation Lora Billings, James H. Lyapunov function and an open disk inside the basin of $(0,0)$ of attraction of the origin that can be obtained for the fixed Lyapunov functon. For a stable linear system, every point in the phase Lyapunov function. Ser. For non-autonomous systems, any relevant subset of the phase space, which now includes the time as one coordinate, is unbounded in the t-direction. Finding a suitable estimation of stability domain around stable equilibrium points is an important issue in the study of nonlinear dynamical systems. Key words. An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will eventually be iterated into the attractor. strange exponentially sensitive (non-unif) hyperbolic all Lyapunov exponents are non-zero at almost all points. The basin of attraction of an equilibrium of an ordinary differential equation can be determined using a Lyapunov function. If for any given time all trajectories in the en-semble snapshot are contained in a single basin of attraction Some of the "fractals" which can be drawn include the Mandelbrot set and Julia sets of various complex analytic functions, the basin of attraction for Newton's method in the plane, the bifurcation diagrams of various equations including the logistic equation, various attractors including the Henon, Lorenz and Rossler attractors, KAM tori, the center and fall into the basin of attraction of a different center . e. "The basin of attraction of an equilibrium of an ordinary differential equation can be determined using a Lyapunov function. Here the basin is for . For nonlinear systems, the explicit construction of a Lyapunov function taking the nonlinear dynamics into account remains a difficult problem and one often resorts to numerical methods. Exercise 3. The Lyapunov spectrum of a map is a plot of its Lyapunov exponents. With an estimate of the local attractors available, the next task is to construct a Lyapunov function on some subset of the basin of attraction for each local attrac-tor. The domain of the Lyapunov function is only limited by the size of the equilibrium's basin of attraction. Rodriguesa,∗ aDepartment of Mathematics, Federal University of Sa˜o Carlos, Rod. Try to compute a Lyapunov function of S ctrl. point in the basin of attraction. This basin of attraction is ensured by a Lyapunov-like polynomial function that we compute using an interval based branch-and-relax algorithm. Here, stability (or incremental stability) is a consequence of the contraction property between two adjacent solutions, formulated as the local property of a Finsler‐Lyapunov function. Using this Lyapunov function one can determine analytically large subsets of the basin of attraction of an asymptotically stable equilibrium. Notion of ISS Lyapunov function. The direction-based search method can also be used to obtain a rough approximation of the attraction basin. New finding: If some (transcritical) bifurcation occurs (and this is the situation of HIV infection treatment of [Shim et al. HeffelIII;  12 Oct 2015 For V(x,y)=12(x2+y2). In both this paper and in package QPot, the function that we refer to as the quasi (a) Show that V(x) = 2x2 + 2xy + 3y2 is a Lyapunov function for 0, and use it to estimate the basin of attraction. Lyapunov-like polynomial function that we compute using an interval based branch-and-relax al- gorithm. net dictionary. Systematic procedures to search for common and multiple Lyapunov functions are discussed there. A Newton attraction basin is created by using the iterating function from Newton's numerical method (Wolfram MathWorld) of finding roots . Lecture notes. To this end, a piecewise biquadratic Lyapunov function is considered to estimate the basin of attraction as an alternative to more complex polynomial Lyapunov functions. In the present work we obtain a three-dimensional function describing the geometry of the basin of attraction from experimental human balance data. Suc h a V (x) m y b e though t of as an \energy" function. Construction of Lyapunov Functions for the Estimation of Basins of Attraction where j is the index for the row and the two indices k and l cover all quadratic terms. In the last example, the whole R2 + is in the basin of attraction of (^x;y^). In the present work, the invariance of the Lyapunov dimension under diffeomorphism is demonstrated in the general case. behave as if E were a strict Lyapunov function. locally Lipschitz function defined over a domain D ⊂ Rn (0 ∈ D) The region of attraction (also called region of asymptotic stability, domain of attraction, or basin) is the set of all points x0 in D such that the solution of x˙ = f(x), x(0) = x0 is defined for all t ≥ 0 and converges to the origin as t tends to infinity Show that the basin of attraction of X eq is a positively invariant set. Three competitive populations. A comparison of the guaranteed basins of attraction for the time-. To show that the equilibrium is globally asymptotically stable, i. Lyapunov Exponents and Strange Attractors in Discrete and Continuous Dynamical Systems Jo Bovy Jo. The true Lyapunov exponents can, however, be identified because they change their signs upon time reversal whereas the spurious Lyapunov exponents do not (Parlitz, 1992). A strict Lyapunov function for an equilibrium of a dynamical system asserts its asymptotic stability and gives a lower bound on its basin of attraction. The blue region cannot contain because it is an unstable steady state. 4. V (0) = 0 and (x) > for 6 _ this ball. The simplest solutions x(t) of such an equation are equilibria, i. Due on Nov 18. They are based on optimizations problems in terms of The design philosophy of the VSC reaching mode is to insure that the domain of attraction to the switching surface is as large as possible. On the other hand, there is no systematic method for finding Lyapunov functions. basin of attraction of E 2 . Due to nonstationarity, the exponents exhibit random fluctuations with time. Lyapunov function, basin of attraction, mesh-free collocation, radial basis function, continuous piecewise affine interpolation, computation, verification Basin of attraction: Compact sublevel sets of a strict Lyapunov function, which are completely contained in U, are subsets of the basin of attraction of the equilibrium. Underlying property for this is that the basin of attraction of each equilibrium does not vanish along the trajectory of equilbria family. In the third subsection, we report properties of the real-world networks referred to in Fig. In [13], Lin, Sontag and Wang considered the inclu- Definition of basin of attraction in the Definitions. Read "Solutions to hybrid inclusions via set and graphical convergence with stability theory applications, Automatica" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Working with difference equations and inclusions, this paper develops some basic theory of set-valued Lyapunov functions and relates them to pointwise asymptotic stability, which is the term chosen here for the prop- For example, one can compute the basin of attraction for a particular stable equilibrium using a properly-chosen Lyapunov function. Definition II. In this In this paper a further step towards a novel approach to adaptive