(You'd better add some friction in order to get an attractor, but I
assume you know that already.)

There is this semi-obvious generalization of your two latest attempts:

dot(x)=v
dot(v)=-x-s
dot(s)=c
dot(c)=-s

Except that, by being four-dimensional, it violates your second
requirement. But I doubt that you can do much better, since it's
awfully hard to embed a circle in a line. (Which argument does not
constitute a proof that you can't do better, mind you: There is extra
room along the x and v axes, though I don't see how you can utilize
that room.)

There is this semi-obvious generalization of your two latest attempts:

dot(x)=v
dot(v)=-x-s
dot(s)=c
dot(c)=-s

Except that, by being four-dimensional, it violates your second
requirement. But I doubt that you can do much better, since it's
awfully hard to embed a circle in a line. (Which argument does not
constitute a proof that you can't do better, mind you: There is extra
room along the x and v axes, though I don't see how you can utilize
that room.)

By setting the scales correctly you can represent this as motion on a
torus in 3D. That's the classical way to think of the above system.

Hanche-Olsen is right, it is not an attractor as it is written, but is
that what really matters?

Dear group,

I want to rewrite the nonautonomous equation for the forced pendulum

ddot(x) + x = sin(t)

as an autonomous system of first order ODE's. My question
concerns the best
formulation for the forcing term, where best is defined as
- The solution is simple
- The result is a system of three first order ODE's.
- The system produces a nice "attractor" (the forced pendulum
is an example;
in reality I am interested in nonlinear systems).

The most naive approach

dot(x) = v
dot(v) = -x - sin(z)

dot(z) = 1

has the problem that it does not produce a classical
attractor, as z is a
linear increasing function.

On the other hand

dot(x) = v
dot(v) = -x - Im(z)
dot(z) = i z

does what I want but I would like to prevent imaginary numbers
(this would
only confuse the students). Lastly, the formulation

dot(x) = v
dot(v) = -x - z
dot(z) = sqrt(1-z^2)

formally works but does not work with a numerical
approximation due to the
sqrt.

- The system produces a nice "attractor" (the forced pendulum
is an example;
in reality I am interested in nonlinear systems).

There's no attractor here. The solutions all have
x^2 + v^2 -> infty as t -> (+/-) infty.

If you write p = exp(x) cos(t) and q = exp(x) sin(t),
then

dot(p) = v p - q
dot(q) = v q + p
dot(v) = - ln(p^2 + q^2)/2 + q/sqrt(p^2 + q^2)

It only fails at p=q=0.

ddot(x) + a dot(x) - x + x^3 = F cos(omega t).

Wil wrote:

I'm not sure what function ddot(x) and dot(x) are.

dot = derivative with respect to t
ddot = second derivative with respect to t

Dear Lou and Harald,

There is this semi-obvious generalization of your two latest attempts:

dot(x)=v
dot(v)=-x-s
dot(s)=c
dot(c)=-s

Except that, by being four-dimensional, it violates your second
requirement. But I doubt that you can do much better, since it's
awfully hard to embed a circle in a line. (Which argument does not
constitute a proof that you can't do better, mind you: There is extra
room along the x and v axes, though I don't see how you can utilize
that room.)

By setting the scales correctly you can represent this as motion on a
torus in 3D. That's the classical way to think of the above system.

It just seems a bit contrived to have to add two extra dimensions to the
system to represent a periodic forcing. This is for a course on Chaotic
Processes and we want to use this to demonstrate that a periodic forcing
changes the 2D system to a 3D system, so that chaotic motion becomes
possible. But perhaps I am trying to oversimplify the situation, and should
the system indeed be regarded as four-dimensional. Can you point me to some
examples of this approach in literature?

Robert Israel wrote:

- The system produces a nice "attractor" (the forced pendulum
is an example;
in reality I am interested in nonlinear systems).

There's no attractor here. The solutions all have
x^2 + v^2 -> infty as t -> (+/-) infty.

I was aware of that, I had taken the first second order system and added a
periodic forcing term. Perhaps I should have used the equations for the
forced double well oscillator

ddot(x) + a dot(x) - x + x^3 = F cos(omega t).

If you write p = exp(x) cos(t) and q = exp(x) sin(t),
then

dot(p) = v p - q
dot(q) = v q + p
dot(v) = - ln(p^2 + q^2)/2 + q/sqrt(p^2 + q^2)

It only fails at p=q=0.

I checked with Maple, and this is indeed consistent with the equations for
the forced pendulum. I do not understand how it works though. Can you
expand a bit on how you arrived at this expression? Would this approach
also work for a general forced second order equation ddot(x) + f(dot(x), x)
= cos(t)?

Hi,

ddot(x) + a dot(x) - x + x^3 = F cos(omega t).

I'm relatively new to the forum, and had just been following along.
This is a basic question... I'm not sure what function ddot(x) and
dot(x) are. Can anyone give a quick summary of what they are? At
least the name would help, so I can go and look it up. Thanks.

Wil

Robert Israel wrote:

- The system produces a nice "attractor" (the forced pendulum
is an example;
in reality I am interested in nonlinear systems).

There's no attractor here. The solutions all have
x^2 + v^2 -> infty as t -> (+/-) infty.

I was aware of that, I had taken the first second order system and added a
periodic forcing term. Perhaps I should have used the equations for the
forced double well oscillator

ddot(x) + a dot(x) - x + x^3 = F cos(omega t).

If you write p = exp(x) cos(t) and q = exp(x) sin(t),
then

dot(p) = v p - q
dot(q) = v q + p
dot(v) = - ln(p^2 + q^2)/2 + q/sqrt(p^2 + q^2)

It only fails at p=q=0.

I checked with Maple, and this is indeed consistent with the equations for
the forced pendulum. I do not understand how it works though. Can you
expand a bit on how you arrived at this expression? Would this approach
also work for a general forced second order equation ddot(x) + f(dot(x), x)
= cos(t)?

israel@math.ubc.ca wrote:
Wil wrote:

I'm not sure what function ddot(x) and dot(x) are.

dot = derivative with respect to t
ddot = second derivative with respect to t

One might also mention that the notation comes from TeX.


Martin

--
Quidquid latine scriptum sit, altum viditur.

Ahh, that clears it up a lot. thank you.

Wil

Martin Eisenberg wrote:
israel@math.ubc.ca wrote:
Wil wrote:

I'm not sure what function ddot(x) and dot(x) are.

dot = derivative with respect to t
ddot = second derivative with respect to t

One might also mention that the notation comes from TeX.


Martin

--
Quidquid latine scriptum sit, altum viditur.