What is the intuition behind equilibrium point?

Question

Consider a linear first order $2\times 2$ system of different equation with constant co-efficient given by

$\cfrac{dx}{dt}=Ax$

Equilibrium point of the system is defined to be the point at which $A x_0=0$ or in other words gradient equals zero.

What is the intution behind considering only such points? And how it will give information about dynamic of the system.

I understand that equilibrium point are points that is invariant with time , but my question is how is helps in understanding the dynamics of the system .

Answer 1

Definition: An equilibrium point (or equilibrium) of a dynamical system generated by an autonomous system of ordinary differential equations (ODEs) is a solution that does not change with time.

For example:

Each motionless pendulum position in the above Figure corresponds to an equilibrium of the corresponding equations of motion, one is stable, the other one is not (called unstable). Geometrically, equilibrium points are points in the system's phase space.

The concept of equilibrium of a dynamical system is associated with the attractors of the system, which characterize the long-term behavior. Equilibrium points represent stationary conditions for the dynamics of a system. The stability of a system is intimately connected to its equilibrium state. If a system in equilibrium is disturbed slightly, then if it is stable it tends to return to or oscillate about its original equilibrium state. An unstable system tends to continue to move away from its original equilibrium state when perturbed from it.