Lyapunov time This has profound implications for understanding the predictability and information scrambling properties of black holes. However, measures in terms of 2-foldings and 10-foldings are sometimes found, since they correspond to the loss of one bit of information or one digit of precision respectively. Apr 30, 2024 · Exercise 9 3 2 Plot the bifurcation diagram and the Lyapunov exponent of the following discrete-time dynamical system for r> 0: (9. Theory provides methods to calculate Lyapunov times, but the calculations can be unstable and producing a consistent, repeatable Lyapunov Theory for Discrete Time Systems This work contains a collection of Lyapunov related theorems for discrete time systems. nonuniform discrete domains or domains consisting of Oct 16, 2023 · The Lyapunov spectrum of recurrent neural networks is calculated and analytical approximations through random matrix theory are provided. It is defined as the inverse of a system's largest Lyapunov exponent. In this section, we provide a brief introduction to a few of the basic ideas and results in the context of linear Dec 16, 2020 · "By convention, it is defined as the time for the distance between nearby trajectories of the system to increase by a factor of e. The proposed framework broadly consists of a learner, which uses a gradient-based method for updating the parameters of the Lyapunov function and policy, and a verifier, which produces counterexamples (if . Its main purpose it to collect in a self contained document part of the Lyapunov theory in discrete time, since, in the literature, there does not seem to be a unique work which contains these results and their proof, apart from [2], which deals with discrete time The Lyapunov exponent is defined as the exponential rate at which nearby trajectories diverge. In this chapter the jdf=dxj > concept is generalized to higher dimensional maps and flows. Moreover, it will be shown that quadratic Lyapunov functions suffice. Lyapunov Stability The stability of solutions to ODEs was first put on a sound mathematical footing by Lya-punov circa 1890. 1. The Lyapunov time (the quantity reciprocal to the maximum Lyapunov exponent) characterizes the timescale of predictable dynamics. 3. 李雅普诺夫时间 （Lyapunov time）是一个 数学 用语，是指一个 动态系统 出现 混沌 特性所需要的时间 [1]。 李雅普诺夫时间的命名是为纪念 亚历山大·李亚普诺夫。 TheLyapunov time(the quantity reciprocal to the maximum Lyapunov exponent) characterizes the timescale of predictable dynamics. Its final statement for linear time invariant systems is elegant and easily tested using MATLAB Mar 20, 2017 · Below is the log distance of initially closely separated points against time in seconds for each system (obtained through simulation by slightly perturbing initial states and averaging over many orbits). In this article we discuss and characterize Lyapunov stability for linear systems. May 30, 2019 · In wikipedia text Lyaponov time is defined as inverse of the largest Lyapunov exponent. Stretching implies a positive Lyapunov exponent while A universal scaling behavior of the Lyapunov time with the nonintegrability strength is observed for the quasi-integrable regime. Sep 4, 2021 · Consider the continuous-time system (14. The Lyapunov exponents are asymptotic in time and aver-aged over an attractor. The signs of the spectrum of Lyapunov exponents helps de ne a dynamical system. M. In other words, the tajectory of the process Sep 19, 2003 · 5. For one dimensional maps the exponent is simply the average log over the dynamics (chapter 4). The main results and proofs are presented in … Nov 21, 2015 · Among other techniques (e. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. Then, employing state space reconstruction (delay coordinates), two approaches for estimating Lyapunov exponents from time series are presented Jan 1, 2022 · An overview of stability conditions in terms of the Lyapunov matrix for time-delay systems is presented. Our interest in Lyapunov equation stems from control and filtering applications rather than stability Lyapunov Equation & Linear Quadratic Regulator (LQR) Problem "For the continuous-time system, x Ax Bu ; find a linear feedback control u( t ) L x ( t ) that minimizes ( Lyapunov Stability Theorems Definition: Lyapunov Function A continuously differentiable function V(x) is called a Lyapunov Function for the following autonomous (or time-invariant) system if Feb 1, 2022 · Therefore, the Lyapunov time defined as T L ≡ μ − 1, is a local time-scale in phase space and represents the characteristic time of predictable dynamics in chaotic domains. 8. 