The** key difference** between fixed point and equilibrium point is that **fixed point is useful to find the steady-state of a system, whereas equilibrium point is the state at which the system does not change as the system variables are change**d.

Fixed point and equilibrium point are useful terms in mathematics to identify the steady-state of a desired physical system.

### CONTENTS

1. Overview and Key Difference

2. What is Fixed Point

3. What is Equilibrium Point

4. Similarities Between Fixed Point and Equilibrium Point

5. Side by Side Comparison – Fixed Point vs Equilibrium Point in Tabular Form

6. Summary

## What is Fixed Point?

The fixed point of a function in mathematics is an element of the domain of that function that can be mapped to itself through the function. In other words, “c” is a fixed point of the function “f” when f(c)=c. This is also known as fixpoint or invariant point. Therefore, f(f(…f(c)…))=f^{n}(c)=c which is an important terminating concern regarding the recursively computing “f”. We can name a set of fixed points as a fixed set.

Let us consider an example in order to understand this phenomenon. If we take “f” in real numbers by f(x) = x^{2} – 3x +4, then 2 is a fixed point of “f” because f(2) = 2. However, all functions do not have fixed points. E.g. when f(x) = x + 1, it has no fixed points because “x” never equals “x +1” for any real number. Considering the graphical terminology, a fixed point “x” refers to the point (x,f(x)) which is on the line y = x. In other words, the graph of “f” contains a point in common with that line.

Fixed points are periodic points having their period equal to one. Considering the projective geometry, the fixed points of a projectivity are named as double points. According to the Galois theory, the series of fixed points of a set of field automorphisms is named as a fixed field of that set of automorphisms.

There are different applications of fixed points, including economics, physics, programming language compilers, type theory, the vector on PageRank values of all web pages, the stationary distribution of Markov chain, etc.

## What is Equilibrium Point?

An equilibrium point is a constant solution to a different equation in mathematics. This term comes mainly under differential equations in mathematics. We can classify the equilibria by observing the signs of the eigenvalues of the linearization of the equations about the equilibria. In other words, we can categorize equilibria by evaluating the Jacobian matrix at the equilibrium points of the desired system, followed by finding the resulting eigenvalues. There, we can determine the behaviour of the system in the neighbourhood of the equilibrium points quantitatively by finding the eigenvector(s) that are associated with the eigenvalues.

We can say an equilibrium point is hyperbolic when none of the eigenvalues has zero real part. However, if all eigenvalues have a negative real part, then the equilibrium becomes a stable equation. Similarly, if there is a positive real part, then the equilibrium becomes unstable. Moreover, if there is at least one negative real part and at least one positive real part in eigenvalues, then the equilibrium obtains a saddle point.

## What are the Similarities Between Fixed Point and Equilibrium Point?

- These points may not be stable.
- Both points are described for a steady-state condition of a system.

## What is the Difference Between Fixed Point and Equilibrium Point?

The terms fixed point and equilibrium point are used in mathematics. The key difference between fixed point and equilibrium point is that fixed point is useful to find the steady-state of a system, whereas equilibrium point is the state at which the system does not change as the system variables are changed.

## Summary – Fixed Point vs Equilibrium Point

Fixed point and equilibrium point are useful terms in mathematics to identify the steady-state of a desired physical system. The key difference between fixed point and equilibrium point is that fixed point is useful to find the steady-state of a system, whereas equilibrium point is the state at which the system does not change as the system variables are changed.

##### Reference:

1. “Fixed Point (Mathematics).” *Wikipedia*, Wikimedia Foundation, 26 Mar. 2021, Available here.

##### Image Courtesy:

1. “Fixed point example” By Krishnavedala – Own work (CC0) via Commons Wikimedia

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