Chaos in new, simple, symmetrical 3D ODE
Arne Dehli Halvorsen
A new, simple attractor has been discovered which exhibits rotational
symmetry around the axis x, y, z
(This means that in its definition, x is to y and z
as y is to z and x
as z is to x and y)
The definition:
dx/dt = -ax-4y-4z-y*y
dy/dt = -ay-4z-4x-z*z
dz/dt = -az-4x-4y-x*x
(a from ~1.2 to ~3)
View this way :
px = x-0.5*(y+z)
py = 0.866*(y-z)
J. C. (Clint) Sprott has run an automated search for such examples,
and found a large number of such symmetrical attractors (see:
http://sprott.physics.wisc.edu/fractals/symmetry/
)
Has anyone else studied this class of ODEs?
Or the symmetrical class one would get by rotating the Lorenz equation
so that its repellent points are at: (0,0,0), (0,1,0), (0,0,1)?
Arne
Lou Pecora
In article <33019a72...@nntp.sn.no>,
Arne.Dehli...@computas.no (Arne Dehli Halvorsen) wrote:
> A new, simple attractor has been discovered which exhibits rotational
> symmetry around the axis x, y, z
>
> (This means that in its definition, x is to y and z
> as y is to z and x
> as z is to x and y)
>
> The definition:
>
> dx/dt = -ax-4y-4z-y*y
> dy/dt = -ay-4z-4x-z*z
> dz/dt = -az-4x-4y-x*x
>
> (a from ~1.2 to ~3)
Is this the same as the homo(hetro?) clinic orbit, for example in
Golubitsky's papers?
Lou Pecora
code 6343
Naval Research Lab
Washington DC 20375
USA
== My views are not those of the U.S. Navy. ==
== No spaming or solicitations -- both are illegal at this site.
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Arne Dehli Halvorsen
On 14 Feb 1997 12:22:31 GMT, pec...@zoltar.nrl.navy.mil (Lou Pecora)
wrote:
>In article <33019a72...@nntp.sn.no>,
>Arne.Dehli...@computas.no (Arne Dehli Halvorsen) wrote:
>
>> A new, simple attractor has been discovered which exhibits rotational
>> symmetry around the axis x, y, z
(snip)
>
>Is this the same as the homo(hetro?) clinic orbit, for example in
>Golubitsky's papers?
I don't know, and I could not decide on the basis of what I found by
searching the net for (+Golubitsky homoclinic heteroclinic) - what I
found was beyond my (low) level of mathematical sophistication.
The attractor that I found (by semirandom setting of coefficients) was
not designed to have any special mathematical properties beyond being
symmetric with respect to cyclical interchanges of x, y, z, and having
a simple set of equations - thus having symmetry over 120-degree
rotations (around the axis x=y=z) like the Lorenz attractor has
symmetry over 180-degree rotations around the axis (z=0).
If it has any special properties, I would love to hear about them...
Arne D Halvorsen
R J Morris
Arne Dehli Halvorsen wrote:
>
> I don't know, and I could not decide on the basis of what I found by
> searching the net for (+Golubitsky homoclinic heteroclinic) - what I
> found was beyond my (low) level of mathematical sophistication.
>
> The attractor that I found (by semirandom setting of coefficients) was
> not designed to have any special mathematical properties beyond being
> symmetric with respect to cyclical interchanges of x, y, z, and having
> a simple set of equations - thus having symmetry over 120-degree
> rotations (around the axis x=y=z) like the Lorenz attractor has
> symmetry over 180-degree rotations around the axis (z=0).
>
> If it has any special properties, I would love to hear about them...