1 Abstract Lyapunov and storage functions In general, Lyapunov functions are real-valued functions of system’s state which are mono tonically non-increasing on every signal from the system’s behavior set. In this paper, we have shown that using a new, simplified method, it is possible to estimate full spectrum of n Lyapunov exponents by integration of $$(n-1)$$ ( n - 1 ) perturbations May 1, 2019 · In this paper, we develop the Lyapunov–Razumikhin method to finite-time stability (FTS) and finite-time contractive stability (FTCS) of time-delay systems. And guess what? Anything that messes with the balanced system messes with this Lyapunov time thing too. The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. Then the theoretical results are applied to FTS and FTCS for a class of linear time-varying (LTV) time-delay system. 1) is asymptotically stable if and only if all the eigenvalues of A are in the open left half plane. ” It designates a period, when a certain process (physical, mechanical, quantum, or even biological) moves beyond the bounds of precise (or probabalistic) predictability and enters a chaotic mode. [2][3] There are many cases where a short Lyapunov time, TL, is associated with the body moving on a trajectory that crosses the orbit of a large perturber, i. We introduce a linear transfer operator-based Lyapunov measure as a new tool for stability verification of stochastic systems. [1][2] In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is where is a Hermitian matrix and is the conjugate transpose of , while the continuous-time Lyapunov equation is . 3 Lyapunov Stability of Linear Systems In this section we present the Lyapunov stability method specialized for the linear time invariant systems studied in this book. The Fermi-Pasta-Ulam-Tsingou-like lattices are exemplified by using the method, which agree well with molecular dynamical simulations for the cases of Lyapunov Time In modern physics, primarily exploring “very unbalanced conditions” and chaotic systems, there is a technical term - “Lyapunov time. Lyapunov exponents, which provide aqualitative andquantitative characterization ofdynamical behavior, are related to the exponentially fast divergence or convergence ofnearby orbits in phase space. Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by Converse theorems If A is stable, there exists a quadratic Lyapunov function V (z) = zT P z that proves it, i. Also, stability is defined at a time instant t0. (8. , what are now known as first and second Lyapunov methods), he introduced a new tool to analyze the stability of solutions of linear time-varying systems of differential equations, the so-called characteristic numbers, now commonly and appropriately called Lyapunov exponents. The dependency of attractor dimensions and entropy rates on coupling strength and input fluctuations is identified and a point symmetry of the Lyapunov spectrum is revealed. To open a window on such long-term behavior, we compute the entire Lyapunov spectrum of a forced 6 days ago · A Lyapunov function is a scalar function V (y) defined on a region D that is continuous, positive definite, V (y)>0 for all y!=0), and has continuous first-order partial derivatives at every point of D. Sep 4, 2021 · The idea behind Lyapunov's "direct" method is to establish properties of the equilibrium point (or, more generally, of the nonlinear system) by studying how certain carefully selected scalar functions of the state evolve as the system state evolves. For an n-dimensional system there are n such rates of divergence. Aleksandr Mikhailovich Lyapunov[a][b] (Алекса́ндр Миха́йлович Ляпуно́в, 6 June [O. General Properties of Lyapunov Exponents A measure of chaos (how sensitive to initial conditions?) Lyapunov exponent is a generalization of an eigenvalue Average the phase-space volume expansion along trajectory 2-D example: Circle of initial conditions evolves into an ellipse Area of ellipse: A = pd1d2 / 4 Where d1 = d0el1t is the major axis And d2 = d0el2t is the minor axis Magnitude and Jan 1, 2021 · The notion of stability that we discuss has been introduced in 1882 by the Russian mathematician A. First, by introducing the notions of uniformly exponentially stable and uniformly exponentially expanding functions, some Lyapunov differential inequalities based characterizations With these de nitions, we can now de ne a time-varying function that will indirectly allow us show that a system is stable. The time required for that to happen is called the “event time” (TE). Chapter 7 Lyapunov Exponents Lyapunov exponents tell us the rate of divergence of nearby trajectories—a key component of chaotic dynamics. The Lyapunov time of a dynamical system is a timescale after which it is unpredictable, i. 1) where f: [0,∞)× D→ Rn is piecewise continuous in t and locally Lipschitz in x on [0,∞)× D, and D⊂Rn is a domain that contains the equilibrium point of origin x=0. By convention, it is defined as the time for the distance between nearby trajectories of the system to increase by a factor of e. In particular, the theory is applied to Averaging Theory and Separation of time scales. More gener ally, stotage functions are real-valued functions of system’s state for which explicit upper bounds of increments are available. 