Question about third order for Polynomial Sprott

[mbharat]

Hello,

What is the order of the 60 coefficients? That is, I am assuming for the first dynamical equation, we have:

x' = P0 + P1x + P2x^2 + P3x^3 + P4xy + P5xz + P6x^2y + P7xy^2 + P8x^2z + P9xz^2 + P10y + P10y^2 + P11y^3 + P13yz + P14y^z + P15yz^2 + P16z + P17z^2 + P18z^3 + P19xyz

The other two equations should obviously have a similar format to the above (assuming it is correct).

Thanks.

Bharath

[Jaxan]

I don't understand your question...

but i think your formula is correct (but the term with P14 seem to have a typo...). This polynomial is also described in Julien Sprotts book: http://sprott.physics.wisc.edu/sa.htm
and yes the y and z formula's have the same structure

NB: the x' means "the new value of x" and not the deravative (ie. speed) of x. Not sure why the author of the chaoscope manual mixed up the conventional notation...

[mbharat]

I don't understand your question...

but i think your formula is correct (but the term with P14 seem to have a typo...). This polynomial is also described in Julien Sprotts book: http://sprott.physics.wisc.edu/sa.htm
and yes the y and z formula's have the same structure

NB: the x' means "the new value of x" and not the deravative (ie. speed) of x. Not sure why the author of the chaoscope manual mixed up the conventional notation...

Thanks for the response!

First, my question means is the third order polynomial represented by:

x' = P0 + P1x + P2x^2 + P3x^3 + P4xy + P5xz + P6x^2y + P7xy^2 + P8x^2z + P9xz^2 + P10y + P11y^2 + P12y^3 + P13yz + P14y^2 + P15yz^2 + P16z + P17z^2 + P18z^3 + P19*XYZ (1)

OR any other possible representation:

x' = P0 + P1x + P2x^2 + P3x^3 + P4xy + P5xz + P6x^2y + P7xy^2 + P8x^2z + P9xz^2 + P10*XYZ + P11y + P12y^2 + P13y^3 + P14yz + P15y^2 + P16yz^2 + P17z + P18z^2 + P19z^3 (2)

Note that I moved the XYZ term to the "middle" (P10) in (2). This completely changes the Pxx numbering. I couldn't find the full expression in Sprott's book but I made an educated-guess based on Sprott's discussion and wanted to make sure. Also, I fixed the typo in P14 . So, I will assume (1) is correct.

x' is defined as the the new value of x because you use a numerical method (like Runge-Kutta?) to approximate the derivative?

Anyway, my basic problem is that chaoscope claims there are no attractors for this system:

x'(t) = 28.0602*x(t) - 4.26471*x(t)^3 + 70.028*y(t)
y'(t) = 7.0028*x(t) - 7.0028*y(t) - 14.7059*z*(t)
z'(t) = 55*y(t)

But simulating this system in Mathematica gives rise to a nice attractor:

In[1] := chaosScopeSeconds =
NDSolve[{x'[t] == 28.0602*x[t] - 4.26471*(x[t])^3 + 70.028*y[t],
y'[t] == 7.0028*x[t] - 7.0028*y[t] - 14.7059*z[t],
z'[t] == 55*y[t], x[0] == 0.1, y[0] == 0.1, z[0] == 0}, {x, y,
z}, {t, 0, 50}, MaxSteps -> Infinity]
In[2] := ParametricPlot[
Evaluate[{x[t], z[t]} /. chaosScopeSeconds], {t, 0, 50},
PlotRange -> All, AspectRatio -> 1, AxesLabel -> {"x(t)", "z(t)"}]

I would like to use chaoscope to render the attractor, I just wanted to make sure that I got the order of the coefficients in the polynomial correct. Here is the chaoscope code:

info {

version "1.00"

author "Bharathwaj Muthuswamy"

date "01/13/2010"

}

attractor {

type polynomial_sprott

order 3

iterations 50

parameters <0,28.0602,0,-4.26471,0,0,0,0,0,0,70.028,0,0,0,0,0,0,
0,0,0,0,7.0028,0,0,0,0,0,0,0,0,-7.0028,0,0,0,
0,0,-14.7059,0,0,0,0,0,0,0,0,0,0,0,0,0,55,0,0,0,0,0,0,0,0,0>

}

view {

mode gas

width 640

height 480

scale 0.69931965895706

model_scale 1.70424196140688

origin <0,0,0>

translation <0,0,0>

rotation <-0.714026751930084, 0.586192342565187, -0.382811095771939, 2.1489955242846>

peak 90038.921875

gamma 0.2

contrast 4.51851851851852

gradient {

variable color

colors <0, 1, 1, 1,

1, 0, 0, 0>

}

}

I didn't adjust the scale, model_scale, rotation and peak values. I am hoping they don't affect the numerical simulation.