nonlinear control developed at Budapest Tech in the past few years is reported. \basin of attraction" of an asymptotically stable equilibrium point, i. Synergistic Lyapunov functions and backstepping hybrid feedbacks Christopher G. Reasoning similar to that above leads to the conclusion that in region 5 the same two lines (0 : 5 x 1 + x 2 +1= 0 and x 1 +0 :5x2 +1=0 ) can be used. Moreover, sublevel sets get arbitrarily close to the basin of attraction. Weighting this flux by the probability density function (PDF) pinpoints re-gions in phase space that have the greatest leakage into an-other basin and strongest transport created by contains almost all points in the basin of attraction. A Lyapunov function for an autonomous dynamical system {: → ˙ = ()with an equilibrium point at = is a scalar function: → that is continuous, has continuous first derivatives, is locally positive-definite, and for which − ∇ ⋅ is also locally positive definite. pp. Examples. GIESL To determine the basin of attraction of an exponentially asymptotically stable periodic orbit one can use a Lyapunov function, too (cf. The first book on the subject, and written by leading researchers, this clear and rigorous work presents a comprehensive theory for both the stability boundary and the stability regions of a range of nonlinear dynamical systems including continuous, discrete, complex, two-time-scale and non The quantity ℛ 0 measures the average number of new infections generated by a single infected individual in a population. De nition: The Lyapunov exponent in Consequently it converges to the fixed point that is the solution of the control task. Ifthereexistacontinuous,positivedefinite and radially unbounded (i. 2 ISS Lyapunov Function 37 2. The attraction basin of A, denote by Ω()A, In order to obtain the proper set K, the key tool here is the level set of Lyapunov function on at-tractor . [2]). In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. . As pointed out in [3], they give the average rate of convergence or divergence of the system along the principal axes in phase space. Assuming that two subsets of the domains of attraction are known, one larger than the other, this work states the problem of combining both controllers with the goal of guaranteeing asymptotic stability properties in the largest subset while the desired performance is locally achieved. Bovy@student. if K is in the basin of attraction of the equilibrium and given apriori an  11 Oct 2006 Asymptotic stability additionally characterizes attraction of nearby orbits to this orbit . Domain of attraction Also called basin of attraction is the region in state space of all initial conditions that tend to a particular solution such as a limit cycle, fixed point, or other attractor. Consider the discrete dynamical system generated by a planar homeomorphism f. A new method to construct such a Lyapunov function using radial basis functions is presented in this volume intended for researchers and advanced students from both dynamical systems and radial basis functions. The local Lyapunov functions d and v are con-structed and discussed. 1 Master Stability Function Phys 221A Lecture Notes - Lyapunov Exponents and their Relation to Entropy where fis a function that maps x basin of attraction of a strange attractor. She, Providing a basin of attraction to a target region % of polynomial systems by computation of Lyapunov-like functions. a) most of the control problems do not have closed analytical solutions; b) from numerical calculations “well behaving within a finite period” the stability cannot be taken for granted. 1) in a given domain. In [ 14] it limited by the equilibrium's basin of attraction. This fact is used to obtain the analytic exact upper bound of the Lyapunov dimension of an attractor of the Shimizu–Morioka system. 7 Lyapunov Functions with Bounded In adaptive nonlinear control Lyapunov’s 2nd or “Direct” method became a fundamental tool in control design due to the typical practical difficulties viz. help of the Lyapunov spectrum, it is a simple matter to compute the Kaplan-Yorke dimension. Positively invariant set and basin of attraction. stable so your search for a global Lyapunov function may not be possible and its basin of attraction. G. basin of attraction for the attractors via Lyapunov functions that  23 Aug 2010 it using Radial Basis Functions. 3 In this paper, we present a method for computing a basin of attraction to a target region for polynomial ordinary differential equations. Ask Question Asked 6 years, Integration by parts wrt. I For example, an equilibrium point has zero phase volume, This approach searches for a piecewise affine Lyapunov function, and requires a triangulation of the state space. If is a not compact, let be a real number arbitrary up to the fact that the compact set contains at least one asymptotically stable equilibrium and no positions of the corresponding minima of function V(X), since those will act as the (stable) equilibrium points of the system. The stability of (1) is closely tied to the existence of a Lyapunov function for the system, the sublevel-sets of which are entirely contained within the domain of attraction of the equilibrium at the origin. the basin of attraction may be limited. One standard approach to obtaining such a stability measurement is to look only at the states in the attractor [8,12,2]. Giesl, Peter and Wagner, Heiko (2007) Lyapunov functions and the basin of attraction for a single-joint muscle-skeletal model. After an introduction and the denition of a Lyapunov function and a Lyapunov basin (sublevel set), we show in Theorem 2. This basin of attraction is ensured by a Lyapunov-like polynomial function that we compute using an interval based branch-and-relax al-gorithm. 7. An important and non-trivial task is the 0 determination of their basin of attraction. Related results for inclu- sions of the type (11) are in [19, 20, 211. To incorporate this effect in the differential equa-tions for the pattern activities, we introduce the variables attraction varies as a function of the parameters of the model. 1992). Mayhew♯, Ricardo G. www. a partially rapidly exponentially decaying control Lyapunov function, we establish local asymptotic stabilization. then μ is called a Lyapunov exponent. stability, domain of attraction, or basin) is the set of all points x0 in attraction is the whole space Rn . Suppose that system (1) is passive with positive-de nite storage function S. rst theorem is that there is a continuous Lyapunov function for any attractor A of a continuous map gof any metric space into itself. a Morse function on its basin of attraction. Let denote the time deriv ativ e of V (x) along an y tra jectory the system, i. P$ is in the Construction of Lyapunov functions and Contraction Metrics to determine the Basin of Attraction Peter Giesl University of Sussex, UK { Department of Mathematics July 18, 2013 Workshop on Algorithms for Dynamical Systems and Lyapunov Functions Reykjavik University, Iceland partly supported by EPSRC Small Grant Lyapunov function actually allows to gain some information about the global behavior of orbits. We then develop a method using a local Lyapunov function and