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Chapter 13: Problem 9

If the eigenvalues and eigenvectors of a planar, autonomous, linear system are complex, then there are no straight line solutions. Use \([\mathrm{v}, \mathrm{e}]=\mathrm{eig}(\mathrm{A})\) to demonstrate that the eigenvalues and eigenvectors of the system \(x^{\prime}=2.9 x+2.0 y, y^{\prime}=-5.0 x-3.1 y\) are complex, then use pplane6 to show that solution trajectories spiral in the phase plane.

Short Answer

Expert verified
The eigenvalues are complex, indicating spiral trajectories in the phase plane.

Step by step solution

01

Formulate the System in Matrix Form

The given system of equations is \( x' = 2.9x + 2.0y \) and \( y' = -5.0x - 3.1y \). We can represent this system in matrix form as \[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 2.9 & 2.0 \ -5.0 & -3.1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \].
02

Calculate the Eigenvalues

The eigenvalues of the matrix \( A = \begin{pmatrix} 2.9 & 2.0 \ -5.0 & -3.1 \end{pmatrix} \) are determined by solving the characteristic equation \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix. Calculate the determinant of \( \begin{pmatrix} 2.9 - \lambda & 2.0 \ -5.0 & -3.1 - \lambda \end{pmatrix} \).
03

Solve the Characteristic Equation

The characteristic equation becomes: \((2.9 - \lambda)(-3.1 - \lambda) - (2.0)(-5.0) = 0\). Expanding this we get: \[\lambda^2 - (2.9 - 3.1)\lambda + (2.9)(-3.1) + 10 = 0\]. Simplifying, we have \[\lambda^2 + 0.2\lambda + 0.98 = 0\].
04

Find the Eigenvalues from the Quadratic Formula

Solve \( \lambda^2 + 0.2\lambda + 0.98 = 0 \) using the quadratic formula \[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \], where \( a = 1, b = 0.2, c = 0.98\). Calculate \( b^2 - 4ac = 0.04 - 3.92 = -3.88 \), which is negative, indicating the eigenvalues are complex.
05

Interpret Eigenvalues and Determine Nature of Solutions

Since the eigenvalues found are complex, this confirms that any trajectories will not follow straight lines, as complex eigenvalues result in trajectories that spiral in the phase plane.

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Key Concepts

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

Linear Systems
In mathematics, a linear system is a collection of linear equations that share a common set of variables. Such systems can be represented in matrix form, which allows for a clearer visualization and simplicity in calculations. This is especially useful when dealing with multiple linear equations simultaneously.

A linear system can usually be expressed using matrices, where each equation in the system corresponds to a row in the matrix. For example, the system given in the exercise involves two equations represented in matrix form as \( \begin{pmatrix} x' \ y' \end{pmatrix} = A \begin{pmatrix} x \ y \end{pmatrix} \), with a specific matrix \( A \).

One key trait of linear systems is that they overlap with other mathematical topics, like linear algebra and differential equations. These systems often require solving for variables or understanding the behavior of the system's solutions. Understanding the composition of linear systems helps lay the foundation for more advanced topics in mathematics.
Autonomous Systems
An autonomous system, in the context of differential equations, refers to a system where the rules governing the behavior of the system do not explicitly depend on time. This means that the system's behavior is solely determined by the current state, making calculations simpler in some respects.

Autonomous systems are vital in studying dynamic systems such as those that model physical phenomena. Because they don't change over time unless the state changes, they provide a reliable model of behavior under constant conditions.

For example, in the given exercise, the system \( x' = 2.9x + 2.0y \) and \( y' = -5.0x - 3.1y \) is autonomous because the equations lack a time variable. Exploring these kinds of systems can reveal behaviors like equilibrium points and consistent patterns over time.

Understanding autonomous systems is crucial for analyzing trajectories and predicting system behavior in mathematical modeling.
Differential Equations
A differential equation involves equations that describe how one variable changes with respect to another, typically involving derivatives. These equations are powerful tools in modeling situations where change is fundamental, such as physics, engineering, and economics.

In the context of the exercise, the differential equations \( x' = 2.9x + 2.0y \) and \( y' = -5.0x - 3.1y \) are involved in determining how the variables \( x \) and \( y \) change with respect to time. Solving these equations provides insights on how variables evolve and interact over time.

Differential equations can be categorized into linear and nonlinear types. Linear differential equations, like those in the exercise, have solutions that can often be superimposed. This property helps in analyzing and solving complex systems more effortlessly.

Mastering differential equations paves the way to understanding a vast range of scientific and engineering problems.
Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors are fundamental concepts in linear algebra, often described in relation to matrices. They are essential for understanding the dynamics of systems represented in matrix form.

