This is the first of your videos that I’ve watched. It is very clear and easy to follow. I’ll back up and watch the series from the beginning. Excellent work!
at 2:23 you said it is quite easy to do this for discrete dynamical systems, my question does this same approach for computing the Lyapunov exponents works for continuous dynamical systems? Thanks
I follow this paper: www.sciencedirect.com/science/article/abs/pii/0167278985900119 There is associated MATLAB code that is easy to implement: uk.mathworks.com/matlabcentral/fileexchange/4628-calculation-lyapunov-exponents-for-ode
quick question, why take the limit as n -> infinity for the lyapunov exp? is it just to satisfy the equality of delta n = delta 0 e^lambda*n? because I thought there can be many lyapunov exponents for a given dynamical system.
The limit provides an average that gets rid of transients. Yes, there are many Lyapunov exponents for a dynamical system and this comes from the dependence on the initial condition. If you start on a fixed point you will get a different Lyapunov exponent than on a 2-cycle or on a transient orbit. Typically people refer to "the Lyapunov exponent" of a chaotic system as that of "almost every" initial condition, which is a transient orbit over the attractor. This is where the limit comes in as you get an average over the attractor (see ergodic theory).
Yes, just make t an autonomous variable. In continuous time you extend the system to x_1 ' = f(x_1,x_2) and x_2' = 1, where x_2 = t. You do the same basic idea for discrete time.
I follow this paper: www.sciencedirect.com/science/article/abs/pii/0167278985900119 There is associated MATLAB code that is easy to implement: uk.mathworks.com/matlabcentral/fileexchange/4628-calculation-lyapunov-exponents-for-ode
@@jasonbramburger I have tried the matlab code for my model but getting wrong output. For example, at a certain parameter value I have limit cycle therefore the lyapunov spectrum for my 3D model should be (0,-,-) right? However using the code i am getting (+,-,-).
This is the first of your videos that I’ve watched. It is very clear and easy to follow. I’ll back up and watch the series from the beginning. Excellent work!
Thanks for watching!
These videos are incredibly interesting, thanks for your work!
Glad you like them!
at 2:23 you said it is quite easy to do this for discrete dynamical systems, my question does this same approach for computing the Lyapunov exponents works for continuous dynamical systems? Thanks
I follow this paper: www.sciencedirect.com/science/article/abs/pii/0167278985900119
There is associated MATLAB code that is easy to implement: uk.mathworks.com/matlabcentral/fileexchange/4628-calculation-lyapunov-exponents-for-ode
brilliant continue sir👍
Very helpful, thx
What is the name of theorem that you mentioned at @5:31?
Oseledets Theorem: en.wikipedia.org/wiki/Oseledets_theorem
quick question, why take the limit as n -> infinity for the lyapunov exp? is it just to satisfy the equality of delta n = delta 0 e^lambda*n? because I thought there can be many lyapunov exponents for a given dynamical system.
The limit provides an average that gets rid of transients. Yes, there are many Lyapunov exponents for a dynamical system and this comes from the dependence on the initial condition. If you start on a fixed point you will get a different Lyapunov exponent than on a 2-cycle or on a transient orbit. Typically people refer to "the Lyapunov exponent" of a chaotic system as that of "almost every" initial condition, which is a transient orbit over the attractor. This is where the limit comes in as you get an average over the attractor (see ergodic theory).
thanks, this video was really helpful!@@jasonbramburger
Nice explanation! If we have a time series data point i.e. (t,x) where x is the signal and t is the time. Can we get the Lyapunov exponent for this ?
Yes, just make t an autonomous variable. In continuous time you extend the system to x_1 ' = f(x_1,x_2) and x_2' = 1, where x_2 = t. You do the same basic idea for discrete time.
How to find lyapunov exponent for 3D continuous model?
I follow this paper: www.sciencedirect.com/science/article/abs/pii/0167278985900119
There is associated MATLAB code that is easy to implement: uk.mathworks.com/matlabcentral/fileexchange/4628-calculation-lyapunov-exponents-for-ode
@@jasonbramburger thanks a lot
@@jasonbramburger I have tried the matlab code for my model but getting wrong output. For example, at a certain parameter value I have limit cycle therefore the lyapunov spectrum for my 3D model should be (0,-,-) right? However using the code i am getting (+,-,-).
What is the name of theorem that you mentioned at @5:31?