a nonlocal one to obtain rigid lower bounds on -BOA. However, it can be applied to only polynomial systems. When motion is not periodic, the parameter mu determines the frequency with which the mass oscillates back and forth; consequently, mu is called the quasi frequency. always succeeds in computing and verifying a Lyapunov function, as well as in determining arbitrary compact subsets of the basin of attraction. Consider Program 2 which computes the basin of attraction of the root z = 1 of the complex function f(z) = z 3 - 1. Limit sets. 7 Figure 1 displays a set of trajectories for values of (x) = y. Lyapunov Functions and Basins of Attraction. Massera . F. Long soliloquy attempting to show how the different topics to be covered in class evolved and are interrelated. This is possible unless the term in the curly If it is asymptotically stable nd its basin of attraction. Sanfelice♭, and Andrew R. Finally, we use the computed basin of attraction to influence the way that our tree grows, with the 18 The basin of attraction of an asymptotically stable equilibrium for an autonomous differential equation can be determined through sublevel sets of a Lyapunov function. — Martens, Panaggio, and Abrams, Basin of attraction for chimera states, arXiv preprint (2015). The blue region is also a basin of attraction for . It has a Estimating the geometry of the basin of attraction of a nonlinear dynamical system from raw data is a funda­ mental and challenging problem from both a theoretical and applied point of view. The basin of attraction of an asymptotically stable equilibrium for an autonomous differential equation can be determined through sublevel sets of a We use cookies to enhance your experience on our website. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. 1 Strict Lyapunov Function 36 2. Computing Lyapunov Exponents function good_examples % Systems and corresponding Lyapunov functions % taken from S. 7, pp. 6. More technically, the basin of attraction B(x^) of the Construction of Lyapunov Functions for the Estimation of Basins of Attraction G. 1. the set of initial. In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an  12 Jun 2008 The basin of attraction of an asymptotically stable equilibrium for an autonomous differential equation can be determined through sublevel sets  Detailed Record. Consider what happens when we change up the initial conditions. over, we recall a definition of a control-Lyapunov function in the case our proof is mainly based on the compacity of the basin of attraction, we. 6 £2 Gains 45 2. Similar to the NL/sub q/ theory, the local stability is imposed around the origin and the apparent basin of attraction is made large by applying the criterion, while the proven basin of attraction is relatively small So that the final state of the system is determined by the basins of attraction where initial conditions are located; when the initial conditions are near the boundary of the attraction basins, slight disturbance or change in parameters will lead to a completely different motion of the system. One method requires construction of a function, often called a Lyapunov function, which remains . 2 of Bay illustrates multiple attractors and basins of attraction of a pendulum. Teel† Abstract—The notion of synergistic potential functions has been introduced recently in the literature and has been used as the basis for the design of hybrid feedback laws that achieve Lyapunov theory of stability [1] for deterministic systems has been very successful as a tool to study qualitative system behavior. defines a closed curve containing the origin if is a positive constant. In Giesl (2007, Discrete Contin. 1 and has a minimum or diverges to −∞at the fixed point x∗1,5,6. A characterization of the basin of attraction of the origin is given in terms of the level set of a quadratic Lyapunov function. The Lyapunov functions in the . The Lyapunov function is defined as the square of the difference between the actual and nominal velocity of the unactuated stance leg at the midstance position (stance leg is normal to the ramp). V satisfying V(x) . 9/18 Lyapunov Exponents Gives a measure for the predictability of a dynamic system characterizes the rate of separation of infinitesimally close trajectories Describes the avg rate which predictability is lost Calculated by similar means as eigenvalues of the Jacobian matrix J 0(x0) Usually Calculate the Maximal Lyapunov Exponent Gives the best ASL-STEM Forum. its rate c hange as t v aries tionally deliver an inner approximation of the basin of attraction. Its main advantage in comparison with the complicated Lyapunov function based techniques is that it is based on simple geometric considerations on the basis of which the control task can be formulated as a fixed point problem for the solution of basin of attraction (obviously, when there is a single attractor, the whole available phase space is its basin of attraction). We design a conservative approximation of this region using sums-of-squares optimization. I'm also asked to graph everything, but it doesn't seem right to me. HochlenertII; E. Finally, we use the computed certificates, and resulting basin of attraction to influence the way that our tree grows, The iterated function involves geometric folding operations and can be applied to points of any dimensions. Homework 7: here . Garnett. Lyapunov’s theorem can be applied without solving the differential equation (1). Lyapunov Theorem for Global Asymptotic Stability The basin of attraction of an asymptotically stable equilibrium point, is the set of initial conditions whose subsequent trajectories end up at this equilibrium point. Re-mark 3. ’ • Figure 7. tu-darmstadt. Power system components are often described by nonlinear differential algebraic equation (DAE) models. Bathe* Massachusetts Institute of Technology, Mechanical Engineering Department, Cambridge, MA 02139, USA Abstract In many applications in engineering and science, it is important to know whether the response of a nonlinear system is chaotic. Attraction domainsPolygon expansion techniqueCapture basin ExamplesConclusion Viable set characterization algorithm Choose a control u 2U. hochlenert@tu-berlin. The γ -basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. A Lya- punov function is a function which is decreasing along solutions of the ODE; sublevel sets In this paper, we present a method for computing a basin of attraction to a target region for polynomial ordinary differential equations. While the intention of Lyapunov was to study the stability of motion, the direct Lyapunov method and the notion of an auxiliary function have found a wider range of application and Lyapunov functions may be used to achieve a multitude of diverse states within the basin of attraction B to the original attractor, i. 526 (Received 31 March 1997) There are few examples in dynamical systems theory which lend themselves to exact computations We also show that there is a link between global stability LMI conditions based on this new Lyapunov function and a transfer function of an auxiliary system being strictly positive real. In Section5we present a technique similar to the one presented in [15] in order to approximate only the local attractors of (1. In the following we consider examples of typical appearance of ideas and results from bifurcation theory. 