📚 Topic Summary
Planar systems are sets of differential equations that describe how two variables change with respect to time. Nullclines are curves where one of the derivatives is zero, indicating where the variable is not changing. Phase portraits are graphical representations of the system's behavior, showing trajectories that solutions follow in the phase plane. Analyzing nullclines and phase portraits helps us understand the stability and long-term behavior of these systems.
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Nullcline |
a. A graphical representation of the trajectories of a dynamical system. |
| 2. Phase Portrait |
b. A point where the derivatives of both variables are zero. |
| 3. Equilibrium Point |
c. A curve where one of the derivatives in a system of differential equations is zero. |
| 4. Trajectory |
d. The path a solution follows in the phase plane. |
| 5. Planar System |
e. A set of two differential equations describing the rates of change of two variables. |
(Match the terms with the definitions: 1-c, 2-a, 3-b, 4-d, 5-e)
✍️ Part B: Fill in the Blanks
Complete the following paragraph:
A _______ is a curve where $\frac{dx}{dt} = 0$ or $\frac{dy}{dt} = 0$. The points where the nullclines intersect are called _______ points. A _______ shows the direction of the vector field and the trajectories of the system. The behavior of trajectories near an equilibrium point determines its _______. Analyzing these portraits helps us understand the _______ behavior of the system.
(Answers: nullcline, equilibrium, phase portrait, stability, long-term)
🤔 Part C: Critical Thinking
Consider the following system of differential equations:
$\frac{dx}{dt} = x(1 - x - y)$
$\frac{dy}{dt} = y(0.75 - y - 0.5x)$
Describe how you would find the nullclines for this system, and what information the nullclines provide about the behavior of the system.