# What is Lyapunov: Definition and 32 Discussions

Lyapunov is a Russian surname that is sometimes also romanized as Ljapunov, Liapunov or Ljapunow. Notable people with the surname include:

Prokopy Lyapunov (d. 1611), Russian statesman
Zakhary Lyapunov (d. after 1612), Russian statesman, Prokopy Lyapunov's brother
Mikhail Lyapunov (1820–1868), Russian astronomer
Aleksandr Lyapunov (1857–1918), son of Mikhail, Russian mathematician and mechanician, after whom the following are named:
Lyapunov dimension
Lyapunov equation
Lyapunov exponent
Lyapunov function
Lyapunov fractal
Lyapunov stability
Lyapunov's central limit theorem
Lyapunov time
Lyapunov vector
Lyapunov (crater)
Sergei Lyapunov (1859–1924), son of Mikhail, Russian composer
Boris Lyapunov (1862–1943), son of Mikhail, Russian expert in Slavic studies
Alexey Lyapunov (1911–1973), Russian mathematician

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1. ### A Does the Maximum Lyapunov exponent depend on the eigenvalues?

I am currently reading this paper where on page 8, the authors say that: This correlates with Figure 8 on page 12. Does it mean that there is a real correlation between eigenvalues and Lyapunov exponents?
2. ### I Lyapunov Stability for a nonlinear system

I am trying to understand attracting, Liapunov stable, asymptotically stable for given coupled system. I don't have any Liapunov function. Just two coupled systems such as ##\dot{x} = y##, ##\dot{y} = -4x## or sometimes normal systems ##\dot{x} = -x##, ##\dot{y} = -5y## How can I approach...
3. ### I How to check chaotic system using Lyapunov

Greetings! Hey, can anyone help me? I need an explanation how can Lyapunov help me to check the system weather it is chaotic or not. Let say I have this equation Rossler System Eq.(1) So how can you tell that the system have chaotic behavior or not? Does it depends on parameters? or from...
4. ### I Attractors of a time series

Hi, I am a beginner and I don't speak very well... So I'm really sorry for my poor scientific language... I work on 1-Dimension time series of a same system measured at different periods. In these periods, time series have different chaotic characteristics as their lyapunov exponent are...
5. ### A Lyapunov coefficient for function f(x) = 1/(1+x)

May I use function f(x) = 1/(1+x) to investigate chaos? I am trying to understand chaos using this function, but things are not going as I expected... Could you please advice me how I can calculate Lyapunov coefficient for this function?
6. ### I Understanding Lyapunov Exponent: Why Do We Use an Exponential Function?

Hey guys, I need your help. I am not sure if this is the right part of the forum to ask this question. So I started reading papers about the Lyapunov Exponent, but there is something I do not understand in the formula. Why ? It seems logical that we want because we want to get the Exponent...
7. ### I Variational Equations, Chaos Indicators

I work with an electromagnetic molecule trap, and I'd like to determine which orbits are chaotic. To this end, I intend to study the evolution of a perturbation on a trajectory with time. I'd like to compute something called the fast lyapunov indicator for various trajectories y(t), where I...
8. ### I Lyapunov coefficient of Lorenz System

Hi. I'm trying to get the Lyapunov coeficient for a Lorenz System (namely, a laser, using the Lorenz-Haken model) but I'm not getting the plots that would be expected. This is how two trajectories with near identical initial conditions behave (only one of the variables): And here's the...
9. ### Lyapunov exponents of a damped, driven harmonic oscillator

Homework Statement I am supposed to calculate Lyapunov exponent of a damped, driven harmonic oscillator given by ## \ddot{x} + 2\beta \dot{x} + \omega_0^2 x = fcos(\omega t)## Lyapunov exponent is ## \lambda ## in the equation ## \delta x(t) = \delta x_0 e^{\lambda t} ## The attempt at a...
10. ### Difference between Lyapunov and linear stability criteria

Dear all, Consider the connection of two electrical circuits. Both circuits, Z1 and Z2, are stable and only one of them is non-passive. I.e., the eigenvalues are located in the LHP but Re{Z2(jw)}<0 in a frequency range. For studying the closed-loop stability, you represent the linear system by...
11. ### Lyapunov exponent -- Numerical calculations

In computational physics is very often to calculate largest Lyapunov exponent. If largest Lyapunov exponent ##LE## is positive there is chaos in the system, if it is negative or zero there is no chaos in the system. But what can we say about some certain value of ##LE##. For example...
12. ### MHB Constructing Lyapunov function for system of ODEs

**Background:** I have been working on this problem for my research for months now, and I am in dire need of help. That is why I have come here to seek help. I have a system of nine ODEs that describe the dynamics of HIV and Tuberculosis co-infection in a population. The disease-free...

Homework Statement I can't seem to figure out how this next step of this derivation for equation 2.33 was produced. This is a graduate level textbook on Adaptive Backstepping.
14. ### MHB Interval of stability, Lyapunov exponent

For the function fc(x)= (6/x) + (x/2) -c, generate an estimate of the Lyapunov exponent for at least one c value chosen from each of the following intervals : (note 0 <= c <= 3) a) the interval of stability of the fixed point b) the interval of stability of the 2-cycle c) the interval of...
15. ### Energy of a pendulum (variable length, Lyapunov)

Hello, question about the energy of a variable length pendulum. Suppose you have a pendulum in the standard sense where θ is the angle, and we let the length r be a function of time r = r(t). What is the energy of the pendulum? So far, I have determined that kinetic energy is =...
16. ### Differental equation system and Lyapunov stability

