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. $$ d x / d t=(2+x)(y-x), \quad d y / d t=y\left(2+x-x^{2}\right) $$

Short answer

Based on the given system of two first-order differential equations, identify the critical points and their stability by analyzing direction fields and phase portraits. Critical Points: 1. (-2, y) for any y 2. (x, x) for any x 3. (1±√3, 0) Stability (by analyzing direction fields and phase portraits): 1. Asymptotically stable, stable, or unstable (depends on the specific plots) 2. Asymptotically stable, stable, or unstable (depends on the specific plots) 3. Asymptotically stable, stable, or unstable (depends on the specific plots)

Step-by-step solution

Find the critical points

To find the critical points, we need to find the values of x and y where the given system of differential equations equals zero simultaneously. So set the equations equal to zero and solve for x and y: $$ (2+x)(y-x) = 0, \quad y\left(2+x-x^{2}\right) = 0 $$ We have two cases here: 1. On the first equation, we have \((2+x)(y-x) = 0\). Two cases are possible: - \((2+x) = 0\), so \(x = -2\). In this case, \(y\) can have any value because \((y-x) = y-(-2) = y+2\) will be non-zero. - \((y-x) = 0\), so \(y = x\). In this case, \(x\) can have any value because \((2+x)\) will be non-zero. 2. On the second equation, \(y(2+x-x^2) = 0\). Two cases are possible: - \(y = 0\). In this case, the first equation becomes \((2+x)(-x) = 0\), so \(x=-2\) or \(x=0\). - \((2+x-x^2) = 0\). This is a quadratic equation in \(x\), so we can solve for \(x\) using the quadratic formula. In this case, \(x = 1 \pm \sqrt{3}\). So, the critical points are \((-2, y)\) for any y, \((x, x)\) for any x, and \((1\pm\sqrt{3},0)\).

Draw a direction field for the system

Since this step requires a computer, use software like MATLAB, Mathematica, or Python (with matplotlib library) to draw the direction field. This will give you a visual representation of how the solutions to the differential equations change with different initial conditions.

Draw a phase portrait for the system

Like in Step 2, use the software of your choice to draw a phase portrait for the system. This will give you a good idea of the trajectories of the solutions in the phase plane.

Determine the stability and classify each critical point

With the help of the directional field and the phase portrait plotted in Steps 2 and 3, we can analyze the behavior of the system around the critical points. Observe how the trajectories move towards or away from the critical points. Based on this information, classify the critical points as asymptotically stable, stable, or unstable. Use the classification based on the portraits to determine the stability of each critical point: - Asymptotically stable: The solutions approach the critical point as time goes to infinity. - Stable: The solutions remain close to the critical point, but they do not necessarily approach it. - Unstable: The solutions move away from the critical point as time goes to infinity. With these classifications in hand, you should be able to determine the stability and type of each critical point.

Key concepts

Equilibrium Solutions

Equilibrium solutions in differential equations, often referred to as 'critical points', are the heartbeats of a system where the rate of change becomes zero. To find these, the strategy is akin to solving a puzzle — equations are set to zero and the values of variables that satisfy these conditions are teased out.

Take our exercise as an example. The system of equations given by \( d x / d t=(2+x)(y-x) \) and \( d y / d t=y(2+x-x^2) \) forms a canvas where the equilibrium solutions are the points where the figurative paint doesn't budge. By solving \( (2+x)(y-x) = 0 \) and \( y(2+x-x^2) = 0 \) simultaneously, we unveil a landscape of critical points, including \( (-2, y) \) for any \( y \), \( (x, x) \) for any \( x \), and \( (1\pm\sqrt{3},0) \). These points tell us where the system can 'rest' and are fundamental in sketching the overall behavior of the system.

Direction Field

The direction field, sometimes known as a slope field, is a way to visualize the behavior of solutions to a differential equation without actually solving it. Picture a grid where at each point, a small line segment is drawn with a slope corresponding to the differential equation's value at that point.

In our exercise, crafting a direction field for the system with a computer, such as using Python's matplotlib library or MATLAB, grants us a constellation of arrows. These arrows not only form patterns that whisper the paths of potential solutions as they slide through each point, but also give immediate visual insight into the system's dynamics. It is like laying down trajectories on a road map, with each indicating the compass direction in which a particle would move if it were at that point.

Phase Portrait

The phase portrait is a step beyond the direction field; it's like watching the flow of a river from above. Each stream curve represents the trajectory of a solution to the differential equations over time. Using the same computer software that brought the direction field to life, the phase portrait can be rendered, showing not just trajectories but also how they evolve.

When observing the phase portrait in our task, we would see curving lines delineating the fate of a particle zooming through our phase plane. It's a topographical tour through time, revealing if our particle tours endlessly in a loop, spirals inwards like a whirlpool to an equilibrium solution, or jets off the page, never to return.

Stability in Differential Systems

Stability in differential systems provides us with a weather forecast of sorts — telling us whether a small initial disturbance will grow into a storm, stay a mild breeze, or dissipate altogether. Analyzing the direction field and phase portrait created for our given system, we search for signposts of stability near the critical points.

Asymptotically stable points attract nearby solutions like magnets, with particles eventually settling down at these restful points. If the phase portrait shows pathways looping back to the same spot, these are stable points — not quite magnets, but points that solutions want to dance around. Should the paths diverge away, like rockets escaping a planet's gravity, we've got ourselves an unstable point. These descriptors are the language we use to qualify the equilibria and are vital for understanding the long-term behavior of the system.