Question

When you portray sufficient solution trajectories in the phase plane so as to determine all of the important behavior of a planar, autonomous system, you have created what is called a "phase portrait." In Exercises 15 - 22 , use pplane 6 to create a phase portrait for the indicated system on the prescribed display window. Take special notice of where solution curves end, as reported in the message window. $$ \begin{aligned} &x^{\prime}=y \\ &y^{\prime}=\left(1-y^{2}\right) y-x \\ &-3 \leq x \leq 3,-3 \leq y \leq 3 \end{aligned} $$

Short answer

Use PPLANE6 to plot the system and identify key behaviors from the generated phase portrait.

Step-by-step solution

Analyze the Given System

We are given a system of differential equations where \( x' = y \) and \( y' = (1-y^2)y - x \). Our task is to create the phase portrait for this system within the window defined by \(-3 \leq x \leq 3\) and \(-3 \leq y \leq 3\). The phase portrait will show all important behaviors of the system.

Utilize Software Tool PPLANE6

Open the PPLANE6 software tool, which is designed for analyzing planar autonomous systems. Enter the differential equations \( x' = y \) and \( y' = (1-y^2)y - x \) into the corresponding fields of the program.

Set Up Display Window

Configure the display window in PPLANE6 to fit the dimensions \(-3 \leq x \leq 3\) and \(-3 \leq y \leq 3\) as specified in the problem. This ensures we visualize the behavior of the system within the provided limits.

Generate the Phase Portrait

Use PPLANE6 to plot the phase portrait. The software will draw solution trajectories starting from various initial conditions within the display window. Observe how the curves behave, including where they start and end, and the overall flow direction.

Identify Critical Points and Directions

Look for critical points (equilibrium points) in the phase portrait, which are points where \( x' = 0 \) and \( y' = 0 \). Also, note the direction of flow near these points and in various regions of the plane. This is essential for understanding the system dynamics.

Interpret Solution Curves and Behavior

Examine the solution curves in the phase portrait. Investigate how they behave near critical points, where they converge or diverge, and any cyclic behavior that may be present in the system.

Document Observations

Record key observations from the phase portrait, including the locations of equilibrium points, the nature of the flows (e.g., clockwise or anti-clockwise), and any stable or unstable behaviors. This helps in summarizing the characteristics of the system.

Key concepts

Planar Autonomous Systems

Planar autonomous systems are a fascinating type of mathematical model used frequently in the study of dynamical systems. An 'autonomous' system means that the system does not rely on an external time variable. This makes the equations time-independent and focuses solely on how variables interact with each other. These systems are often expressed as:

• First-order differential equations
• Consist of two variables, usually denoted by \( x \) and \( y \)
• The system evolves in a plane, hence the term 'planar'

A common way to analyze such systems is by creating a phase portrait, where trajectories of solutions are plotted on a plane, showing how the system behaves over time. This tool helps visualize how changes in one variable affect the other, revealing the system's overall dynamics.

Differential Equations

Differential equations are equations involving derivatives of a function, expressing how a change in one quantity depends on other quantities. Specifically, in planar autonomous systems, the equations take the form \( x' = f(x, y) \) and \( y' = g(x, y) \). Here, \( x' \) and \( y' \) represent the rate of change of \( x \) and \( y \) respectively.

• They describe the behavior of a system continuously over a plane.
• These equations form the backbone of creating a phase portrait.
• Helps in predicting future behavior of the system by analyzing how current rates of change lead to future positions.

For our exercise, the differential equations are \( x' = y \) and \( y' = (1-y^2)y - x \), which define how \( x \) and \( y \) move within our specified limits.

Critical Points

Critical points, also known as equilibrium points, are crucial for understanding the behavior of a system. These points occur where the system's derivatives equal zero — meaning the system is at rest or in equilibrium. For a system defined by \( x' \) and \( y' \), a critical point satisfies:

• \( x' = 0 \)
• \( y' = 0 \)

In our case, setting \( x' = y = 0 \) and \( y' = (1-y^2)y-x = 0 \) helps us locate these points. Finding these critical points involves solving these equations simultaneously.
Once found, examining the nature of these points (whether they are stable or unstable, nodes or spirals) provides insight into the stability of the system and how other trajectories behave relative to these points.

Solution Trajectories

Solution trajectories are lines or curves in the phase plane representing the path that the system's state follows over time. They are integral to understanding phase portraits as they illustrate the flow of the system's solutions.

• Trajectories indicate how the values of \( x \) and \( y \) evolve.
• Starting at different initial conditions, they provide a visual map of the system's dynamic behavior.
• Help identify trends such as cyclical paths, convergence to critical points, or divergence away from them.

Tracking these trajectories offers a comprehensive view of how a system behaves under various conditions, highlighting areas of attraction or repulsion, and displaying potential periodic orbits inherent in the system.