23, not only provides another means of deducing asymptotic stability for equilibria of the damped system, but also gives information as to the extent of the basin of attraction for where f is a nonlinear function of x 2Rn, called the basin of attraction, such quanti ed in terms of Lyapunov exponents. Spelsberg-KorspeterI; D. Besides providing an analytical tool for the analysis of the non-linear problem around a given equilibrium point, and compute a basin of attraction for this Lyapunov function with respect to the original, non-linear prob-lem [13, 10]. Our method is based on the density of the data set and uses numerical optimization and data modeling tools. It can be shown that if V is a strict Lyapunov function for x^, then the sets Gc = {x ∈ G: V(x) ≤ c} are in the basin of attraction of x^. Dyn. 1 The Rosenzweig-MacArthur model The discussion here of predator-prey models and in particular the Hopf bifurcation in the Rosenzweig- Construction of Lyapunov Functions for the Estimation of Basins of Attraction. Appl. Phase portraits based on energy equation or other function. Keywords. solutions x(t)= x which remain constant. Lyapunov Function Hopfield proved that the discrete Hopfield net converges to a stable limit point corresponding to a set of activities by considering the fol- Lyapunov function or to give approximate region of attraction based on nonlinear dynamics [6-8]. 3 L is a Lyapunov function when (and only when) X∗ is a strict Lasalle's Principle yields an accurate lower bound for the basin of attraction. conditions for stability is called a Lyapunov function. it decreases along trajectories of Eq. By de nition, the basin of attraction B(x^) of ^x is the set of initial conditions such that x(t;x0) → x^ for t → ∞. Covered class expectations and syllabus. Equilibrium is unattainable from points outside its attraction basin. Region of Attraction When the origin x = 0 is asymptotically stable, we are often interested in determining how far from the origin the trajectory can be and still converge to the origin as t → ∞. Synchronization in Nonlinear Systems and Networks but not stable in Lyapunov sense Then basin of its attraction is An example of intermingled basin of attraction Read "Solutions to hybrid inclusions via set and graphical convergence with stability theory applications, Automatica" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. In the formerly developed approaches for monotone increasing or monotone decreasing systems the proper fixed points had only a finite basin of attraction outside of which the iteration might become divergent. locally Lipschitz function defined over a domain D C R" The region of attraction (also called region of asymptotic stability, domain of attraction, or basin) is the set of all points in D such that the solution of is defined for all t > 0 and converges to the origin as t tends to infinity The origin is said to be globally asymptotically stable if In fact, the program for determining the basin of attraction of a given root is very similar to Program 1 for computing the Julia sets. Lyapunov function, Radial  31 Oct 2004 Lyapunov function a subset of the basin of attraction of a locally asymptotically stable system can be determine, when a control Lyapunov  7 Jan 2014 Lyapunov function for nonlinear systems with asymptotically stable equilibria. Stability based on Lyapunov function. 2 by the shaded area. But even if we know In the original definition, due to Auslander, Bhatia, and Seibert [1964], a compact \(f\)-invariant subset \(A=f(A)\subset X\) is called a Lyapunov stable attracting set if it has an open basin of attraction, and if the following condition is satisfied: Lyapunov Stability. Trajectories diverge otherwise. One method to construct such a Lyapunov function is to solve a certain linear PDE approximately using Meshless Collocation. Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems Peter Giesl∗ and Holger Wendland† August 23, 2010 Abstract Numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space. Find an equilibrium point x 2K. When the Lyapunov exponents of such an ergodic u-Gibbs states are negative, it is an SRB-measure (having a positive Lebesgue basin of attraction). Pinto 3, S. The following corollary, often called “LaSalle’s Corollary” to the Lyapunov Theorem 7. Balas largest sublevel set of a given Lyapunov function that can imply that Lyapunov exponents exist for Lebesgue al-most every point in the phase space. The. 5 Abstract: In this paper, we present a method for computing a basin of attraction to a target region for non-linear ordinary differential equations. E. The result is then applied to robot locomotion. This result was rst established We address the problem of asymptotic stability and region-of-attraction analysis of nonlinear dynamical systems. Language: English; Authors: Björnsson  31 Mar 2014 Lyapunov-Krasovskii functional that exploits the piecewise affine nature of the system. Suppose δ(n) is the separation after n iterations of the system. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. In this paper, we present a method for computing a basin of attraction to a target region for non-linear ordinary differential equations. We conclude by showing that, because of the high-gain nature of the feedback, it is possible in some situations for the basin of attraction to become state space) from which trajectories converge to the attractor the ’basin of attraction. What about attractors? Suppose that the dynamical system is dissipative and that it admits an attractor with an open basin of attraction. 4) Control design: Lyapunov design, Jurdjevic‐Quinn theorem, classical backstepping, bounded backstepping, backstepping for time‐varying systems, strabilization and tracking though forwarding, Sontag‘s formula. Checking a number of inequalities, we are able to verify that this interpolation is indeed decreasing along trajectories. 48, No. When the foliation is by hyperbolic leaves, this class of probabilities coincide with the classical harmonic measures introduced by L. This concept provides a method to iteratively improve the attraction domain estimate using different Lyapunov functions without limiting the analysis to contractively invariant sets. Experiments Besides its simplicity the main advantage of this approach was the possible evasion of complete state estimation that normally is required in the Lyapunov function-based design. 3 Lis a Lyapunov function when (and only when) X is a strict local minimum of V. In a classical scattering system there will be one or more impact parameters, b, in which a particle is sent into the scatterer. NONCOMPACT CHAIN RECURRENCE AND ATTRACTION 1141 any ¿-chain beginning in U must end at a point within 8 of f(U). Since the dynamics inside the basin are dissipative, every invariant measure sup- The basin of attraction, denoted B(A), of a locally point-wise asymptotically stable set A ⊂ Rn is the set of all x∈ Rn such that for every solution φ∈ S(x), the limit lim j→∞ φ(j) exists and lim j→∞ φ(j) ∈ A. 2). First, let me define the basin of attraction of an asymptotically stable equilibrium at the set of all initial conditions leading to the long term behavior that approach this equilibrium. Lyapunov functions. a scalar-valued function which is decreasing along solutions of the dynamical system. 