# I How to get a Chaos Sequence from this Equation?

#### ohaited

Hi, I have this equation about Rossler system, describe as Eq. 1. Given that the chaotic behavior of the system for parameter values a=b=0.2 and c=5.7.

How can I calculate the chaotic sequence for this equation below.

The equation also referred from http://mathworld.wolfram.com/RoesslerAttractor.html

Your help is highly appreciated.

P/S: I'm a student that doesn't have any physics background. But I'm willing to learn more about chaotic behavior and system. Thanks!

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#### m4r35n357

How can I calculate the chaotic sequence for this equation below.
P/S: I'm a student that doesn't have any physics background. But I'm willing to learn more about chaotic behavior and system. Thanks!
Firstly, this is a mathematical problem, not a physics one. It is a system of "ordinary differential equations" or ODEs. To solve it you "just" need to integrate the equations on the right hand side.

The standard answer here would be to use a standard ODE solver in Octave, Sage, Scipy, whatever (which will probably user Runge-Kutta 4 - RK4 under the covers).

Here is a different answer - I have written open source software that will perform this calculation (as well as many other chaotic systems) here. Invoke the Python version as follows:

Code:
./tsm-mp.py rossler 16 10 0.01 1001 0.0 -6.78 0.02 .2 .2 5.7
This performs a simulation to 16 decimal places and 10th order Taylor approximation, with 1001 points starting from x=0.0, y = -6.78, z = 0.02, and parameter values a=b=0.2 and c=5.7.

The program is pure Python3 i.e. no external dependencies. Learning how the software works is even more interesting than the Rossler attractor IMO - It is far superior to RK4 ;)

Last edited:

#### ohaited

Firstly, this is a mathematical problem, not a physical one. It is a system of "ordinary differential equations" or ODEs. To solve it you "just" need to integrate the equations on the right hand side.

The standard answer here would be to use a standard ODE solver in Octave, Sage, Scipy, whatever (which will probably user Runge-Kutta 4 - RK4 under the covers).

Here is a different answer - I have written open source software that will perform this calculation (as well as many other chaotic systems) here. Invoke the Python version as follows:

Code:
./tsm-mp.py rossler 16 10 0.01 1001 0.0 -6.78 0.02 .2 .2 5.7
This performs a simulation to 16 decimal places and 10th order Taylor approximation, with 1001 points starting from x=0.0, y = -6.78, z = 0.02, and parameter values a=b=0.2 and c=5.7.

The program is pure Python3 i.e. no external dependencies. Learning how the software works is even more interesting than the Rossler attractor IMO - It is far superior to RK4 ;)

Hey, Thanks for responding! Actually, I want to create a cryptosystem using a chaotic map. So I choose this Rossler system to implement in cryptography field.

Okay,

1.) How to integrate by using Right Hand Side? Hee, I need some "kickstart" to refresh my knowledge in maths.

2.) RK4, Yeah I've tried that one by simulate an example I got from youtube : (Oh is it true how he demonstrate the RK4? Or do you have any references on RK4 for beginners)

3.) I'm sorry sir, but I'm a bit lost on your answer right after the code for phyton. (I'm sorry, but I really need to know about it)

Oh, so that I need to run the phyton script in your github, which one should I choose (Oh ya, I'm quite new with the github things)

Thank you sir, I hope you can consider my questions.

#### m4r35n357

Oh, so that I need to run the phyton script in your github, which one should I choose (Oh ya, I'm quite new with the github things)

Thank you sir, I hope you can consider my questions.
You will need to use git in a shell and do this:
Code:
git clone https://github.com/m4r35n357/ODE-Playground
Then go into the directory it makes, and type the command I gave above.

Then you can go and learn about crypto coding ;)

#### ohaited

./tsm-mp.py rossler 16 10 0.01 1001 0.0 -6.78 0.02 .2 .2 5.7
Oh wait, one question. What does those numbers after tsm-mp.py rossler do? I mean it represent which represent the variables in RK4 for rossler part?

#### Mark44

Mentor
How can I calculate the chaotic sequence for this equation below.
In post 1, your first equation is shown as $x = -y - z$, which is incorrect. From the website you gave, this equation should be $\dot x = -y - z$. The three equations are for the derivatives of x, y, and z, with respect to t.