3 ) over varying r. Sep 1, 2020 · A non-zero value of the maximum Lyapunov exponent typically indicates the chaotic type of motion, and the exponent equal to zero signals its regular type. Discrete-time results all linear quadratic Lyapunov results have discrete-time counterparts the discrete-time Lyapunov equation is AT P A − P + Q = 0 李雅普诺夫时间 （Lyapunov time）是一個 數學 用語，是指一個 動態系統 出現 混沌 特性所需要的時間 [1]。 李雅普诺夫时间的命名是為紀念 亞歷山大·李亞普諾夫。 Jan 1, 2021 · Lyapunov’s theory for characterizing and studying the stability of equilibrium points is presented for time-invariant and time-varying systems modeled by ordinary differential equations. The goal is to generalize and extend these types of equations and subsequent analysis to dynamical systems on domains other than R or Z, e. , all systems with constant and periodic coefficients are regular) and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically Lyapunov stable. Interestingly, the scaling exponent of the Lyapunov time is the same as the thermalization time, which indicates a proportional relationship between the two timescales. S. 1) x (t) = A x (t) We have already established that the system (14. Mar 24, 2024 · Here's where Mr. Theory provides methods to calculate Lyapunov times, but the calculations can be unstable and producing a consistent, repeatable Feb 1, 2022 · Therefore, the Lyapunov time defined as T L ≡ μ 1, is a local time-scale in phase space and represents the characteristic time of predictable dynamics in chaotic domains. The paper is organized as follows. Lyapunov in his doctoral thesis; hence it is often referred to as Lyapunov stability. A system with one or more positive Lyapunov exponents is defined to The RHS represents the time-average of the eigenvalue A. For most flows of practical importance, the FTLE varies as a function of space and time. A system with one or more positive Lyapunov exponents is defined tobe A Lyapunov function for an autonomous dynamical system { g : R n → R n y ˙ = g ( y ) {\displaystyle {\begin {cases}g:\mathbb {R} ^ {n}\to \mathbb {R} ^ {n}&\\ {\dot {y}}=g (y)\end {cases}}} with an equilibrium point at is a scalar function that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative is non positive (these conditions are Lyapunov stability is named after Aleksandr Mikhailovich Lyapunov, a Russian mathematician who defended the thesis The General Problem of Stability of Motion at Kharkov University (now VN Karazin Kharkiv National University) in 1892. Lyapunov Theory for Discrete Time Systems This work contains a collection of Lyapunov related theorems for discrete time systems. As the sphere is evolved, it can be stretched in some directions and squeezed in others. e, a trajectory that exhibits a large radial variation. " Okay, but in a Lorentz attractor, the distance between Sep 29, 2022 · The Lyapunov exponents 11, 12 of the dynamical system can be estimated from the time series, in addition to the critical transition 13, 14 and the sensing phase coherence 15. We revisit the canonical continuous-time and discrete-time matrix algebraic and ma-trix di erential equations that play a central role in Lyapunov based stability arguments. A nonsmooth Lyapunov-based analysis is used to prove semi-global asymptotic tracking of the desired trajectory. Imagine a patient with a serious head injury or a nasty chest infection—now the doctor's dealing with a system on the brink of chaos, with time running out fast. A consequence of this is that Aug 1, 2025 · This paper studies a Lyapunov characterization for prescribed-time stability, a novel set of prescribed performance functions, and investigates their … We propose a novel method to learn Lyapunov functions and stabilizing controllers, represented as neural networks (NNs) with ReLU activation functions, for discrete-time nonlinear systems. However in spatially extended systems, it is important to describe the spatial evolution as well. , chaotic. 