Homework Statement Example: x'=y-x^3 y'=-x-y^3 Homework Equations The Attempt at a Solution Linear system x'=y y'=-x Is stable because Det(P-\lambdaE)=\lambda2+1 \lambda1,2=+-i So if I am not mistaken,than Ishould use Lyapunov stability,because the linear system is stable and I can't say...
17. ### Proving Lyapunov Stability for $$\dot{x} = Ax + B(t)x$$

A question I am doing hints that the solution (y,\dot{y}) = (0,0) of \ddot{y} - \frac{2}{t}\dot{y} + y = 0 is unstable. I believe (although I am not 100% sure) that is true however I am struggling to prove it. I can rewrite the equation as a system of equations in matrix form to get \dot{x} =...
18. ### Constructing Lyapunov functions using eigenvectors

Homework Statement How can I produce a Lyapunov function using the eigenvalues and vectors x'=-x+y y'=-x Homework Equations The Attempt at a Solution So I got the matrix using jacobian and I got the matrix -1 1 -1 0 then i found the eigenvalues to be λ_1= (-1+sqrt3...
19. ### Lyapunov Exponent: Calculate for Linear Map xn+1= rxn

Homework Statement Calculate the Lyapunov exponent for the linear map xn+1= rxn. Homework Equations λ = Lyapunov Exponent λ = \lim_{n \rightarrow \infty} \begin{bmatrix}\dfrac{1}{n} \sum_{i = 0}^{n - 1} ln|f'(x_i)| \end{bmatrix} The Attempt at a Solution f'(x) = r. λ =...
20. ### How Does the Lyapunov Equation Determine Matrix Stability?

The matrix \mathbf{B}satifies the following Lyapunov equation \begin{gathered}\mathbf{A}^{T}\mathbf{B}\end{gathered}+\mathbf{BA}=-\mathbf{I} prove that necessary and sufficient condition generating a symmetric and positive determined \mathbf{B}is that all of the eigen values of...
21. ### Show that the system has no closed orbits by finding a Lyapunov

Show that the system has no closed orbits by finding a Lyapunov ... Homework Statement I'm at the point in the problem where I need constants a and b satisfying ax2(y-x3) + by2(-x-y3) < 0 and ax2+bx2 > 0 for all (x,y)≠(0,0). Homework Equations Just in case you're wondering...
22. ### Lyapunov stability, mathematics vs reality

Homework Statement I have not been doing Lyapunov for a while and when doing an ordinary Lyapunov problem the other day, I ran into a funny situation. The correct way of doing it: \begin{align} \dot{e} &= \frac{1}{L}(u - R(e + x_{ref})) \\ V(e) &= \frac{1}{2}Le^2 \\ \dot{V} &= Le\dot{e} = Le...
23. ### Lyapunov function - stability analysis

Homework Statement State the strongest stability property of the system (stable, asymptotically/exponentially): \begin{align} \dot{x_1} &= x_2 \\ \dot{x_2} &= -x_1 e^{x_1 x_2} \end{align} Homework Equations With the Lyapunov function candidate: V(x) = \frac{1}{2}(x_1^2...
24. ### Lyapunov Function: Show 0 is Stable

Assume that f(0) = 0 and Df(0) has eigenvalues with negative real parts. Con- struct a Lyapunov function to show that 0 is asymptotically stable.
25. ### : Lyapunov Equation for backward continuous-time Kalman Filter

URGENT: Lyapunov Equation for backward continuous-time Kalman Filter Hi, Consider a continuous Kalman filter running backward in time as desired in a "two-filter" smoother. What would be the form of Lyapunov equation for this backward-time filter? Given a system: dx/dt = Fx + Gv, and...
26. ### Lyapunov and behavior of a linearized system

Homework Statement Examine the behavior in the neighborhood of the origin. Derive the given Lyapunov V(x,y). Show the region \frac{dV}{dt} = 0 and relate it to your analysis. Homework Equations \dot{x} = x^3 - xy \dot{y} = -y + y^2 + xy - x^3 As given by the text: V(x,y) = \frac{x^4}{4} +...
27. ### Lyapunov exponent - order of magnitude

Hello, Analyzing data from a chaotic pendulum, I calculated the Lyapunov exponent to be somewhere around 10^5 . While my gut tells me something is wrong with this number , i failed to find any information regarding the order of magnitude of Lyapunov exponents and their meaning. Can someone...
28. ### Why is the sum of Lyapunov exponents negative in dissipative systems?

Hi, the discussions in this forum have always been a great help to me as it seems there's always someone who's answered my question. However, this time I'm still puzzled. People often talk about the sum of the Lyapunov exponents of a dynamical system (i.e. adding the exponents from each...
29. ### How Do You Compute Lyapunov Exponents in MATLAB?

Homework Statement http://courses.ncsu.edu/py411/lec/001/hw6.pdf #1 part B specifically. The Attempt at a Solution For #1 part B I am confused. I have already done part A with MATLAB, and am having syntax issues with trying to compute the Lyapunov exponent(which is the last part of...
30. ### Solving Lyapunov Equation: Need Help Finding Q

I had a thread earlier on the notation but it hasn't really helped me progress on the question. I'm really confused on how to start this, am I expected to find Q myself? As in choose any random Q? I need the Q to maximise rho, and so I think I need to find a Q where q1 is as large as possible...
31. ### Lyapunov Theory: Find Unique Equil Pt.

Lyapunov Theory: Please Help! Homework Statement If the origin x=0 is globally asymptotically stable equilibrium point of the system then it must be the _________ equilibrium point of the system. Homework Equations None The Attempt at a Solution This is an objective/one word answer.
32. ### Nonlinear backstepping method and Lyapunov theory

Hi Everybody, I have just finished the control systems components of the engineering degree and found it very enjoyable. I'm now undertaking a holiday project to create a quad rotor aircraft. There is a surprising amount of information on these aircraft on the internet and while most of it is...