BTW, the programming language is python (like the snake), not phyton.

#### m4r35n357

Oh wait, one question. What does those numbers after tsm-mp.py rossler do? I mean it represent which represent the variables in RK4 for rossler part?
Have you looked at the line you quoted back to me? I did miss one parameter, the step size is 0.01.

#### ohaited

Have you looked at the line you quoted back to me? I did miss one parameter, the step size is 0.01.
Oh, okay! Thanks. Now I get it. Yeah gotta do a little research for what those variables does. Thanks :)

#### ohaited

Have you looked at the line you quoted back to me? I did miss one parameter, the step size is 0.01.
Oh does all variables declared can be use in other system from your script?

#### ohaited

You will need to use git in a shell and do this:
Code:
git clone https://github.com/m4r35n357/ODE-Playground
Then go into the directory it makes, and type the command I gave above.

Then you can go and learn about crypto coding ;)

Sir, why does it shows in 4 columns with 10 starting point. as i initialise it with 10 starting point. What does those columns means?

#### ohaited

Here is a different answer - I have written open source software that will perform this calculation (as well as many other chaotic systems) here. Invoke the Python version as follows:

Code:
./tsm-mp.py rossler 16 10 0.01 1001 0.0 -6.78 0.02 .2 .2 5.7
This performs a simulation to 16 decimal places and 10th order Taylor approximation, with 1001 points starting from x=0.0, y = -6.78, z = 0.02, and parameter values a=b=0.2 and c=5.7.

The program is pure Python3 i.e. no external dependencies. Learning how the software works is even more interesting than the Rossler attractor IMO - It is far superior to RK4 ;)
Hey sir, I got this as output (in the picture below) I wanna ask, what's with the four columns. what does it represent? As I changed the 1001 points starting, to 10 points. Can you clear me up?

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#### m4r35n357

Hey sir, I got this as output (in the picture below) I wanna ask, what's with the four columns. what does it represent? As I changed the 1001 points starting, to 10 points. Can you clear me up?
That looks right. the first three numbers are x, y, z, and the fourth is time.

BTW looks like you have '.' on your path, you did not use './tsm-mp.py'!

#### ohaited

That looks right. the first three numbers are x, y, z, and the fourth is time.

BTW looks like you have '.' on your path, you did not use './tsm-mp.py'!
Oh okay then! That's the for the columns.

Oh btw, I've tried to execute using ./ but doesn't work. Look at this picture

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#### m4r35n357

Oh okay then! That's the for the columns.

Oh btw, I've tried to execute using ./ but doesn't work. Look at this picture
View attachment 235532
Ah, that is your operating system - don't worry about it then ;)

There are now some things you can do on a unixy os, but they depend on pipes. I don't know if cmd.exe can use them. You will also need matplotlib installed. Then firstly you can try:
Code:
./tsm-rossler-dbg 16 10 0.01 150001 0.0 -6.78 0.02 .2 .2 5.7 | ./plotAnimated.py 1 -20 30
without the './' I suppose. If the pipes work, you should see a progressive data plot of the x, y, z coordinates.

If this does not work, I would suggest you try this on a spare Linux box or virtual machine - there is really no excuse for not having one or the other these days IMO ;) It should also work on a mac.

#### m4r35n357

Not sure how to add a picture to an existing post, but this is what you should see.

x=green, y=yellow, z=cyan.

Once this is working you can think about plotting the trajectories in animated 3D, but you need either visual python or Pi3d to display them.

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#### ohaited

Not sure how to add a picture to an existing post, but this is what you should see.

x=green, y=yellow, z=cyan.

Once this is working you can think about plotting the trajectories in animated 3D, but you need either visual python or Pi3d to display them.

Hey man, I know this is quite sometimes but can you help me with this error below? Actually I just got it a few days ago.

P/S: I really don't know how to solve this. I've tried to install back again my Python but it doesn't make any changes. Really appreciate some help right here.

Thank you in advance :)

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#### m4r35n357

P/S: I really don't know how to solve this. I've tried to install back again my Python but it doesn't make any changes. Really appreciate some help right here.
Did you follow the instructions in the README for gmpy2? You need to set up a virtual environment, or install the dependencies yourself. Does this make sense?

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