In essence, an eigenvalue is a scalar that indicates how much an eigenvector is stretched or compressed during a linear transformation. An eigenvector, on the other hand, is a direction along which this transformation occurs.

In the problem, we compute eigenvalues by solving the characteristic equation derived from the system matrix. The presence of complex eigenvalues, as found in the solution, implies that the system exhibits oscillatory behavior, such as spiraling trajectories in the phase plane.

Comprehending eigenvalues and eigenvectors is crucial as they offer insights into the stability and behavior of systems, including those modeled by differential equations. This understanding allows for predictions of system responses to various conditions, which is invaluable in engineering and physics.

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Most popular questions from this chapter

In Exercises \(5-7\), find the eigenvalues and eigenvectors with the eig and null commands, as demonstrated in Example 4 of Chapter 12. You may find format rat helpful. Then enter the system into pplane6, and draw the straight line solutions. For example, if one eigenvector happens to be \(\mathbf{v}=[1,-2]^{T}\), use the Keyboard input window to start straight line solutions at \((1,-2)\) and \((-1,2)\). Perform a similar task for the other eigenvector. Finally, the straight line solutions in these exercises divide the phase plane into four regions. Use your mouse to start several solution trajectories in each region. $$ \begin{aligned} &x^{\prime}=6 x-y \\ &y^{\prime}=-3 y \end{aligned} $$

If a system has two distinct negative eigenvalues, then both straight line solutions will decay to the origin with the passage of time. Consequently, all solutions will decay to the origin. Enter the system, \(x^{\prime}=-4 x+y, y^{\prime}=-2 x-y\), in pplane6 and plot the straight line solutions. Plot several more solutions and note that they also decay to the origin. Select Solutions \(\rightarrow\) Find an equilibrium point, find the equilibrium point at the origin, then read its classification from the PPLANE6 Equilibrium point data window.

In contrast to Exercises \(31-34\), consider the system $$ \begin{aligned} &x^{\prime}=y+a x^{3} \\ &y^{\prime}=-x \end{aligned} $$ for the three values 0,10 and \(-10\) of the parameter \(a\). a) Show that all three systems have the same Jacobian matrix at the origin. What type of equilibrium point at \((0,0)\) is predicted by the eigenvalues of the Jacobian? b) Use pplane6 to find evidence that will enable you to make a conjecture as to the type of the equilibrium point at \((0,0)\) in each of the three cases. c) Consider the function \(h(x, y)=x^{2}+y^{2}\). In each of the three cases, restrict \(h\) to a solution curve and differentiate the result with respect to \(t\) (Recall: \(d h / d t=(\partial h / \partial x)(d x / d t)+(\partial h / \partial y)(d y / d t))\). Can you use the result to verify the conjecture you made in part b)? Hint: Note that \(h(x, y)\) measures the distance between the origin and \((x, y)\). d) Does the Jacobian predict the behavior of the non-linear systems in this case?

If a system has one negative and one positive eigenvalue, then one straight line solution moves toward the origin and the other moves away. Consequently, general solutions (being linear combinations of straight line solutions) must do the same thing. Enter the system \(x^{\prime}=9 x-14 y, y^{\prime}=7 x-12 y\), in pplane6 and plot the straight line solutions. Plot several more solutions and note that they move toward the origin only to move away at the last moment. Select Solutions \(\rightarrow\) Find an equilibrium point, find the equilibrium point at the origin, then read its classification from the PPLANE6 Equilibrium point data window.

The system of differential equations: $$ \begin{aligned} &x^{\prime}=\mu x-y-x^{3}, \\ &y^{\prime}=x, \end{aligned} $$ is called the van der Pol system. It arises in the study of non-linear semiconductor circuits, where \(y\) represents a voltage and \(x\) the current. It is in the Gallery menu. a) Find the equilibrium points for the system. Use pplane6 only to check your computations. b) For various values of \(\mu\) in the range \(0<\mu<5\), find the equilibrium points, and find the type of each, i.e, is it a nodal sink, a saddle point, ...? You should find that there are at least two cases depending on the value of \(\mu\). Don't worry too much about non-generic cases. Use pplane6 only to check your computations. c) Use pplane6 to illustrate the behavior of solutions to the system in each of the cases found in b). Plot enough solutions to illustrate the phenomena you discover. Be sure to start some orbits very close to \((0,0)\), and some near the edge of the display window. Put arrows on the solution curves (by hand after you have printed them out) to indicate the direction of the motion. (The display window \((-5,5,-5,5)\) will allow you to see the interesting phenomena.) d) For \(\mu=1\) plot the solutions to the system with initial conditions \(x(0)=0\), and \(y(0)=0.2\). Plot both components of the solution versus \(t\). Describe what happens to the solution curves as \(t \rightarrow \infty\).
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