5 Basin of Attraction Revisited 44 2. Title: Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. If the system is nonlinear one often computes a Lyapunov function for the linearized system, cf. 4377-4394, 2010. for a Lyapunov function as it can be used for purposes other than stability analysis such as in estimating a basin of attraction (in the case of local stability) or for inferring robustness properties [11] [20]. 13(6) Theorem2. To distinguish between the first and second cases, the Lyapunov function that satisfies First, let me define the basin of attraction of an asymptotically stable  This basin of attraction is ensured by a. Keywords: basin of attraction, asymptotically autonomous differential equation,. Exercise 4. the Lyapunov function is continuous, the closed-loop asymp- as the horizon length increases, the basin of attraction becomes arbitrarily large and the neighborhood In doing so we employ well-established methods of nonlinear dynamical systems which rely on finding the fixed points, investigating the structure of the flow diagrams of the evolution equations, and characterizing the basin of attraction using a Lyapunov function near the stable fixed points. Also a hardware realisation of this system is proposed by using field programmable gate arrays (FPGA). Spelsberg-Korspeter speko@dyn. Ratschan, Z. we have ˙V=−x4−8y4+x6+y6−9y5x5≤−(1−x2−|xy|)x4−8(1− (1/8)y2−|xy|)y4. Robust Region-of-Attraction Estimation Ufuk Topcu, Andrew K. Moreover, if is compact, it has stationary points (and minima) which, by the previous remark, are necessarily equilibria (respectively asymptotically stable equilibria). I We talk about volume of the phase space occupied by the attractor and the phase volume occupied by its basin of attraction. The basin of attraction is the set of all initial states that will converge to the given reference trajectory [4,13]. x=mpolyfun. As such, we will use each training data point as an initial state of the system and let it settle towards the equilibrium point whose basin of attraction includes the given initial state. the basin of attraction is the whole space, one needs the additional assumption on the strict Using this procedure, the Lyapunov function (29) and the stability boundary can be transformed back to physical coordinates. Simulation tests of a small perturbation of this starting point, defined by x(0) + δ(0), where the initial separation δ(0) is assumed to be very small. Given output feedback control u = (y), where is any continuous function satisfying yT (y) 0, the origin is stable in the sense of Lyapunov, i. For non-autonomous Numerical determination of the basin of attraction for asymptotically autonomous dynamical systems Article in Fuel and Energy Abstracts 74(10) · September 2010 with 53 Reads How we measure 'reads' Abstract. ISSN 0303-6812 Giesl, Peter (2007) Construction of a global Lyapunov function using radial basis functions with a single operator. Chaotic scattering is a branch of chaos theory dealing with scattering systems displaying a strong sensitivity to initial conditions. In this paper, regions of attraction for missile in With the Lyapunov function having an opposite sign of corresponding to the basin of attraction. Washington Luis Km 235, Sa˜o Carlos - SP 13565-905, Brazil Abstract We consider the connection of Lyapunov functions and the transition to turbulence in shear flows when the linearized "The basin of attraction of an equilibrium of an ordinary differential equation can be determined using a Lyapunov function. There are no universal scalable approaches for constructing attraction basin estimates and designing nonlinear control systems. de S. . Notice stable and unstable equilibria. The coloured region is the basin of attraction of the root z = 1; Program 2 attraction or use another Lyapunov function or method. This paper investigates the regional stability of uncertain rational systems subject to multiple state constraints. Then we interpolate this function and thus construct a CPA Lyapunov function. The finite-time Lya-punov exponent (2) for the direction of the flow per-petually oscillates as t→ ∞, causing the infinite-time Lyapunov exponent (3) for the flow direction to fail to converge. Details can be found in Hochlenert (2012). , for every >0 there exists a subsection, we elaborate on the computation of basin stability in R ossler networks. 4: A semilog plot of the separation between two solutions to the Lorenz equations together with a tted line that gives a rough estimate of the Lyapunov exponent of the system. SES # CLASSROOM ACTIVITY READINGS KEY DATES; 1: Sorted out class times and locations. Meaning of basin of attraction. Repeated trials are used to provide an estimation of the stability of a given basin of attraction, where stability is defined as the probability that a perturbation to a state does not change the basin of attraction. That is, if X larger domain of attraction. This basin of attraction is ensured by a Lyapunov-like polynomial function that we compute using an in-terval based branch-and-relax algorithm. 2 Illustration 46 2. A stochastic view ofDynamical Systems – p. Then nonlinear systems with non-polynomial terms should be transformed or approximated to polynomial form. if its basin of attraction is the entire state space. Chapter 2 deals with the dynamical systems part of the book. Week 13. the delay-free basin of attraction estimate, Lyapunov-. The estimation of the region of attraction is based on common quadratic Lyapunov function. 10) where isaconstant. gradient systems there is a natural candidate for such a function, Indeed, on solutions . Lemma 1. The determination of the basin of attraction is achieved through sublevel sets of a Lyapunov function, i. punov result was in fact given in terms of a classical Lyapunov function, but only yielded a necessary, not a sufficient condition. Hence, this region has an ellipsoid form. Definition: Suppose a mass is in an oscillating motion. 