Jeyhun's Notes

9/26/2025

Understanding the Logistic Map and Fixed Points

Have you ever wondered how complex behaviors in nature, like population growth, can emerge from surprisingly simple rules? Enter the Logistic Map, a mathematical model that’s a cornerstone in the study of chaos theory and complex systems.

It’s defined by the recursive equation:

$x_{t+1} = R \, (x_t - x_t^2)$

Here, $R$ is the growth rate, and $x_t$ represents the population (or system state) at a given time $t$ . The magic of this simple equation is how it can produce a stable, predictable outcome with one set of $R$ values, and complete chaos with others!

What is a Fixed Point?

In the context of the logistic map, a fixed point is a value that, once reached, remains constant in all subsequent iterations. It’s a point of stability where the system settles. Mathematically, a fixed point, $x^*$ , satisfies the condition:

$x^* = R (x^* - {x^*}^2)$

This means the value $x^*$ at time $t$ results in the exact same value at time $t+1$ .

Finding the Fixed Point: Step-by-Step

The quiz example provides a perfect scenario to demonstrate how a system finds its stability.

The Task:
We are given $R = 2.5$ and an initial value $x_0 = 0.2$ . We need to find the fixed point.

Method 1: Iterative Calculation (The Simulation Approach)

We’ll start with $x_0 = 0.2$ and calculate $x_1, x_2, x_3, \dots$ until the value stops changing significantly.

Iteration ( $t$ )	Calculation: $x_{t+1} = 2.5(x_t - x_t^2)$	Result ( $x_{t+1}$ )
t=0	$x_1 = 2.5 \times (0.2 - 0.2^2) = 2.5 \times (0.2 - 0.04)$	$x_1 = 0.4$
t=1	$x_2 = 2.5 \times (0.4 - 0.4^2) = 2.5 \times (0.4 - 0.16)$	$x_2 = 0.6$
t=2	$x_3 = 2.5 \times (0.6 - 0.6^2) = 2.5 \times (0.6 - 0.36)$	$x_3 = 0.6$

Since $x_2$ and $x_3$ are the same, the system has converged! The fixed point is 0.6.

The visual from the NetLogo simulation (which models a similar logistic system) confirms this behavior: the population quickly rises from its initial state to a stable maximum.

Method 2: Analytical Calculation (The Fixed Point Formula)

To be absolutely sure, we can use the fixed point formula $x^* = R (x^* - {x^*}^2)$ and solve for $x^*$ using $R = 2.5$ .

Set the equation:

$x^* = 2.5 (x^* - {x^*}^2)$
Divide both sides by $x^*$ (assuming $x^* \ne 0$ ):

$1 = 2.5 (1 - x^*)$
Divide by $2.5$ (or multiply by $0.4$ ):

$\frac{1}{2.5} = 1 - x^*$

$0.4 = 1 - x^*$
Solve for $x^*$ (rearrange):

$x^* = 1 - 0.4$

$\mathbf{x^* = 0.6}$

Both methods confirm that the system stabilizes at 0.6.

Conclusion

This simple problem beautifully illustrates how a deterministic equation—the Logistic Map—can lead to a stable, predictable outcome, a fixed point. For $R=2.5$ and $x_0=0.2$ , the system quickly approaches and settles at 0.6 (Option b in the quiz).

The Logistic Map is a foundational tool in understanding how complex behaviors emerge from simple rules. If you increase the $R$ value, you start to see fascinating phenomena like period-doubling and eventually, chaos!

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Biology, Complexity, Math, Statistics, Systems