Just quickly before I continue...
Ed Lorenz had three differential equations...
dx/dt = o(y-x)
dy/dt = rx - y - xz
dz/dt = xy - bz

But what do they mean? Did they actually have something to do with the weather? And how did he work them out? (Note: the "o" should be a "sigma")

DIFFERENCE EQUATIONS

If you have a difference equation x_n+2 + ax_n+1 + bx_n = 0 is it always possible to write this as an ODE? Like x" + ax' + bx = 0.
Solving such an ODE would be hard!

You could substitute x = x_0 e^st. But where did the "s" come from? Is it a parameter that satisfies s^2 + as + b = 0; this coming from the auxillary equation from the ODE? If it is then the general solution is x = x_0 e^st.

From the auxillary equation this implies that there are two linearly independent solutions. With that in mind and knowing x = x_0 e^st how would you go about substituting this back into the difference equation???

I heard that you try x_n = x_0 L^n (where L is lambda).

But if you tried this then you would get x_0L^n+2 + ax_0L^n+1 + bx_0L^n = 0

The general solution is of the form x_n = Ar^2 + Br^2 where A and B are constants. I am getting the idea that you may treat difference equations like ODE's. Is this a correct assumption?

Say you wanted to solve x_n+2 - 3x_n+1 + 2x_n = 0 with initial conditions x_0 = 0 and x_1 = 1

The auxillary equation is L^2 - 3L + 2 = 0 which is a quadratic with roots L = 2 and L = 1.

So the general solution will be x_n = A.2^n + B.1^n
Using the initial conditions...
0 = A + B
1 = 2A + B } => A = 1 and B = -1

The solution is x_n = 2^n - 1

But finding the solution of a difference equation by the above method was also covered by the iterative method. Is both methods okay? Is the method outlined above more practical? I would think so...

THE LOGISTIC EQUATION

The general form of the logistic equation is x_n+1 = ax_n(1-x_n). I have seen this too, as a 1st order differential equation! But not as a difference equation.

Looking at this we would expect x to be positive since this equation can govern population and we cannot have negative population. This restricts x_n to [0,1] If x_n is greater than 1 then x_n+1 is negative and we do not want that!

I would like to know how you get the fact that it has a maximum value of x = 1/2 resulting in the fact that if a > 4 then x_n+1 is negative again.


FIXED POINTS

What is a fixed point? Is it that if we follow say the logistic equation above through each iterate x_n+1 -> x_n -> x_n-1 ...and they all equal each other then does the equation map to a fixed point X.

So if it tends to X under iteration we say that the fixed point X is attracting. and if it moves away from X then it is repelling.

So it would be efficient to find all the attractors of a nonlinear system?

Finally, could someone explain why for a logistic map, the fixed points satsify X = f(X) = aX(1-X) and that when you solve it there is one fixed point for a<1 and two for a>1 and how can you determine whether these are attractors or repellors?

Thanks.

Edward Lorenz was modeling a simplified version of the weather. It takes into account the fact that warm air rises, causing a low pressure area, causing air motion. He had done that previously for a plane but his famous model was done on a rotating sphere, including coriolis effects.

DIFFERENCE EQUATIONS
For linear difference equations, yes, the "characteristic equation" method works nicely, just as it does for differential equations.

For linear equations with constant coefficients, it is simpler to use the characteristic equation than to use "iteration", again just like differential equations.

"What is a fixed point? Is it that if we follow say the logistic equation above through each iterate x_n+1 -> x_n -> x_n-1 ...and they all equal each other then does the equation map to a fixed point X."

A fixed point for a set of differential equations is a point r* = (x,y,z) where dx/dt = dy/dt = dz/dt = 0. It is fixed in the sense that the velocity is zero; once you are in this point, you're never getting out again.

"So if it tends to X under iteration we say that the fixed point X is attracting. and if it moves away from X then it is repelling."

That is correct, there are stable fixed points and unstable fixed points. The stability can be determined by linearizing the set of differential equations around the fixpoint. This always yields exponential factors: linearizition gives you equations of the form dx/dt = &alpha;*x => x = exp(&alpha;t). If &alpha; < 0 then the fixed point is stable (close to the fixpoint your differential equation drives it back to the fixpoint) for &alpha; > 0 it is unstable.

"So it would be efficient to find all the attractors of a nonlinear system?"

That depends on what you want to do, and on the initial conditions of the system.

First of all, you assume that your differential equations (dynamics) are such that they are capable of travelling through the entire phasespace, i.e. that you can end up in a fixed point if one exists. This assumption is refered to as ergodicity. But this is not always true, for example, consider the (undamped) harmonic oscillator: there is a fixed point (&Theta; , v) = 0 but you never reach it unless you start with v = 0 in &Theta; = 0.

In some cases it will be sufficient to find the attractors of the nonlinear system, mostly people assume that "for t -> oo , you end up in a fixed point".

Bye!

Crisp

Thankyou guys for your feedback. As usual, very informative.

for example, consider the (undamped) harmonic oscillator: there is a fixed point (È , v) = 0 but you never reach it unless you start with v = 0 in È = 0.