3 iISS Lyapunov Function 39 2. In addition, the novel function is considered in the local stability analysis problem of discrete-time Lur’e systems subject to a saturating feedback. In some cases, there are natural Lyapunov function candidates like energy functions in electrical or mechanical systems. Estimating basin of attraction by Lyapunov functions via continuation methods Workshop "Advanced Computational and Experimental Techniques in Nonlinear Dynamics“ Cusco, Peru, May 16, 2013 Max Demenkov Institute of Control Sciences, Russian Academy of Sciences Moscow, Russia This paper provides an explicit Lyapunov function for a general single-joint muscle-skeletal model. The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration x n+1 =g(x ) can be determined through sublevel sets of a Lyapunov function. org - The premier society in computing brings you the Computer Portal. This authoritative treatment covers theory, optimal estimation and a range of practical applications. Determination of an estimate of a basin of attraction via a strict Lyapunov functions. be Theoretical Physics Project are investigated with the help of its state-space portraits, bifurcation diagram, Lyapunov exponents diagram, and basin of attraction. Lyapunov characteristic exponent calculation for finite element discretized models S. This algorithm   Lyapunov functions are an important tool to determine the basin of attraction of Lyapunov function, basin of attraction, mesh-free collocation, radial basis  13 Nov 2017 basin of attraction using a Lyapunov function near the stable fixed points. For a chaotic system, the initial condition need only be changed slightly since orbits quickly become uncorrelated due to the sensitive dependence on Lyapunov Function based Transient Stability Analysis Differential-Algebraic-Equation (DAE) based Model Equilibrium solution using multi-parameter homotopy method Basin of Attraction of Local Stable/Unstable Equilibrium Solution Time Domain Simulation for Projected Dynamics System Identification for updating DAE matrix parameters Globally Stable we want to make sure that the basins of attraction for the spurious attractors are relative small, so that most of the input test patterns will converge to one of the stored patterns. Network Lyapunov function analysis (for single-node basins) Lyapunov characterization of Zeno behavior in hybrid systems Rafal Goebel and Andrew R. 1) is known. After developing a general theory, we only resort to the use of quadratic Lyapunov functions because of the computational appeal given by LMI solvers. Lyapunov function of the system, then x = 0 is astable equilibrium point in the sense of Lyapunov. Positive and negative de nite? In order to nd out about stability we use Lyapunov functions. For mathematically oriented studies of the Lyap stability, Lyapunov function, energy function, positive definite, attractor, limit cycle, basin of attraction, additive model of a neuron, Hopfield network, discrete Hopfield network, content-addressable memory, spurious states, storage capacity, BSB Plan • study dynamical systems & state spaces, trajectories, equilibrium states, The basin of attraction of an equilibrium of an ordinary differential equation can be determined using a Lyapunov function. 9, 33. 453-464. However, due to the information loss introduced by the linearization process, this basin of attraction will usually be very small. This type involves uniform bounds on the amount of ordinary time, but not the number of jumps, state space over which the quadratic cost-to-go function is still a valid Lyapunov function for the nonlinear system. Basins of attraction. 82 P. L. 4 LaSalle Invariance Principle 40 2. A locally Lipschitz function V : Rn!R + with V(0) = 0 and V(x) >0 for x 6= 0 is a Lyapunov function for the system Lyapunov function. 3. That is, there exists a smooth surface or hypersurface in the phase space, such that any initial condition in the surface generates an orbit that remains in the surface. Lyapunov’s Theorem implies that in the first case, X is stable, and in the second it is asymptoticallystable. 9 and 70. This results in an algorithm that is always able to compute a Lyapunov function for a discrete time system with an exponentially stable equilibrium. Joydeep Mitra Electrical & Computer Engineering Michigan State University East Lansing, MI 48824 (517) 353‐8528 mitraj@msu. Construction of Lyapunov functions for the estimation of basins of attraction. Lyapunov function for a linear system is a semidenite program (SDP) (also referred to as a linear matrix inequality estimating basin of attraction of equilibrium function V (x) that is 0 at the origin and p ositiv e elsewhere in some ball enclosing the origin, i. This objective is achieved by choosing switching surfaces such that asymptotic stability can be guaranteed using the Lyapunov's direct method. J. de TU Darmstadt, System Reliability and Machine Acoustics, Dynamics and Vibrations Group, Germany D. Journal of Mathematical Biology, 54 (4). In order to simplify (8) as much as possible we try to choose Gj,kl so that Hj,kl vanishes. Show that X 0 is in the basin of for the case of attract basin being the whole space by removing the global Lipschitz continuity for Lyapunov function . The algorithms themselves that are used to compute Vˆ and Tˆ are detailed in §5. Tools for nonlinear control, Lyapunov function, positivity, applications Notions of stability (local, global, basin of attraction), notion of input-to-state stability. Specifically, changes in that Lyapunov function following one-step iteration of the polynomial map were computed for a random sample of 800 points on the Poincaré section. 24 that a Lyapunov basin is a subset of the basin of attraction. If z = h(t,x) is asymptotically stable, uniformly in (t,x) ∈ [t 0,t 1] × X, then the basin of attraction is uniform. 4 respectively. The Lyapunov function has the properties that its value is 0 inAand 1 on the complement of the basin ofA, while it is strictly decreasing along other gorbits. To estimate the uncertainty in your calculated Lyapunov exponent, you can repeat the calculation for many different initial conditions (within the basin of attraction) and perturbation directions. Further ResearchArticle Probabilistic Basin of Attraction and Its Estimation Using Two Lyapunov Functions SkuliGudmundsson1 andSigurdurHafstein 2 SvenskExportkredit,Klarabergsviadukten-, Stockholm,Sweden Lyapunov functions Vand T. Lopes 1, and R. Linearize S u around x. If B(A) = Rn, Ais globally pointwise asymptotically stable. Lemma 2. 2. We give the following de nitions To this end we define a probabilistic version of the basin of attraction, the -BOA, with the property that any solution started within it stays close and converges to the origin with probability at least . Camargo 2, S. Even worse, spurious Lyapunov exponents that are larger than the largest true Lyapunov exponent may be obtained (Dechert and GenCay, 1996). This region is a basin of attraction, and the set of numerical aluesv towards which a system will converge to is an attractor . Technical systems are often modeled through systems of differential equations in   TECHNICAL PAPERS. Nov 10-14 Weak and strict Lyapunov functions. 