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Chapter 7: Problem 12

A system of differential equations is given. (a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the direction of motion. (b) Obtain an expression for each equilibrium. \(p^{\prime}=2 q-1, \quad q^{\prime}=q^{2}-q-p\)

Short Answer

Expert verified
The equilibrium point is \((-\frac{1}{4}, \frac{1}{2})\).

Step by step solution

01

Identify the Nullclines

For the system of differential equations, the nullclines are found by setting the derivatives to zero. This means solving for when \( p' = 0 \) and \( q' = 0 \). Thus, \( 2q - 1 = 0 \) and \( q^2 - q - p = 0 \).
02

Solve for Nullclines

From \( 2q - 1 = 0 \), we solve for \( q \): \( q = \frac{1}{2} \). For \( q^2 - q - p = 0 \), rearrange to find \( p \): \( p = q^2 - q \). These expressions define the nullclines.
03

Find Equilibrium Points

To find the equilibria, we solve the system by setting \( p' = 0 \) and \( q' = 0 \) simultaneously. Substitute \( q = \frac{1}{2} \) from the first nullcline into the second equation: \( p = \left( \frac{1}{2} \right)^2 - \frac{1}{2} = \frac{1}{4} - \frac{1}{2} = -\frac{1}{4} \). The equilibrium point is \((p, q) = \left(-\frac{1}{4}, \frac{1}{2}\right)\).
04

Plot Nullclines and Equilibria on the Phase Plane

Plot the nullcline \( q = \frac{1}{2} \) as a horizontal line, and the nullcline \( p = q^2 - q \) as a parabola. Mark the equilibrium point \((-\frac{1}{4}, \frac{1}{2})\) on the graph.
05

Determine Direction of Motion on the Phase Plane

Examine the sign of the derivatives \( p' \) and \( q' \) around the equilibrium point. If \( q > \frac{1}{2} \), \( p' > 0 \); if \( q < \frac{1}{2} \), \( p' < 0 \). Check \( q' \) by plugging points into \( q^2 - q - p \) to determine the arrows' directions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Phase Plane Analysis
Phase plane analysis is a useful graphical tool to understand systems of differential equations. It involves plotting solution trajectories in a plane where each point represents the state of the system at a given time.

In a phase plane, the axes usually represent the variables of the system. In our context, these are the variables \(p\) and \(q\).

By analyzing the phase plane, one can see how the system evolves over time. It visually displays equilibria, nullclines, and trajectories of motion. This helps in understanding the dynamic behavior even without solving the system analytically.

When constructing a phase plane:
  • Draw nullclines, where changes in the system variables become zero.
  • Identify equilibrium points, where the system resides in a balanced state.
  • Denote the direction of trajectories using arrows to indicate motion.
Equilibrium Points
Equilibrium points are the heart of understanding stability in a system of differential equations. They occur when both time derivatives stop changing, i.e., \( p' = 0 \) and \( q' = 0 \).

Finding equilibrium points involves solving these equations simultaneously. Once found, these points represent states where the system doesn't evolve if not perturbed.

For our system:
  • The equilibrium point is \( (p, q) = \left(-\frac{1}{4}, \frac{1}{2}\right) \).
  • This means at this point, neither \( p \) nor \( q \) changes over time.
Studying equilibria reveals whether these points are stable (small disturbances won't change the system) or unstable (small disturbances lead to vastly different behavior).

Understanding stability requires further analysis, often involving linearization or other advanced techniques.
Nullclines
Nullclines are critical curves in a phase plane where the rate of change of one of the system's components is zero.

In practice, they provide vital information about the system's behavior:
  • They form boundaries separating areas where the derivative of a variable alters sign.
  • Intersections of nullclines often reveal equilibrium points.
For our system of equations, nullclines are derived by setting \( p' \) and \( q' \) to zero:
  • \( 2q - 1 = 0 \) or \( q = \frac{1}{2} \).
  • \( q^2 - q - p = 0 \) gives \( p = q^2 - q \).
These define the nullclines, represented by a horizontal line and a parabola, respectively, in the phase plane.

Understanding nullclines is central to phase plane analysis, as they visually separate regions of differing dynamic behavior.
Direction of Motion
The direction of motion on a phase plane shows how the system's state changes over time.

Analyzing the sign of derivatives in various regions provides insight into the movement direction:
  • When \( q > \frac{1}{2} \), the derivative \( p' > 0 \), suggesting \( p \) increases.
  • Conversely, if \( q < \frac{1}{2} \), then \( p' < 0 \), indicating \( p \) decreases.
These directional insights are depicted as arrows on phase planes. Checkpoints help verify correctness by plugging into the equations and observing if the changes agree with intuition.

Direction of motion provides a dynamic snapshot of the system's pathways, crucial for predicting long-term behaviors, and assessing stability around equilibrium points.

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