What is the Difference Between Fixed Point and Equilibrium Point?

The terms "fixed point" and "equilibrium point" are often used interchangeably in mathematics and physics, as they both describe a state where a system does not change over time. However, there are some subtle differences between the two concepts:

• Fixed Point: A fixed point of a function is an element of the domain of that function that, when applied to the function, yields the same element. In other words, if $$f(c) = c$$, then $$c$$ is a fixed point of the function $$f$$. Fixed points are useful for finding the steady-state of a system.
• Equilibrium Point: An equilibrium point is a state at which the system does not change as the system variables evolve over time. Equilibrium points represent the simplest solutions to differential equations. In the context of discrete-time dynamical systems, equilibrium points occur when the system remains in one place, i.e., $$Tx = x$$.

While both fixed points and equilibrium points describe a steady-state condition of a system, they are different perspectives on the same concept. Fixed points are more relevant in mathematical analysis, whereas equilibrium points are often used in the context of physical systems and their stability.

Comparative Table: Fixed Point vs Equilibrium Point

The difference between a fixed point and an equilibrium point lies in their definitions and the context in which they are used. Here is a table summarizing the key differences:

In summary, fixed points are specific solutions to mathematical equations, while equilibrium points are states where the rates of change of the variables in a system become zero. Both concepts are used to describe steady-state conditions of a system, but they are derived from different mathematical contexts.