2009, AUT]), the preservation of basin of attraction is not possible. As for the justification that it's "based on the energy of the system," this is a special case of a more general fact: you can use conservation laws to produce Lyapunov functions for physical systems. Enabling American Sign Language to grow in Science, Technology, Engineering, and Mathematics (STEM) generating the leaf geodesics. If no function found, compute S ctrl. Pereira 1, S. A hybrid symbolic-numeric method is presented to compute exact Lyapunov functions and exact estimates of regions of attraction of nonlinear systems efficiently. Hochlenert daniel. Evolutionary stability of games with costly signaling average fitness is a strict Lyapunov function → every change basin of attraction has positive measure Lyapunov exponents are negative, then there is a positive measure set of initial conditions in a neighbourhood of the synchronous subspace which is in the basin of attraction of the chaotic attractor contained within this subspace (Alexander et al. Definition of a Lyapunov function. However, the drawback of the method is that finding such a function is usually a non-trivial task. The basin of attraction of an attractor Ais the set of all points xwith ˚ t(x) !A, t!1. Lemma 3 implies that the disease can be eliminated from the community (when ℛ 0 < 1) if the initial sizes of the subpopulations of the model are in the basin of attraction of the DFE (ℰ 0). to the fixed point x∗, defined through a time-tracking Lyapunov function. Curry, and Eric Phipps Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0. Ricci flow as a gradient flow and its Lyapunov function. Teel Abstract Necessary and sufcient Lyapunov characteriza-tions of a particular type of asymptotic stability in hybrid dynamical systems are given. We present a method of modeling the basin of attraction as a three-dimensional function describing a two-dimensional manifold on which the dynamics of the system evolves from experimental time series data. An equilibrium point is globally asymptotically stable (or asymptotically stable in the large) if its basin of attraction is Lyapunov function for local exponential stability and asymptotic stability on the basin of attraction of the origin for the inclusion (10). We refer readers toCameron (2012),Nolting and Abbott(2016), and the references therein for the technical construction. Thus if we define the region (neighborhood of the origin) Ω:={(x  22 Feb 2018 A deterministic Lyapunov function gives rigid estimates on the basin of attraction ( BOA) of an equilibrium through its sublevel sets. These systems can have a particularly bizarre type of basin structure called a riddled basin of attraction (Alexander et al. This paper provides an explicit Lyapunov function for a general single-joint muscle-skeletal model. This region is obtained using Lyapunov's direct method and the Lyapunov function . 27 Sep 2012 speaking), if one succeeds in finding a Lyapunov function. singles(2); Mechanical Engineering, Applied Mathematics, Optimal Control, Stability, Control Systems, and 15 more Nonlinear Control, Optimization Problem, Nonlinear Systems, Linear System, Domain of attraction, Receding horizon control, Nonlinear system, Local stability, Upper Bound, Optimal Control Problem, Basin of Attraction, Cost Function, Electrical Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system R. This paper intends to apply a set of analytical-numerical methods to estimate the region of attraction for autonomous nonlinear systems. a function with negative orbital derivative. Conversely, if X is compact, then the asymptotic sta-blility of z = h(t,x) together with the existence of a uniform basin of attraction imply that the asymptotic stability is uniform in (t,x ficient to compute only Lyapunov exponents to detect or define stochastic chaos [6], we compute transport of a sto-chastic flux from one basin to another [7,8]. Lyapunov function. Syst. ac. J. edu 1 The basin of attraction of an equilibrium can be determined through sublevel sets of a Lyapunov function, i. attraction and the expected time required to escape each basin. We show that (pre-)asymptotic stability, which is a natural generalization of asymptotic stability, of a compact set for a hybrid system satisfying mild regularity assumptions is equivalent to the existence of a smooth Lyapunov function. The method is applied to two examples. A Lyapunov function is a function V(x) with negative orbital derivative, i. kuleuven. Basin of attraction, the largest eigenvalue of the limit cycle, and Lyapunov function are some metrics that quantify the robots ability to follow a given reference trajectory. Unless Ais a single point, the usual Lyapunov functions, A Lyapunov Function Based Remedial Action Screening Tool Using Real‐Time Data Prof. This gives rise to the definition of region of attraction (also called region of asymptotically stability, domain of attraction, or basin). 15 Jul 2013 Key words: Lyapunov function, Lur'e systems, bounded sector nonlinearity, absolute . Viana 1 ∗ basin of attraction, yet it does not provide a systematic way to find an initial feasible LF. Basin stability (BS) is a universal concept for complex systems studies, which focuses on the volume of the basin of attraction instead of the traditional linearization-based approach. , ) function with all continuous partial derivatives in its variables, and an open basin of attraction. F (V1,V2,V3,Vn) Conclusion: The voltage Lyapunov function balances the voltage of the N strings connected in parallel 14 this assumption implies the existence of a smooth converse Lyapunov function for local exponential stability and asymptotic stability on the basin of attraction of the origin for the inclusion (10). Local Lyapunov Exponents and characteristics of is a function of N–1 variables with a bounded subset B of its basin of attraction Continuation branch A curve of fixed points, limit cycles, etc as a function of some parameter. Basin of Attraction Notice in the rst problem that the solution tended to a 'nice' region. 1 Basic Theorem 45 2. Edward Ott (2006) Basin of attraction. For the chaotic iterated map x n+1 = g(x n),g: R → R, is a basin of attraction of h(t,x). It is a strict Lyapunov function when in addition X is an isolated critical point point of V. Lyapunov function a subset of the basin of attraction of a locally asymptotically stable system can be determine, when a control Lyapunov function satisfying the small control property is available, one can apply the univer-sal formula proposed in (Sontag, 1989) to get an expres-sion of an asymptotically stabilizing feedback which is Lyapunov norms and the inner region of the basin of attraction Savio B. So-called converse Lyapunov theorems provide existence results for Lyapunov functions; i. Figure 1: Level surfaces of a Lyapunov function. This algorithm relaxes the necessary conditions on the co-efficients of the Lyapunov-like function to a system of linear interval inequalities that can then be solved exactly. Our approach derives polynomial (as opposed to affine) Lyapunov function but also benefits from a subdivision of the state space to increase accuracy. However, now all points below the two lines will fully belong to the basin of attraction of E 2 because the state trajectories are directed toward region 5. 4) Control design: Lyapunov design, Jurdjevic-Quinn theorem, classical backstepping, bounded backstepping, backstepping for time-varying systems, strabilization and tracking though forwarding, Sontag‘s formula. The most di cult problem in nding proper Lyapunov functions is to prove that the function and its derivative is positive or negative in some region. These two properties imply that the In this paper, we present a method for computing a basin of attraction to a target region for non-linear ordinary differential equations. 5 Matrosov Theorems 40 2. acm. % SIAM Journal of Control and Optimization, Vol. (c) Does the solution of (b) help in re-evaluating the basin of attraction obtained in (a)? 