Question

(a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and portrait for the system. (c) From the plot(s) in part (b) determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. $$ d x / d t=-(x-y)(1-x-y), \quad d y / d t=x(2+y) $$

Short answer

Short Answer: There are three critical points: (0,0), (0,1), and (1,-2). By analyzing the direction field and phase portrait plot, we can determine the stability and type of each critical point. For example, if the trajectories around a critical point converge, it is asymptotically stable; if they diverge, the point is unstable; and if the system neither converges nor diverges, it is stable. The specific stability and type (node, saddle point, focus, center, or spiral) will depend on the results of the direction field and phase portrait analysis.

Step-by-step solution

Find critical points

To find critical points, set both dx/dt and dy/dt equal to 0, then solve the system of equations: $$ -(x-y)(1-x-y) = 0 \\ x(2+y) = 0 $$ Solve the system of equations to find the critical points. There are two cases: - \(\boldsymbol{x = 0}\): In this case, the first equation becomes: $$ 0-y(1-y)=0 \\ y(1-y)=0 $$ We have two values for y: \(y=0\) and \(y=1\). So, two critical points are (0,0) and (0,1). - \(\boldsymbol{2+y = 0}\): In this case, \(y=-2\). The first equation becomes: $$ -(x+2)(1-x+2)=0 \\ (x+2)(1-x+2)=0 \\ (x+2)(3-x)=0 $$ We have one value for x: \(x=1\). So, one critical point is (1,-2). In conclusion, the critical points are (0,0), (0,1), and (1,-2).

Draw the direction field and portrait using a computer

Use Python, MATLAB, or other software to draw the direction field for the given system of differential equations. Observe the behavior of the system around the critical points (0,0), (0,1), and (1,-2). The plot should show the direction of the system at each point in the xy plane. You can also add some trajectories that show the path of the system over time. Additionally, include portraits for the system, displaying phase portraits for different initial conditions.

Determine stability and classify critical points

From the direction field and phase portrait plot obtained in Step 2, observe the behavior of the system around each critical point and determine their stability and type: - (0,0): If the trajectories converge to this point, it is asymptotically stable; if they diverge from the point, it is unstable; if the system neither converges nor diverges, it is stable. - (0,1): Repeat the analysis for point (0,1). - (1,-2): Repeat the analysis for point (1,-2). Check the type of each critical point, e.g., saddle point, node, focus, center, or spiral. Finally, summarize the stability and type for each critical point, e.g., (0,0) is asymptotically stable and a node, (0,1) is unstable and a saddle point, and (1,-2) is stable and a node. These classifications will vary based on the results of the direction fields and phase portraits.

Key concepts

Understanding Direction Fields

Imagine you're walking through a field where the wind blows in different directions at each point on the ground. Similarly, in mathematics, a direction field is a visual representation of a differential equation. It shows you how a function evolves over time, given its slopes in the xy-plane.

For the exercise at hand, each point on the plot of the direction field represents a possible state of the system given by \( dx/dt = -(x-y)(1-x-y) \) and \( dy/dt = x(2+y) \). By following the 'wind’—or, more scientifically, the slope indicated at each point—you can predict the system's behavior from different starting conditions. This tool is particularly useful for visualizing how solutions to the system might behave without actually solving the equation analytically.

Using software to create direction fields helps students to comprehend the global dynamics of the system, offering them insights into how a system reacts over time and will also aid in identifying types of equilibrium points.

Phase Portrait Analysis

A phase portrait is a bit like a family album of a dynamical system – it's a graphical representation showing all possible trajectories or orbits of the system. Each 'photo'—or plot—captures different moments, illustrating various initial conditions in a two-dimensional state space.

When you sketch a phase portrait, you're drawing several solution curves that represent how the system's states evolve through time, given varying initial states. The exercise's system sketches these portraits for \( dx/dt = -(x-y)(1-x-y) \) and \( dy/dt = x(2+y) \), helping students to see where the system tends to go over a long period of time.

Phase portraits are especially valuable in visualizing complex behaviors such as spirals, nodes, and saddles that can emerge in non-linear systems. Students can use these portraits to intuitively understand the flow within the system and to closely study the system's behavior near critical points.

Equilibrium Solutions

In the quest to understand dynamic systems, students will encounter the term equilibrium solution, which refers to a state where all change has ceased, and the system is at rest. These are the 'eye of the storm' points in the phase portrait. Equivalent to a ball settling at the bottom of a valley, these points occur where \( dx/dt = 0 \) and \( dy/dt = 0 \), meaning that the system has no tendency to change its state.

For the given exercise, solving the system of equations \( -(x-y)(1-x-y) = 0 \) and \( x(2+y) = 0 \) brings us the critical points: (0,0), (0,1), and (1,-2). These are the resting 'states' of the system, where the solution curves in the phase portrait intersect. Understanding equilibrium solutions gives students powerful insight into the static states of different systems, and is a critical step in analyzing the overall behavior of the system.

Stability Analysis

When studying equilibrium points of a system like \( dx/dt \) and \( dy/dt \) from the exercise, it's crucial to understand their stability. Think of it as investigating whether a marble will stay put in a dimple in a board, or if it will roll away with the slightest nudge.

Stability Considerations

Observing a direction field or phase portrait will show students how trajectories behave near these equilibrium points. An asymptotically stable equilibrium point attracts nearby trajectories, while an unstable point repels them. If trajectories neither converge to nor diverge away, but stay in proximity, the equilibrium is considered to be stable.

By identifying whether solutions will return to an equilibrium after a small disturbance, stability analysis in differential equations teaches students to predict the resilience of a system's state. It is a foundational aspect of understanding not only the direct behavior of a system but the potential implications of perturbations in real-world applications.