Many, many thanks again for looking into this! Hopefully I don't have any more typos.

Bart

[KayDekker]

Before you do too much head-scratching, I'd suggest that you might care to drop Clint Sprott a quick e-mail asking for clarification of the issue that you don't understand. His e-mail address is sprott@physics.wisc.edu and I think he'd probably be glad to be able to help you.

Or you could take a look at his web page to see if there's anything there that would be useful:
http://sprott.physics.wisc.edu/sprott.htm

[mbharat]

Before you do too much head-scratching, I'd suggest that you might care to drop Clint Sprott a quick e-mail asking for clarification of the issue that you don't understand. His e-mail address is sprott@physics.wisc.edu and I think he'd probably be glad to be able to help you.

Or you could take a look at his web page to see if there's anything there that would be useful:
http://sprott.physics.wisc.edu/sprott.htm

Many thanks for the suggestion, I do know Sprott but didn't thank about emailing him. Anyway, according to Sprott, the format of the equation(s) is actually in Appendix E of his book: http://sprott.physics.wisc.edu/fractals/booktext/ (which I missed while searching his book). So, the equations are:

Case J: D = 3, O = 3, M = 60
X = a1 + a2X + a3X2 + a4X3 + a5X2Y + a6X2Z + a7XY + a8XY2 + a9XYZ +
a10XZ + a11XZ2 + a12Y + a13Y2 + a14Y3 + a15Y2Z + a16YZ + a17YZ2 + a18Z +
a19Z2 + a20Z3

Y = a21 + a22X + a23X2 + a24X3 + a25X2Y + a26X2Z + a27XY + a28XY2 +
a29XYZ + a30XZ + a31XZ2 + a32Y + a33Y2 + a34Y3 + a35Y2Z + a36YZ + a37YZ2
+ a38Z + a39Z2 + a40Z3

Z = a41 + a42X + a43X2 + a44X3 + a45X2Y + a46X2Z + a47XY + a48XY2 +
a49XYZ + a50XZ + a51XZ2 + a52Y + a53Y2 + a54Y3 + a55Y2Z + a56YZ + a57YZ2
+ a58Z + a59Z2 + a60Z3

So, based on my system:
x'(t) = 28.0602*x(t) - 4.26471*x(t)^3 + 70.028*y(t)
y'(t) = 7.0028*x(t) - 7.0028*y(t) - 14.7059*z*(t)
z'(t) = 55*y(t)

the chaoscope project is:
info {
version "1.00"
author "Bharathwaj Muthuswamy"
date "03/05/2010"
}
attractor {
type polynomial_sprott
order 3
iterations 50
parameters <0,28.0602,0,-4.26471,0,0,0,0,0,0,0,70.028,0,0,0,0,0,
0,0,0, 0,7.0028,0,0,0,0,0,0,0,0,0,-7.0028,0,0,
0,0,0,-14.7059,0,0, 0,0,0,0,0,0,0,0,0,0,0,55,0,0,0,0,0,0,0,0>
}
view {
mode gas
width 640
height 480
scale 0.69931965895706
model_scale 1.70424196140688
origin <0,0,0>
translation <0,0,0>
rotation <-0.714026751930084, 0.586192342565187, -0.382811095771939, 2.1489955242846>
peak 90038.921875
gamma 0.2
contrast 4.51851851851852
gradient {
variable color
colors <0, 1, 1, 1,
1, 0, 0, 0>
}
}


But chaoscope still says there are no attractors. Is there a way I can specify a numerical integration method and initial conditions for the ODE instead of attractor search?

Thanks,

Bart