1 An increase in the precision of initial condition 0 by factor 106 ) system only predictable for 2:5 times longer (assuming a tolerance which is 104 0) . [2][3] While See full list on web. Usually, it is challenging to analyze the asymptotic stability of time-varying systems because it is very difficult to find Lyapunov functions with negative definite derivatives We therefore extend the range for which the Lyapunov time was determined to N = 128k using reprehensive N -body solutions, allowing us to test the scaling of the Lyapunov timescale to large N in an actual self-gravitating system. Use lyap to solve the special and general forms of the Lyapunov equation. In this section we will show that this result can be inferred from Lyapunov theory. In particular the proofs developed for Averaging Theory and Separation of time scales departs from those typically used in continuous time systems that are based on twice differentiable change of variables and Kunihiro et al. Main bene t: By looking at how an energy-like function V (a so called Lyapunov function ) changes over time , we might conclude that a system is stable or asymptotically stable without solving the nonlinear di erential equation. [2] The Lyapunov time mirrors the limits of the predictability of the system. g. edu Lyapunov time explained In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. The real-time controller and adaptation laws enable the system to track a desired time-varying trajectory while compensating for the unknown drift dynamics and parameter uncertainties in the control effectiveness. By convention, it is defined as the time for the distance between nearby Jul 1, 1985 · We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Weather system: Lyapunov time (days) of same order as typical relevant time scale. Lyapunov's time comes into play. However, measures in terms of 2-foldings and 10-foldings are sometimes found, since they correspond to the loss of one bit of information or one digit of precision. A universal scaling behavior of the Lyapunov time with the nonintegrability strength is observed for the quasi-integrable regime. The Lyapunov exponent is the timescale's inverse. 6 days ago · Then, a novel adaptive asymmetric time-varying integral barrier Lyapunov control scheme is developed to design the steering controller, longitudinal controller, and yaw moment controller, which is capable of constraining the system states and control signals within the predefined boundary. 1, showing the Lyapunov exponent of Eq. [1] Use The Lyapunov time mirrors the limits of the predictability of the system. While the computation of Lyapunov exponents for given dynamical equations is straight forward, their estimation from time series remains a delicate task. In this page, the Lyapunov exponent is applied to an equation that jumps between stability and instability, between chaos and order — the logistic equation. Lyaponov exponents are Before focusing on the finite time, let us first realize that the basic (infinite-time) Lyapunov analysis can also be applied to time-varying systems: x = f (t, x) We can analyze the stability (local, regional, or global) of this system with very little change. - sreguyal/lyapunov-exponent-estimate May 1, 2025 · For evaluating the efficiency of the proposed Lyapunov optimization algorithms, in addition to changing the control parameters, such as queue weight and penalty factor, and evaluating their effect on the system performance, the results of the Lyapunov-based real-time optimization algorithms are also compared with the numerical results of Abstract. Weaker set-theoretic notion of almost everywhere stochastic stability is introduced and verified, using Lyapunov measure-based Main idea Lyapunov formalized the idea: If the total energy is dissipated, then the system must be sta ble. The Lyapunov time mirrors the limits of the predictability of the system. Wepresent the first algorithms thatllow the estimation of non-negative Lyapunov exponents from an experimental time series. Sep 26, 2003 · Lecture 7: Finding Lyapunov Functions1 This lecture gives an introduction into basic methods for finding Lyapunov functions and storage functions for given dynamical systems. Such methods entail approximating the Reversing the time, we can introduce the scaled Lyapunov exponents for negative time. Use The Lyapunov time mirrors the limits of the predictability of the system. Jan 1, 2024 · The theory of distributed-order Lyapunov stability for both continuous-time and discrete-time systems has undergone improvements, and its applicability has been expanded to more general functions and systems, thereby broadening its range of use. The Lyapunov time reflects the limits of the predictability of the system. In this Chapter, we consider basic analytical methods how Lyapunov timescales can be estimated. Then explain in words how its dynamics change over r. , there exists P > 0, Q > 0 that satis es the (continuous- or discrete-time) Lyapunov equation. We show that the maximal exponent forms geometrical structures in input space, akin to coherent structures in dynamical systems Feb 1, 