Equilibrium Points and Stability Analysis

The video tutorial reviews types of graphs and terminology related to one-dimensional autonomous differential equations. It explains how to visualize systems using time series and tangent vs. state graphs, focusing on equilibrium points and derivatives. The tutorial also covers the concept of roots and stability in graphs, and introduces linear stability theory, which classifies equilibrium points based on the slope of tangent lines.

What is a key characteristic of an autonomous differential equation?

• It is always linear.

• It depends on time.

• It is self-governing and does not depend on time.

• It changes based on external factors.

In a time series graph, what is typically plotted on the vertical axis?

• Time

• The rate of change

• The state variable X

• The function f(X)

What does a horizontal line represent in a time series graph?

• A point of instability

• An equilibrium point

• A changing state

• A tangent line

What is the visible derivative in a tangent vs. state graph?

• dx/dt

• The slope of the state axis

• df/dx

• The slope of the time axis

In the context of equilibrium points, what does f(X) = 0 signify?

• The system is unstable.

• The system is decelerating.

• The system is at equilibrium.

• The system is accelerating.

What do equilibrium points look like on a tangent vs. state graph?

• Vertical lines

• Horizontal lines

• X-intercepts

• Y-intercepts

What does linear stability theory help determine about equilibrium points?

• Their time dependence

• Their exact location

• Their rate of change

• Their stability

If df/dx is greater than zero at an equilibrium point, what can be concluded?

• The point is neutral.

• The point is unstable.

• The point is stable.

• The point is semi-stable.

What happens if df/dx equals zero at an equilibrium point?

• The point is neutral.

• Linear stability theory fails to conclude.

• The point is unstable.

• The point is stable.

What is the relationship between dx/dt and f(X) in the context of autonomous differential equations?

• dx/dt is independent of f(X).

• dx/dt is equal to f(X).

• dx/dt is always less than f(X).

• dx/dt is always greater than f(X).