Sensitive dependence on initial conditions

Logistic Function

Malthus' Function

Fixed point of a function

Periodic orbit of a function

# Define all these terms

logistic function is $$f(x) = rx(1-x)$$

The Malthus' function is $$f(x) = rx$$

Sensitive dependence on initial condition:

In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.

### Periodic Orbit of a Function:

**(In simple terms)**

The number of times a given function, $f(x)$, iterates to repeat itself.

### Example:

If the periodic orbit is 2, every 2nd iteration, (once it settles out of its initial value), will have the same value.

If the periodic orbit is 3, then every 3rd iteration.

If the periodic orbit is N, every Nth iteration

Fixed Point:

A fixed point $x_0$ of a function $f$ is a number such that $f(x_0)=x_0$.

Would a fixed point be repulsive or attractive?

The repulsive or attractive nature of the point depends on the slope of the Graph of the function at that point.

A Repulsive point has slope, $m$, values:

$m<-1$ and $m>1$

An Attractive point has slope, $m$, values:

$1>m>-1$

$Definitions:$

(a) Sensitive dependence on initial conditions: a small change in one state of a deterministic nonlinear system can result in large differences in a later state. (also known as the butterfly effect by Ed Lorenz)

(b) The logistic function: $f(x)=rx(1-x)$

(c) Malthus' function: the typical function is $f(x)=rx$ but in other cases this function is used $P(t)=P_0e^{rt}$ for population growth

(d) Fixed point of a function: the x-intercepts are roots of the function; point of intersection; a fixed point $x_0$ of a function $f$ is a number such that $f(x_0)=x_0$

(e) Periodic orbit of a function: if $x_n=x_0$ for some $n$, then the orbit is called periodic.

To expand on these definitions:

a) Sensitive dependence on initial conditions: also known as the butterfly effect, this is the idea that small changes add up over time to completely change the final outcome. It was studied by Ed Lorenz who used it to prove the inaccuracy of long-term weather forecasting.

b) The Logistic Function: $f(x)=rx(1-x)$

These functions will graph as parabolas

c) The Malthusian Function: $f(x)=rx$

These functions will graph as straight lines

d) Fixed Point: Any point at which f(x_{0})=x_{0} ; in other words, when a number you plug into the function returns that same number as the output, you are dealing with a fixed point. These can easily be found by graphing the line $y=x$ on top of your function and looking for points of intersection.

e) Periodic Orbit: If x_{n}=x_{0}

In other words, a periodic orbit occurs when, while performing a functional iteration, outputs continuously repeat themselves (every other iterate is the same, or every three iterates, etc)