Z-transform of this Logistic Difference Equation

[Master1022]

Homework Statement

Find an expression for the Z-transform of the following equation: $$x_{n + 1} = r x_{n} \left( 1 - x_{n} \right)$$

Relevant Equations

z-transform

Hi,

I am trying to work out how I could obtain an expression for a z-transform for the following expression:
x_{n + 1} = r x_{n} \left( 1 - x_{n} \right)

I am hoping to derive X(z) and then use the final value theorem to show agreement with numerically calculated steady state values.

I haven't made it very far because I don't know what to do with a convolution term I have in my z-transform.

Method:
Taking the Z-transform of both sides we get: (we will let \bar{X} = X(z) = Z \{ x_{n} \})
z \bar{X} = r \bar{X} - r \bar{X} * \bar{X}
How would I deal with the convolution term here without explicitly knowing an expression for \bar{X}? Is it possible to proceed with this approach?

Thanks in advance for the help.

 

[rude man]

I think that's a non-linear equation for which the z transform is inapplicable.

 

[Filip Larsen]

The equation is also called the Logistic map and is one of the simplest maps that exhibit chaotic trajectories for r = 4 and other interesting behaviors (like bifurcation) at other values.

While some theoretical insight can be gain via the mathematical tools used to analyze chaotic systems, the z-transform is of no use (as already mentioned). The most accessible way to analyze this particular equation is via numerical methods, that is, simply running the sequences for different values of r and initial values and cataloging the long-term behavior of the time series.

I am a bit puzzled that you are given a homework assignment for making a z-transform of this equation. Can you elaborate on where this comes from?

 

[Master1022]

I am a bit puzzled that you are given a homework assignment for making a z-transform of this equation. Can you elaborate on where this comes from?

Thanks for the replies (both @rude man and @Filip Larsen ). So the original problem was asking us to use numerical simulations and I did those just fine. There was a side question that was asking us to verify that what we saw was what one might expect. The previous topic in this discrete class was on discrete control systems, so I just (naively) tried to apply that material to this content. I haven't studied any chaos theory mathematics yet, so I will just set this to the side for now and re-visit it when I have the required knowledge.

Thanks.

 

[Filip Larsen]

A part of the mathematical tools used for this is stability analysis, which for simple systems and maps can be fairly straight forward. See for instance this page for some examples for the Logistic map (haven't read the page in detail, but the derivations looks similar to what I remember from my old textbooks on non-linear stability analysis).