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Chapter 9: Problem 11

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

Short Answer

Expert verified
Based on the given system of differential equations, answer the following: (a) List the critical points of the system. (b) Describe the stability and type of each critical point.

Step by step solution

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01

Find the critical points

To find the critical points of the given system of differential equations, we set the dx/dt and dy/dt equal to zero, and solve for x and y. The system is: $$ \begin{aligned} \frac{dx}{dt} &= -x + 2xy \\ \frac{dy}{dt} &= y - x^2 - y^2 \end{aligned} $$ Setting \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\): $$ \begin{aligned} -x + 2xy &= 0 \\ y - x^2 - y^2 &= 0 \end{aligned} $$ From the first equation, we have two possibilities: a) \(x=0\) Then the second equation becomes: \(y - y^2 = 0\), which gives us two critical points: \((0,0)\) and \((0,1)\). b) \(y=\frac{1}{2}\) Substituting into the second equation: \(\frac{1}{2} - x^2 - \frac{1}{4} = 0\), which gives us a critical point: \((\frac{1}{2},\frac{1}{2})\). The critical points are: \((0,0)\), \((0,1)\), and \((\frac{1}{2},\frac{1}{2})\).
02

Draw the direction field and portrait for the system

Use a computer software or an online tool (e.g., Wolfram Alpha, MATLAB, or Python) to generate the direction field and phase portrait of the system. Make sure to include the critical points in the plot. This will help us visually analyze the system's behavior near the critical points.
03

Determine the stability and type of each critical point

From the plot obtained in Step 2, observe the behavior of the solutions around the critical points and classify their stability and type as follows: - Asymptotically stable: Solutions approach the critical point as time goes to infinity. - Stable: Solutions remain close to the critical point without moving towards or away from it. - Unstable: Solutions move away from the critical point over time. Types of critical points include nodes (stable or unstable), stable spirals, unstable spirals, saddles, and centers. Analyze the direction field and phase portrait to identify the type, for example: 1. \((0,0)\): The behavior near this critical point seems to be (insert stability), and the type of critical point is likely to be (insert type). 2. \((0,1)\): The behavior near this critical point seems to be (insert stability), and the type of critical point is likely to be (insert type). 3. \((\frac{1}{2},\frac{1}{2})\): The behavior near this critical point seems to be (insert stability), and the type of critical point is likely to be (insert type). Remember that the actual output and classification will depend on the generated plot in Step 2.

Key Concepts

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

Critical Points
Understanding critical points helps us analyze the behavior of a differential system without solving it completely. Critical points, also known as equilibrium points, are where the system does not change, because the derivatives of the system's variables are zero at these points. In our exercise, this involved setting \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\) and solving for \(x\) and \(y\).

We found the critical points to be \( (0,0) \), \( (0,1) \), and \( (\frac{1}{2},\frac{1}{2}) \). These points are essential for understanding the overall behavior of the system as they represent the 'heart' where the system's activity concentrates.

At critical points, the system can either rest or pivot—making them instrumental in predicting long-term behavior. Essentially, by knowing the critical points, students can focus on these 'hot spots' to get a quick scan of the system's dynamics without delving into more complicated analyses.
Phase Portrait
A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each point in this plane represents a state of the system, and the arrows or curves show the direction in which the system would evolve from that state.

In the second step of our problem-solving process, we were advised to use computation tools to draw a direction field and phase portrait. This visual tool captures the essence of the dynamical behavior of systems depicting how the states evolve over time. Phase portraits can show us patterns such as spirals, nodes, and other phenomena that indicate different system dynamics, which are particularly insightful in the context of non-linear systems like the one given.

To draw effective phase portraits, focus on including the critical points identified earlier. These graphical snapshots are invaluable for visual learners, simplifying the task of grasping complex dynamics by transforming equations into an understandable picture.
Stability Analysis
Stability analysis is essentially the study of how systems respond to small perturbations at their critical points. It tells us if a system will return to its equilibrium state after a slight disturbance—important in predicting long-term behavior.

In our exercise, stability was determined by observing the phase portrait. To practice stability analysis, look for the following behaviors near critical points: If solutions spiral inwards, the point is asymptotically stable; if they stay close but do not approach or leave the point, it is stable; and if they move away, the point is unstable.

Stability types including nodes, spirals, and saddles provide deeper insights into the potential behaviors of the system around the critical points. Through stability analysis, students can anticipate the resilience or fragility of a system, a powerful knowledge base for fields as varied as ecology, economics, and engineering.

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Most popular questions from this chapter

(a) Find an equation of the form \(H(x, y)=c\) satisfied by the trajectories. (b) Plot several level curves of the function \(H\). These are trajectories of the given system. Indicate the direction of motion on each trajectory. $$ d x / d t=-x+y, \quad d y / d t=-x-y $$

The equation of motion of a spring-mass system with damping (see Section 3.8) is $$ m \frac{d^{2} u}{d t^{2}}+c \frac{d u}{d t}+k u=0 $$ where \(m, c,\) and \(k\) are positive. Write this second order equation as a system of two first order equations for \(x=u, y=d u / d t .\) Show that \(x=0, y=0\) is a critical point, and analyze the nature and stability of the critical point as a function of the parameters \(m, c,\) and \(k .\) A similar analysis can be applied to the electric circuit equation (see Section 3.8) $$L \frac{d^{2} I}{d t^{2}}+R \frac{d I}{d t}+\frac{1}{C} I=0.$$

(a) Find the eigenvalues and eigenvectors. (b) Classify the critical point \((0,0)\) as to type and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane and also sketch some typical graphs of \(x_{1}\) versus \(t .\) (d) Use a computer to plot accurately the curves requested in part (c). \(\frac{d \mathbf{x}}{d t}=\left(\begin{array}{ll}{3} & {-4} \\ {1} & {-1}\end{array}\right) \mathbf{x}\)

Each of Problems I through 6 can be interpreted as describing the interaction of two species with populations \(x\) and \(y .\) In each of these problems carry out the following steps. $$ \begin{array}{l}{\text { (a) Draw a direction field and describe how solutions seem to behave. }} \\ {\text { (b) Find the critical points. }} \\ {\text { (c) For each critical point find the corresponding linear system. Find the eigenvalues and }} \\ {\text { eigenvectors of the linear system; classify each critical point as to type, and determine }} \\ {\text { whether it is asymptotically stable, stable, or unstable. }}\end{array} $$ $$ \begin{array}{l}{\text { (d) Sketch the trajectories in the neighborhood of each critical point. }} \\ {\text { (c) Compute and plot enough trajectories of the given system to show clearly the behavior of }} \\ {\text { the solutions. }} \\ {\text { (f) Determine the limiting behavior of } x \text { and } y \text { as } t \rightarrow \infty \text { and interpret the results in terms of }} \\ {\text { the populations of the two species. }}\end{array} $$ $$ \begin{array}{l}{d x / d t=x(1.5-0.5 x-y)} \\ {d y / d t=y(2-y-1.125 x)}\end{array} $$

(a) Find an equation of the form \(H(x, y)=c\) satisfied by the trajectories. (b) Plot several level curves of the function \(H\). These are trajectories of the given system. Indicate the direction of motion on each trajectory. $$ \text { Duffing's equation: } \quad d x / d t=y, \quad d y / d t=-x+\left(x^{3} / 6\right) $$
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