Differential Equation

Nice one ... and not an ∫ in sight !

I try to avoid those glorified S's as much as possible.

Come now, Roger. It's just a fancy 'plus' sign! And it sure is preferable to that jagged capital sigma any day, don't you think? And if that weren't enough, consider those shapely curves! No discontinuities there. No sir! Differentiable all the way.

That pretty much sums it up in my (math) book.

Continuity is overated. Fractals aren't continuous and they are just as interesting (if not more). Sigma gets a bad rap, I mean, no one ever complains about Pi (big pi, not little pi). Continuous curves are well fit though. I guess what I'm trying to say is that I have too much free time.

Lorenz is on the other line and is saddened by the fact that you don't seem to find his curvaceous fractals 'interesting.' He reminded me to tell you that fractals need not be discontinuous functions to be fractals. Shall I tell him you prefer angular women as well, and that it's nothing personal?

Hmmm......I didn't know that, though a quick google search confirms what you say. If you have any more information on this I'd appreciate it, I'm interested.

I've contacted Mandlebrot and apologized, I couldn't reach Lorenz because he's moved on to a better place where butterflies can flap their wings free from guilt. I do like sharp features, it certainly wasn't personal.

Non-linear differential equations figure prominently in fractal geometry. Lorenz fractals, for example, derive from three interdependent NLDEs:

These are Lorenz' equations for his famous strange attractor and are basically simplified Navier-Stokes' fluid-dynamics equations. Typical values for the coefficients are P = 10, R = 28, B = 2.667.

Discontinuous functions don't figure as prominently in fractal geometry as you might expect. What does figure prominently usually has these characteristics:

• Cause and effect are generally not proportional and may differ by many orders of magnitude.
• Marked sensitivity to initial conditions. In Lorenz' original climate model a temperature difference of a few millidegrees resulted in major climatic changes several 'model months' downstream (see figure below).
• Nonlinearity.

Lorenz' equations are in three variables with the result that his familiar 'strange attractor' is really a three-dimensional object.

Shown below as a locus of discrete points (little copper balls) to give the viewer a better sense of depth ...

Thanks Europium, I need to think about this a bit to get caught up. I guess my confusion lies in my understanding of fractals. I was under the impression that fractals were named so because they had fractional dimensions. For instance:

Cantor Dust (less than one dimesion):

Cantor Dust (between one and two dimensions):

Cantor Dust (between two and three dimensions)

It seems to me that a continuous curve would be solidly entrenched in a particular dimension (1, 2, 3, etc.) by definition. Then again, I have never read that a fractal couldn't exist at whole number dimesions, I think I just assumed it. It actually would make more sense if they could. Is this line of reasoning ok or am I missing something crucial here? (Sorry to turn a fun conversation serious)

Bifurcation and Lyapunov fractals do not have fractional dimension, for instance, but they are called 'fractals' nonetheless. This is probably due to the gradual generalization of the term to mean a geometric representation of any chaotic system.

But in a strictly canonical sense, fractals are systems that are nowhere differentiable. They are, in a sense, 'fractured.' It is in this sense that Mandelbrot coined the term. It so happens that the Mandelbrot Set has fractional dimension, and it is probably because of this that the popular press associated 'fractal' with 'fractional dimension.'

Now fractals like Lorenz' strange attractor, Lyapunov fractals, logistic maps -- bifurcation fractals -- and their relatives are differentiable in at least parts of their ranges (the Lorenz' fractal being everywhere differentiable), so evidently the term's meaning has been broadened to include these classes as well.

Europium,

I think the core definition of fractals are objects of continuing of complexity as you zoom in on its features or at least some of its features (which is the colloquial way of saying that you can't take the derivative as you indicated earlier). Self similarity seemed like it was part of the definition at one time but now I don't know if that's still the case.

I agree with your explanation as to the origins of the term fractal. Now that you've reminded me I remember hearing that before. Fractured curve is the correct origin.

I think Fractal Dimension is a little bit more important then you are saying. For instance Bifurcation fractals do indeed have fractal dimension:

Book where fractal dimension of bifurcations are discussed

Also Lyapunov fractals have fractal dimension. Perhaps our disagreement in this is stemming from a difference of opinion on the definition of fractal dimension. I'm using a broad definition:

Fractal Dimension Definition

Further insight can be gathered by a discussion of the strange attractor. I've lifted the following from wikipedia:

An attractor is informally described as strange if it has non-integer dimension or if the dynamics on it are chaotic. The term was coined by David Ruelle and Floris Takens to describe the attractor that resulted from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions, but some are like a Cantor Dust, and therefore not differentiable.

The "or" in this statement seems to verify what you say, that some of these fractals have non-integer dimension. Furthermore other things I've read today seem to indicate that something can have fractional dimension and be continuous, so my early assumption was wrong.

I know a lot about fractals but I also don't know a lot about fractals and I find myself a little over my head. Then again, that's when I'm happiest. Feel free to correct me where necessary.

Roger

I'm no expert on fractals and what I don't know about them would probably fill a small library. It helps to discuss them, though. That's what CR4 is 4!

It's frustrating. I'd like to understand Fractals better but I don't have the time to look the stuff up, not to mention I don't know what to look up half the time. I wish there were online courses for stuff like this. Not for a degree, but for a prepackaged curriculum that one could just breeze through in your spare time and learn this stuff.

I remember playing with a discrete function which gave vaguely similar results to your squiggly curve - i.e. a period of stability followed by unpredictable oscillations, or flat-lining, with huge changes resulting from tiny changes in the coefficients (or the starting values?).

It was of the form ...

Z_n+1 = F(Z_n,Z_n-1)

... and it was very simple, but I can't remember what it was. Can you enlighten me, please?

Hi John,

My experience with iterations that depend on the previous two values typically involves the discrete form of a derivative somewhere in the function. PID (proportional + integral + derivative) controllers involve such functions but they typically do not exhibit chaotic behavior. Second-order and higher systems can and often do.

Did the mystery function contain a derivative?

Pretty sure it didn't, but I seem to remember a term in 1/Z

Hmmm...

A function containing a 1/Z term and a Z term might be re-written to contain Z and Z² terms, respectively, to streamline the computation; the motive being that division is more expensive than multiplication in terms of CPU cycles.

But I'm still at loss to justify requiring both a Z_n and a Z_n-1 iteration if a difference is not involved.

Well, thinking about it, I guess could've been something along the lines of

Z_n+1 = f(Z_n) + g(Z_n-1)

... which would be a kind of differentiation if f() or g() produced a negative result.

I've complained about big pi time and time again - more times than I can remember. Doesn't do any good.

OK, Codey - but he did correct himself in #1 .

Hello JohnDG (and everybody else)

Sorry, yes, you're right! I I flicked straight past that and missed it. When I opened the thread again after your post and saw it I knew what you were going to say!

Cheers.........Codey