Phase line

Definition

A phase line is a graphical tool used to study the qualitative behavior of one-dimensional autonomous differential equations. It visually represents equilibrium points and their stability.

5 Must Know Facts For Your Next Test

1. Phase lines help determine the stability of equilibrium points by indicating whether solutions move towards or away from these points.
2. Equilibrium points on a phase line are found where the derivative $\frac{dx}{dt} = 0$.
3. Arrows on the phase line indicate the direction of change for different regions, showing if solutions are increasing or decreasing.
4. Stable equilibrium points attract nearby solutions, while unstable equilibrium points repel them.
5. A semi-stable point can either attract or repel solutions depending on which side they approach from.

Review Questions

• How do you determine if an equilibrium point is stable or unstable using a phase line?
• What does it mean if the arrows on a phase line are pointing towards an equilibrium point?
• Describe how you would construct a phase line for a given autonomous differential equation.

Related terms

Equilibrium Point: A value where the derivative of the function is zero, indicating no change in the state variable over time.

Autonomous Differential Equation: A differential equation where the independent variable (usually time) does not explicitly appear in its expression.

$\frac{dx}{dt}$: $\frac{dx}{dt}$ represents the rate of change of $x$ with respect to time $t$.