**Up:** *Logistic Growth Model*

# One-Dimensional Dynamical Systems

## Part 2: Logistic Growth Model

#### Eventual behavior

In order to describe the long term behavior of iterates, we present a
few more concepts.
**Definition (Fixed Point):** The point *p* is a
*fixed point* of the function *f* if
*f*(*p*) = *p*.

- What are the fixed points of
*f*(*x*) =
*x*^{2}?

What is the relation between the fixed points of the square function
and the intersection points of the graph of the square function with
the special lines *y = x* and *y = -x*?

- Consider the square function on your calculator. Pick a point
"near" (distance less than 0.1 is near enough) a fixed point. Keep
iterating (pushing the squaring button) until you see a pattern for
the long term behavior. Try again with other numbers near each of the
fixed points. Write down what happens for each choice. Does it matter
if the number you pick is less than or greater than the fixed point?

Notice that for the square function, some orbits converge to a point,
a fixed point, and some orbits diverge (get arbitrarily large). Are
there orbits which always stay bounded, but do not converge to just
one point? In general, the answer is yes. You will observe more
complicated long term behavior of orbits in your computer
investigations.

**Definition (Attracting and Repelling Fixed Point):**
A fixed point *p* is easy to detect computationally when orbits
of nearby points converge to *p*. In this case, *p* is
called *attracting*. If the orbits of nearby points move away
from *p*, *p* is called *repelling*. If *p*
is neither attracting nor repelling, *p* is called
*neutral*.

- Which of the fixed points for the square function are attracting?
Which are repelling?

**Up:** *Logistic Growth Model*

*The Geometry Center Home Page*
Written by Hinke Osinga

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Created: Apr 1 1998 ---
Last modified: Wed Apr 8 17:58:50 1998