>
--
Plants For A Future: more than just potatoes
http://www.scs.leeds.ac.uk/pfaf/index.html
Louis M. Pecora
In article <33081e68...@nntp.sn.no>, Arne.Dehli...@computas.no
(Arne Dehli Halvorsen) wrote:
> On 14 Feb 1997 12:22:31 GMT, pec...@zoltar.nrl.navy.mil (Lou Pecora)
> wrote:
>
> >In article <33019a72...@nntp.sn.no>,
> >Arne.Dehli...@computas.no (Arne Dehli Halvorsen) wrote:
> >
> >> A new, simple attractor has been discovered which exhibits rotational
> >> symmetry around the axis x, y, z
> (snip)
>
> >
> >Is this the same as the homo(hetro?) clinic orbit, for example in
> >Golubitsky's papers?
>
> I don't know, and I could not decide on the basis of what I found by
> searching the net for (+Golubitsky homoclinic heteroclinic) - what I
> found was beyond my (low) level of mathematical sophistication.
You'll probably have to do a journal search. Articles containing this
appeared within the last year or so.
--
Louis M. Pecora
pec...@zoltar.nrl.navy.mil
== My views and opinions are not those of the U.S. Navy. ==
== No Spamming or Soliciting -- both are illegal at this site ==
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John Bailey
In article <33019a72...@nntp.sn.no>,
>Arne.Dehli...@computas.no (Arne Dehli Halvorsen) wrote:
>
>> A new, simple attractor has been discovered which exhibits rotational
>> symmetry around the axis x, y, z
>>
>> as y is to z and x
>> as z is to x and y)
>>
>> The definition:
>>
>> dx/dt = -ax-4y-4z-y*y
>> dy/dt = -ay-4z-4x-z*z
>> dz/dt = -az-4x-4y-x*x
If you convert the three symmetrical differential equations to
an analogous difference equation form, you can get striking
chaotic behaviour with a spreadsheet (I used Microsoft Excel)
ODE form
> dx/dt = -ax-4y-4z-y*y
> dy/dt = -ay-4z-4x-z*z
> dz/dt = -az-4x-4y-x*x
> dz/dt = -az-4x-4y-x*x
Difference equation form
x(new)=Ax(old)+Bsum(old)+Cy*y
y(new)=Ay(old)+Bsum(old)+Cz*z
z(old)=Az(old)+Bsum(old)+Cx*x
sum(old)=x(old)+y(old)+z(old)
Typical values for constants: A= 1.03, B= -0.09, C= 1/1000
Plots and and the Excel can be found at
http://www.frontiernet.net/~jmb184/interests/chaos/
Really exciting to be able to play with totally new stuff!!!!
John
Arne Dehli Halvorsen
On Tue, 18 Feb 1997 17:21:45 +0000 (GMT), r...@scs.leeds.ac.uk (R J
Morris) wrote:
>Arne Dehli Halvorsen wrote:
>>
>> If it has any special properties, I would love to hear about them...
>>
>I presume its posible to construct atractors with 5-fold symmetry etc..
By the same means, I'm reasonably sure it can be done in 5 dimensions.
(same kind of equation,
v is to w,x,y,z
as w is to x,y,z,v
as x is to y,z,v,w.... you get the picture)
But can 5-fold symmetric chaos arise in such simple equations with
fewer dimensions?
That is not obvious to me, one way or another.
You could probably design 5-fold symmetric attractors even in 2 -space
ODEs , but they wouldn't be chaotic. I can't see how you could get
5-symmetric chaotic ODEs in 3-space, at least not with reasonably
simple equations.
Arne
hendrik richter
John Bailey wrote:
> ODE form
> > dx/dt = -ax-4y-4z-y*y
> > dy/dt = -ay-4z-4x-z*z
> > dz/dt = -az-4x-4y-x*x
> > dz/dt = -az-4x-4y-x*x
> Difference equation form
> x(new)=Ax(old)+Bsum(old)+Cy*y
> y(new)=Ay(old)+Bsum(old)+Cz*z
> z(old)=Az(old)+Bsum(old)+Cx*x
> sum(old)=x(old)+y(old)+z(old)
> Typical values for constants: A= 1.03, B= -0.09, C= 1/1000
>
> Plots and and the Excel can be found at
> http://www.frontiernet.net/~jmb184/interests/chaos/
>
> Really exciting to be able to play with totally new stuff!!!!