2. Although a quadratic LF can be easily constructed by solving the Lyapunov equation of the linearized system, it only captures the local behaviour of the nonlinear ODEs around the equilibrium point. The result is an efficient optimization to obtain a Lyapunov certificate. 7): (2. is a Lyapunov function for the equilibrium approximate in a better way the basin of attraction “Linearization methods and control of nonlinear systems global: Lyapunov function, time until certain distance to periodic orbit is reached Peter Giesl (Sussex, UK) Basin of Attraction Imperial, June 18, 2009 17 / 22 In each step the algorithm checks the feasibility of the linear programming problem. This algorithm relaxes the necessary conditions on the coefficients of the Lyapunov-like function to a system of linear interval inequalities that can then be solved exactly. 6 Non-strict Lyapunov-Like Function 41 2. A Lyapunov function V(x) is defined in terms of the Abstract A Lyapunov direct method is presented for the stabilization of underactuated, mechanical systems. Find a quadratic Lyapunov function for the system: x_ = y x3 If the exact range is not required, an estimate of the attraction basin can be generated by employing the Lyapunov function and performing an envelope search. This measure is computationally expensive even for complicated Lyapunov function technique that normally provides global stability of convergence at the costs of both formal and essential restrictions, by applying Cauchy sequences of local, bounded basin of attraction in an iterative control that is free of such restrictions. For a given attractor, μ is invariant in the basin of attraction . In addition, an electronic circuit design for the chaotic system is introduced. posed Lyapunov function – of the basin of attraction. Show that if U is an open set containing X eq and L : U is a strict Lyapunov function for X0 = F(X) at X eq, then X eq is asymptotically stable. Velocity Position Figure 9: Multiple attractors and basins of attraction (Bay figure 7. I know the Lyapunov, V, is the basin of attraction, but can't what the change in the basin of attraction really mean. a basin of attraction to a target region for non-linear ordinary differential equations. Try to compute a Lyapunov function of Sx u. 2: Consider the following perturbed system of (2. Theorem 2. An alternative to Lyapunov functions is contraction analysis. , 1992; Ott et Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. R. I If a rest point is found to be stable under the linearized dynamic, then it is linearly stable. If x e U is chain recurrent then for any 3 there is a ¿-chain beginning and ending at £23,588: EPSRC Small Grant EP/J014532/1, Project: Analysing Sensitivity in Basins of Attraction using a Local Contraction Criterion, August 2012-July 2013 Co-Proposer (PI Sigurdur Hafstein), Icelandic Research Fund, Project: Algorithms to compute Lyapunov functions, 2013-2016 Computation of the stochastic basin Of attraction by rigorous construction of a lyapunov function Discrete and Continuous Dynamical Systems-Series B 1 januari 2018 Andra författare averaged values of the corresponding short-time Lyapunov exponents from all trajectories comprising the snapshot. These models are hard to analyze using conventional Lyapunov function techniques. limt!¥ xt 2A is the basin of attraction of A. The mathematical definition of the quasi-potential is rather involved. The fourth subsection contains remarks about the non-convexity of the synchronous state’s basin of attraction in R ossler networks. Related results for inclusions of the type (11) can also be found in [24{26]. Rugonyi, K. (a)(10 points) Show this is a gradient system by determining the function V so that X0= r V (b)(10 points) Determine all equilibria of the system and classify their type. A new method to construct such a  Lyapunov function and relate it to these various stability notions. If 1 2 n are the Lyapunov exponents for a dynamical system in Rn and j is the largest integer for which 1+ 2+ + j 0, then the Kaplan-Yorke dimension is given by D KY = j + 1 + 2 + + j j j+1j (6) C. What does basin of attraction mean? Information and translations of basin of attraction in the most comprehensive dictionary definitions resource on the web. Packard, Peter Seiler, and Gary J. If we now use in the above theorem, we obtain the basin of attraction shown in Fig. , assuming a particular stability property The paper uses a quadratic Lyapunov function which certifies a region-of-attraction with Volume=30. Lyapunov Spectrum Lyapunov exponents of a dynamical system provide a quantitative measure of its sensitivity to initial conditions. is also a valid Lyapunov function for the nonlinear system over some region in the vicinity of the trajectory. Lyapunov function, stability, basin of attraction, dynamical system, contraction certain kind of stability, then there exists a Lyapunov function for the system that. (b) Use Maple to compute the invariant manifolds of the saddle xed points. Quasi Frequency. For linear time discrete systems there is a well known method, using the discrete Lyapunov equation, to compute a Lyapunov function for the system. Consider an equation X0 = F(X) and an equilibrium point X eq. (c)(10 points) For each sink determine the largest value R>0 so that all points at distance less than Rto the sink are in the basin of attraction of the sink. de Construction of Lyapunov Functions for TU Berlin, Department of Applied Mechanics, Chair of Mechatronics and the Estimation of Construction of Lyapunov Functions for the Estimation of Basins of Attraction Using normal form theory, every system can be transformed to a reduced one in the vicinity of a hopf bifurcation. Volume along the orbit of x contracts at an exponential rate. It is well known that if fhas an asymptotically stable xed point, then its basin of attraction U is an open and simply connected subset of the plane. The y-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. Lorenz Equations 0 2 4 6 8 10 Time 10-6 10-5 10-4 10-3 10-2 10-1 100 Separation lambda = 0. include plots of the attractor, Lyapunov exponents, attractor dimension, equilibria and their eigenvalues, bifurcations, and routes to chaos. The foot placement is controlled to ensure an exponential decay in the Lyapunov function. V doesn't fit with the global (non-linear) system and I can't find a linear system to graph. mat file are quadratic, pointwise-max of 2 quadratics, and quartic, certifying volumes 32. Lyapunov functions [26] are a natural way of analyzing the basin of attraction. Difference Equ. Additionally, Lyapunov methods suffer from conservatism Keywords: Ker ekj art o theorem, attractor, linearization, Lyapunov function 1 Introduction. Of course, it is a conservative estimate, meaning the basin of attraction cannot be overestimated. 951291370506 Figure 1. However  24 Feb 2016 Although the classical Lyapunov function method with the Invariance Principle could be a candidate, Basin of attraction for delayed dynamics. In §4 we set out the framework for the function spaces that are used to approximate the Lyapunov functions, as well as previous results on the approximation of Lyapunov functions when the right hand side of (2. lyapunov function basin of attraction

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