I am a beginner in Dynamical Systems and Stability Analysis. The theory starts from the definition of steady-state solution (equilibrium) of differential equations, but I cannot understand how the following two explanations of steady-state solution match:
Given an ordinary differential equation $$\frac{dy}{dt}=f(t)$$
- We say $y$ is a steady state solution of the above equation, if $\frac{dy}{dt}=0$.
- The steady state is a state that the behavior of the system is unchanging over time. If a system is in a steady state, then the recently observed behavior of the system will continue into the future.
The first explanation (definition) means the critical point of the following curve, but how does it match the second explanation?
Thank you!
You seem to be confused by two things. First about the difference between the derivative "having a zero" and "being zero". The condition 1 means that the function's derivative is zero, as in "it is zero for all time". Having a critical point doesn't satisfy this.
The second thing you seem to be confused by, is that in addition to having a derivative that is zero with respect to time, it also has to be a solution to the differential equation. An arbitrary $y$ with a zero time derivative isn't a steady state solution unless it is also a solution in the first place.