Is this related to a pendulum which can start pointing straight up. Which is of course the position which you can never reach once you begin the pendulum swinging? (unless you had a damped pendulum) Can you say that position is a fixed point which is unstable and the position of the pendulum hanging straight down is also a fixed point but is stable?

My understanding of a fixed point is not very good. I have been playing around with some cobweb plots. Supposedly it should be showing me that for certain values of a there are fixed points. For example, for 0 < a < 1 the cobweb moves straight to X = (0, 0) and stays there and "never gets out" - to quote Crisp. Is this a stable fixed point?

Then if you have a > 1 the cobweb can tend to X = (0, 0) but this is unstable. However it has another fixed point at X = (1/a) which is stable for 1 < a < 3. Then loses stability again at a = 3! What does all this mean?

At a = 1 there is no fixed points, it keeps going around and around and never re-traced the same path. Is this chaotic behavour?

MORE ON COBWEB PLOTS

If you keep iterating x_n+1 = f(x_n) in the cobweb plot we know that it will never leave [0, 1]. Above I found that it loses stability at a = 3. However just above a = 3, like a = 3.001, the plot does eventually repeat itself (for different values of delta). Is this repeating property tied in with fixed points at all? I mean, if you look at it in this case it does not converge to a fixed point but rather keeps looping around the same path forever. Not exactly a fixed point as above but is it?

Hi oxymoron,

"Is this related to a pendulum which can start pointing straight up. Which is of course the position which you can never reach once you begin the pendulum swinging? (unless you had a damped pendulum) Can you say that position is a fixed point which is unstable and the position of the pendulum hanging straight down is also a fixed point but is stable?"

/me almost chokes on his coffee...

Ofcourse! How could I have forgotten to mention this to clarify the difference between stable and unstable attractors.


Concerning the cobweb plots; I have never heard this term before (but I have never taken a course on non-linear dynamics in English either ;)). I assume it is something like a phaseplot ? If not, open up Maple or Mathematica, and try plotting a "phaseplot" of a set of non-linear differential equations. On these plots you can see the expected motion when starting in a given initial conditions (by simply plotting the velocity vector in every point). This immediatelly shows you stable and unstable fixed points.


To complicate it even further you can have fixed points which are attracting in one direction, but repelling in another. If I remember correctly these are referred to as "saddle points". Anyway, about every decent phaseplot should explain this way better than I do in words.

Bye!

Crisp

Concerning the cobweb plots; I have never heard this term before (but I have never taken a course on non-linear dynamics in English either

You have in another language?? And yes Cobweb Plot <=> Phase Plot

You may be wondering why I am asking all these questions. I have a keen interest in atmospheric dynamics and would dearly enjoy taking a similar course that my university offers. However I have to wait until my 3rd year. So over my summer break I am doing some personal work on nonlinear systems - which inherently leads to chaos (which is a lot of fun!)

So if anyone can help me with my questions it would be much appreciated.

I have also been reading - James Gleick - "Chaos - Making a New Science" which is very helpful but unfortunately does not explain all of the concepts. In the fourth chapter - "A geometry of nature" - he introduces Mandelbrot and his precise definition of dimension.

Gleick writes, "...for the Koch curve, the infinitely extended multiplication by four-thirds gives a dimension of 1.2618..."

"Whoa!", I said. Where did this number come from? Fractional dimensions is something that I have never come across! A few hours later I did eventually find out where this number came from. I think. This is why I am asking for help.

So I found a technique called "box-counting". It says, "Suppose it takes N(E) "objects" of characteristic size E to "cover" a self-similar set at generation n. Then if N(E) "scales" like E^-D. Then...

D = log(N(E))/log(E)

I am having a hard time trying to work out exactly what N(E) and E are going to be. Considering that the Koch Curve is constructed by starting with a unit equilateral triangle then placing smaller triangles at the middle of each side each one-third the size and so on...

I worked out the following information (please check this may be wrong!)
The length of the boundary is 3 x 4/3 x 4/3 x 4/3 x ... oo

So after the nth step the length is 3(4/3)^n (for the unit triangle) ie. the perimeter of the Koch Curve grows by a factor of 4/3. When repeated an infinite number of times the perimeter becomes infinitely long. I am guessing that I am using a scaling factor of 3 here. Is this a correct assumption?? Because 1 -> 4/3 -> 16/9 -> 64/27 ...

So if scale = 3 and we are replacing every side of the triangle by 4 new sides then 3^D = 4 Hence D = log(4)/log(3) = 1.261859507...

Say we had a square. It's side length is squared to get it's area. So N = L^2/E^2 and E = E So D = log(L/E)^2/log(E) = {2log(L) - 2log(E)}/log(E) = 2log(L) = 2. Does this make sense? It doesn't to me!! Please help....

It surely looks convincing, but I have absolutely no idea what to say on the whole fractal thing (you should not overestimate the words "I had a course in non-linear dynamics" ;)).

Bye!

Crisp