## What is an Attracting Fixed Point?

A **fixed point** is a point where f(x) = x. In other words, it’s the set of points for the graph of f that cross the diagonal y = x.

An

**attracting fixed point**is a fixed point that “attracts” the numbers surrounding it: if we start with a number sufficiently close to

*x*, iterating it will always result in convergence to x. A

**repelling fixed point**shows the opposite behavior: it pushes the nearby numbers away. These facts can be described by the behavior of the derivative of the fixed point [1]:

- If the absolute value of the derivative is less than 1, the point is attracting.
- If the absolute value of the derivative is greater than 1, the point is repelling.

## Orbits and Periodic Points

The **forward orbit** (or simply *orbit*) of a point (called the *seed *or *initial point* of the orbit) is the set of places where the point can be moved by group action.

More formally, the **orbit **of point x_{0} under F is defined as the sequence x_{0}, x_{0}, x_{1}, x_{2},… where x_{n + 1} = F(x_{n}) for n ≥ 0.

As an example, let’s say we have a real-valued function f(x) = √x.

If x_{0} = 256 then x_{1} = √256 = 16, x_{2} = √16 = 4; and so on. The next few terms in the orbit are 2, 1.414213562, 1.189207115, 1.090507733, 1.044273783 [2].

Within a dynamical system, there are many orbit types. The orbit of a fixed point is the constant sequence x_{0}, x_{0}, x_{0}, … [3]. **Periodic orbits** are solutions to dynamical systems that repeat over time. Periodic orbits, like fixed points, can also be attracting, repelling, or neutral [4]:

**Attracting periodic orbit:**orbits of nearby points converge to the periodic orbit,**Repelling periodic orbit:**orbits of nearby points move away from the periodic orbit,**Neutral periodic orbit:**orbits of nearby points neither converge to nor move away from the periodic orbit.

## Attracting Fixed Point and the Basin of Attraction

If x_{0} is an attracting fixed point for a function f, then the **basin of attraction **for x_{0} is the set of all points with orbits tending to x_{0} [5]. The **immediate basin of attraction** is the largest interval in the basin that contains x_{0}.

## References

[1] Seidel, P. (2011). Iteration, Fixed Points. Retrieved July 7, 2021 from: https://math.mit.edu/classes/18.01/F2011/lecture3.pdf

[2]

[3] Mace, K. Chaos and Dynamics. Retrieved July 7, 2021 from: https://www.whitman.edu/Documents/Academics/Mathematics/macemk.pdf

[4] Osinga, H. (1998). One-Dimensional Dynamical Systems: Part 4: Linear and Nonlinear Behavior. Retrieved July 7, 2021 from: http://www.geom.uiuc.edu/~math5337/ds/part4/part4_per.html

[5] McKinney, W. (2005). The Schwarzian Derivative & the Critical Orbit. Retrieved July 7, 2021 from: https://ocw.mit.edu/courses/mathematics/18-091-mathematical-exposition-spring-2005/lecture-notes/lecture09.pdf