> John
I have checked the discrete-time system with the given constants for
Lyapunov exponents and it's indeed chaotic: The three LE are
lambda=(0.4848,-0.0072,-0.5115).
If you change the constants slightly to A=1.04, b=-0.09 and c=0.01 you
can even get an hyperchaotic system with two positive LE
lambda=(0.4973,0.0118,-0.5123).
Hendrik Richter
---------------
HTWK Leipzig
Depart. Electr. Engineering
Germany
email:hri...@rz.htwk-leipzig.de
PS: The third eq. should read z(new)=Az(old)+Bsum(old)+Cx*x instead of
z(old)=Az(old)+Bsum(old)+Cx*x
John Bailey
>I have checked the discrete-time system with the given constants for
>Lyapunov exponents and it's indeed chaotic: The three LE are
>lambda=(0.4848,-0.0072,-0.5115).
>If you change the constants slightly to A=1.04, b=-0.09 and c=0.01 you
>can even get an hyperchaotic system with two positive LE
>lambda=(0.4973,0.0118,-0.5123).
>
>
>Hendrik Richter
>---------------
>HTWK Leipzig
>Depart. Electr. Engineering
>Germany
An even simpler version of the symmetrical difference
equation based on symmetrical differential equation model
turns out to be:
x(new)= x(old)-(4*sum(old)-y(old)*y(old))/K
y(new)= y(old)-(4*sum(old)-z(old)*z(old))/K
z(new)= z(old)-(4*sum(old)-x(old)*x(old))/K
where sum(old) = x(old)+y(old)+z(old)
The value of K can range from 5.5 to 11.5
The following are Excel formulas which can be
iterated to produce successive values
of the three variables.
D2=4*SUM(A2:C2)
A3=A2-($D2-B2*B2)/$B$1
B3=B2-($D2-C2*C2)/$B$1
C3=C2-($D2-A2*A2)/$B$1
A three lobed scramble of orbits results.
John
Marc Lefranc
>>>>> "hendrik" == hendrik richter <hri...@et.htwk-leipzig.de> writes:
hendrik> I have checked the discrete-time system with the given
hendrik> constants for Lyapunov exponents and it's indeed chaotic:
hendrik> The three LE are lambda=(0.4848,-0.0072,-0.5115). If you
hendrik> change the constants slightly to A=1.04, b=-0.09 and
hendrik> c=0.01 you can even get an hyperchaotic system with two
hendrik> positive LE lambda=(0.4973,0.0118,-0.5123).
The second exponent should be zero (it corresponds to the direction of
the vector field). The fact that it is positive in one case,
negative in the other is merely due to statistical fluctuations.
--
_____________________________________________________________
Marc Lefranc, Charge de Recherche au CNRS
Laboratoire de Spectroscopie Hertzienne (URA CNRS 249)
Bat P5, UFR de Physique
Universite des Sciences et Technologies de Lille
F-59655 Villeneuve d'Ascq CEDEX (FRANCE)
e-mail: Marc.L...@univ-lille1.fr ; FAX : +33 (0)3 20 33 70 20
_____________________________________________________________
Hendrik Richter
Marc Lefranc wrote:
> The second exponent should be zero (it corresponds to the direction of
> the vector field). The fact that it is positive in one case,
> negative in the other is merely due to statistical fluctuations.
As I know just continous-time system in 3D have always one positive, one
zero and one negative Lyapunov exponent and this cannot be said about
discrete-time systems in 3D in general.
In addition, I don't know what 'statistical' fluctuation means since the
equations describing the system are purely deterministic. The issue that
has to be discussed is numerical reliability (due to round-off errors
ec.) and the convergence (since the LE are define for time to infinity)
of the used algorithm for calculating LE. The used algorithm
(calculating the evolution of the tangent vector and Gram-Schmidt
orthogonalization) is known to have a fast convergence and hence I
assumed that 30000 iterations of the system is sufficient large.
I will proof the calculations again in detail and maybe some other can
make some comments too.