2025 · In this paper, we investigate the fixed-time stability and synchronization issues of stochastic impulsive delay complex networks (SICNs) with delayed … where T is a constant delay time. The largest Lyapunov exponent is as follows [2] Lyapunov proved that if the system of the first approximation is regular (e. In this review article, we first revisit the computation of the Lyapunov spectrum using model equations. We insist that for a uniformly stable equilibrium point x∗ Sep 14, 2018 · In this work, we present the equivalent of many theorems available for continuous time systems. Feb 1, 2024 · We compute how small input perturbations affect the output of deep neural networks, exploring an analogy between deep feed-forward networks and dynamical systems, where the growth or decay of local perturbations is characterized by finite-time Lyapunov exponents. The method has more theoretical importance than practical value and can be used to derive and prove other stability results. On the contrary, finite-time Lyapunov exponents have been proposed to investigate finite-time behav-iors on a chaotic attractor [22,23]. 4. He was the son of the astronomer Mikhail Lyapunov and the brother of the pianist and composer Sergei Lyapunov. For typical nonlinear functions F, analytic solutions are impossible especially in the case where the dynamics are chaotic, and hence numerical methods must be used. This equation is infinite-dimensional in the sense that a continuum of initial conditions over the interval - T < t < 0 is required to specify the behavior. Use The Lyapunov time mirrors the limits of the predictability of the system. Lyapunov Exponents The average rate of divergence or convergence of nearby orbits in space (Wolf 1985). e. Uniform stability is a concept which guarantees that the equilibrium point is not losing stability. The planets' orbits are chaotic over longer time scales, in such a way that the whole Solar System possesses a Lyapunov time in the range of 2~230 million years. I have some difficulties with the units. [1] A. The Lyapunov exponent is a number that measures stability. Explore the intricacies of Finite-time Lyapunov exponent and its role in analyzing complex dynamical systems, revealing patterns and predictability. In some cases, the orbits themselves may change dramatically. Such chaos manifests Umesh Vaidya, Senior Member, IEEE, Abstract—In this paper, we study the stability problem of a stochastic, nonlinear, discrete-time system. Indeed, a chaotic trajectory of any dynamical system cannot be determined on timescales much larger than its Lyapunov time. Jul 1, 1985 · We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. 8) for a finite time Jacobian matrix as the ‘intermediate Lyapunov exponent’, “averaged” over a finite time period. Ulrich Parlitz AbstractLyapunov exponents are important statistics for quantifying stability and deterministic chaos in dynamical systems. Compare this with Fig. They have been also known as local Lyapunov Nov 1, 2014 · The beginning of the 21st century can be characterized as the “time-delay boom” leading to numerous important results. Several Lyapunov-based sufficient conditions for establishing these FTS properties are obtained. In particular, it does not require that trajectories starting close to the origin tend to the origin asymptotically. Lyapunov exponents (lambdas) indicated in the image are obtained through linear regression of the linear parts of the curves. 7) x t = c o s 2 (r x t 1) Figure 9 3 1: Visual output of Code 9. , Lyapunov functions with negative definite derivative), assessing global uniform asymptotic stability 4. Sep 1, 2021 · Abstract This paper studies exponential stability and stabilization of linear time-varying systems without assuming that the coefficient matrix is bounded, which is generally necessary in the existing literature. Lyapunov stability is a very mild requirement on equilibrium points. Oct 23, 2024 · Linear time-varying differential equations that arise in the study of model reference adaptive identification problems are studied in both the continuous-time and the discrete-time frameworks. 25 May] 1857 – 3 November 1918) was a Russian mathematician, mechanician and physicist. 3 The Finite-Time Lyapunov Exponent The finite-time Lyapunov exponent, FTLE, which we will denote by , is a scalar value which characterizes the amount of stretching about the trajectory of point over the time interval [t, t + T]. It is named after the Russian mathematician Aleksandr Lyapunov. Its main purpose it to collect in a self contained document part of the Lyapunov theory in discrete time, since, in the literature, there does not seem to be a unique work which contains these results and their proof, apart from [2], which deals with discrete time May 3, 2023 · Numerical integrations of the Solar System reveal a remarkable stability of the orbits of the inner planets over billions of years, in spite of their chaotic variations characterized by a Lyapunov time of only 5 million years and the lack of integrals of motion able to constrain their dynamics. [3] In all cases, this means that the positions of individual planets along their orbits ultimately become impossible to predict with any certainty. In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. Lyapunov is known for his development of the stability theory of a dynamical Kunihiro et al. Lyapunov Stability Theorems For Non-autonomous (or Time-Varying) Systems Consider the non-autonomous system (L. We can nd the largest Lyapunov exponent = lim lim t!1 Z0!0 t Z(t) ln Z0 Calculating Lyapunov exponents takes a lot of time and computing power. The emphasis in this Introduction to TDSs is on the Lyapunov-based analysis and design of time-delay and sampled-data systems. [12] call the eigenvalues of stability matrix (4. The function Vn : X ! R is a Lyapunov function for a discrete time system if the following conditions hold: Oct 30, 2013 · Lyapunov exponents have been introduced with the goal of characterizing the time evolution of perturbations of lumped dynamical systems. M. The main result of this work is summarized as follows: If a time-varying, stochastic, system admits a stochastic Lyapunov function then the origin is uniformly asymptotically stable in probability under three di erent de nitions (A classical version, an equivalent KL version, and a slightly weaker KL version). To see this, picture a sphere evolving over time through some given set of equations. 3), evaluated at a given instant in time, the ‘local Lyapunov exponents’, and they refer to the set of stability exponents (4. We open by defining local Lyapunov stability for time-invariant systems of the form ̇x(t) = f(x(t)) where we show that the existence of a Lyapunov function or what we sometimes called a Lyapunov stability certificate is sufficient for the stability of the system’s equilibrium point. stanford. By convention, it is defined as the time for the The Lyapunov time of a dynamical system is a timescale after which it is unpredictable, i. Lyapunov bound on reachable set Lyapunov arguments can be used to bound reachable sets of nonlinear or time-varying systems suppose there is a V : Rn → R and a > 0 such that ̇V (z, w) ≤ −a whenever V (z) = b and w ∈ U and define C = {z | V (z) ≤ b} 李雅普诺夫时间（Lyapunov time）是一個數學用語，是指一個動態系統出現混沌特性所需要的時間。李雅普诺夫时间表示系統可預測性的極限。依一般慣例，李雅普诺夫时间大約是指系統軌跡不沿著指數函數軌跡，開始發散的時間。 Apr 19, 2021 · Based on the geometrization of dynamics and self-consistent phonon theory, we develop an analytical approach to derive the Lyapunov time, the reciprocal of the largest Lyapunov exponent, for general nonlinear lattices of coupled oscillators. This theory still dominates modern notions of stability, and provides the foundation upon which alternative notions of stability continue to be built. The efficiency of the 4 days ago · In the context of black holes, a positive Lyapunov exponent signifies chaotic behavior, indicating that even infinitesimally small differences in initial conditions can lead to vastly different outcomes over time. The above result provides a basis to study Lyapunov-Perron regularity for scaled Lyapunov exponents. Within astronomy, systems of orbiting bodies (such as the Earth and the Moon) are subject to such a timescale. Mar 4, 2016 · Positive Lyapunov exponents indicate exponential divergence of neighbouring trajectories and are the most important attribute of chaotic attractors. Chaotic electric circuits (milliseconds) Strogatz Example 9. We assume that for su ciently long times this average is equivalent to an average of A for all possible ows evaluated at the same time. This MATLAB function estimates the Lyapunov exponent of the uniformly sampled time-domain signal X using sampling frequency fs. Estimating largest Lyapunov exponent of time series data using the method of Wolf et al. The original contribution of the article is to present two new strong Lyapunov functions (e. The finite-time Lyapunov exponents are exponential growth rates of small perturbations on the attractor for finite-time intervals. By convention, it is defined as the time for the distance between nearby Sep 12, 2018 · Classical method of Lyapunov exponents spectrum estimation for a n-th-order continuous-time, smooth dynamical system involves Gram–Schmidt orthonormalization and calculations of perturbations lengths logarithms. wiwl fymzk klui nkban huzaj lws reizai chjjhdys zqqh kyltm agupam upx nnzzcw gvhe urkxfor