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Chapter 9: Problem 12

(a) Determine all critical points of the given system of equations. (b) Find the corresponding linear system near each critical point. (c) Find the eigenalues of each linear system. What conclusions can you then draw about the nonlinear system? (d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system. $$ d x / d t=(1+x) \sin y, \quad d y / d t=1-x-\cos y $$

Short Answer

Expert verified
Question: Analyze the behavior of the given system of equations near its critical points and determine the type of critical points. System of equations: $$ (1+x) \sin y = 0, 1-x-\cos y = 0 $$ Answer: The critical points of the given system of equations are \(\mathrm{(-1, \pi n)}\) and \((0, 0)\). The critical points \(\mathrm{(-1, \pi n)}\) will be centers for even integer values of n and saddle points for odd integer values of n. The critical point \((0, 0)\) will be an unstable node.

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01

Finding the critical points of the given system of equations

To find the critical points, we need to solve the following system of equations for x and y: $$ (1+x) \sin y = 0, 1-x-\cos y = 0 $$ Setting each part equal to zero, we get: $$ (1+x) \sin y = 0, 1 - x - \cos y = 0 $$ From the first equation, we obtain \(x = -1\) or \(\sin{y} = 0\). For \(x = -1\), we get \(y=\pi\text{n}\) where n is an integer. From the second equation, we get \(x = 1 - \cos y\). Substituting \(\sin y = 0\) in this equation, we find \(x = 0\), thus obtaining the only other critical point \((0, 0)\). So, the critical points are \(\mathrm{(-1, \pi n)}\) and \((0, 0)\).
02

Linearizing the system near each critical point

We can obtain the corresponding linear system near each critical point by linearizing the given nonlinear system around these points. We do this by computing the Jacobian matrix of the system at each critical point: $$ J(x, y) = \begin{bmatrix} \frac{\partial}{\partial x} [(1+x) \sin y] & \frac{\partial}{\partial y} [(1+x) \sin y] \\ \frac{\partial}{\partial x} [1-x-\cos y] & \frac{\partial}{\partial y} [1-x-\cos y] \end{bmatrix} $$ By taking partial derivatives and substituting the corresponding critical points, we obtain the following Jacobian matrices: For \(\mathrm{(-1, \pi n)}\): $$ J(-1, \pi n) = \begin{bmatrix} 0 & \sin (\pi n) \\ 1 & -\sin (\pi n) \end{bmatrix} $$ For \((0, 0)\): $$ J(0, 0) = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix} $$
03

Finding the eigenvalues of each linear system and analyzing the nonlinear system

We now find the eigenvalues of each Jacobian matrix: For \(\mathrm{(-1, \pi n)}\): The eigenvalues can be found by solving the following determinant equation: $$ \det(J(-1, \pi n) - \lambda I) = \begin{vmatrix} -\lambda & \sin (\pi n) \\ 1 & -\sin (\pi n) - \lambda \end{vmatrix} = \lambda^2 + \sin^2(\pi n) $$ The eigenvalues are \(\lambda_{1,2} = \pm \sqrt{\sin^2(\pi n)}\). For even integer values of n, the eigenvalues are purely imaginary, whereas for odd integer values of n, they are real and opposite signs. This implies that the critical points are either a center or a saddle point, depending on parity of n. For \((0, 0)\): The eigenvalues can be found by solving the following determinant equation: $$ \det(J(0, 0) - \lambda I) = \begin{vmatrix} 1-\lambda & 1 \\ -1 & 1 - \lambda \end{vmatrix} = (1-\lambda)^2 - 1 $$ The eigenvalues are \(\lambda_{1,2} = 1 \pm \sqrt{2}\). Since both eigenvalues are real and positive, this implies that the critical point \((0, 0)\) is an unstable node.
04

Checking conclusions with a phase portrait

By plotting a phase portrait, we can verify our conclusions for the critical points and the behavior of the nonlinear system. A phase portrait for this nonlinear system will show the following: a. The critical points \(\mathrm{(-1, \pi n)}\) will be centers for even integers n and saddle points for odd integers n. b. The critical point \((0, 0)\) will be an unstable node. In cases where the linear system does not provide definite information about the nonlinear system, the phase portrait will be useful for extending the conclusions. However, in this case, our conclusions are supported by the analysis of the eigenvalues.

Key Concepts

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

Nonlinear Systems
Nonlinear systems refer to a set of equations where the change in one variable does not result in a proportional change in another variable. These systems often contain variables that multiply together or appear in functions like sine, cosine, or exponentials.

Unlike linear systems, nonlinear systems can demonstrate a diverse range of complex behaviors such as chaos, multi-stability, and limit cycles. In real-world scenarios, they are used to model phenomena like weather patterns, population dynamics, or even electrical circuits. Understanding these systems is key to analyzing and predicting long-term behavior in dynamic environments.
  • Nonlinear systems are sensitive to initial conditions.
  • They can exhibit multiple equilibrium points, including both stable and unstable points.
  • Sometimes, linear approximations around critical points are used to simplify analyses.
Eigenvalues
Eigenvalues are a fundamental concept when analyzing the stability and behavior of systems of equations. In the context of nonlinear systems, eigenvalues are used to understand the local behavior around critical points after linearization.

To determine eigenvalues, we compute the determinant of the Jacobian matrix minus a scaled identity matrix, setting it equal to zero. The resulting characteristic equation helps identify the eigenvalues for the system.
  • If eigenvalues are both real and of the same sign, the critical point is a node. It could be stable or unstable based on whether they are negative or positive.
  • If eigenvalues are real and of opposite signs, the critical point is a saddle point, indicating instability.
  • If the eigenvalues are complex with non-zero real parts, the system might exhibit spiral or oscillatory behavior.
Jacobian Matrix
The Jacobian matrix is a critical tool for linearizing nonlinear systems around critical points. It consists of all first-order partial derivatives of the system of equations and provides a representation of how small changes in input variables affect the outputs.

For a given nonlinear system, the Jacobian matrix at a critical point helps form the basis for a linear system describing approximated dynamics near that point. Evaluating the Jacobian at these points gives insight into the system's stability characteristics. For the provided system:
  • The Jacobian matrix is evaluated at all critical points calculated from the system equations.
  • By substituting critical points into the Jacobian, we obtain specific matrices that describe the system's local behavior.
  • Jacobian matrices are used for calculating eigenvalues, providing crucial insights into the stability and nature of critical points.
Phase Portrait
A phase portrait is a graphical representation that illustrates the trajectories of a dynamical system in the phase plane. For nonlinear systems, phase portraits help visualize the global behavior of solutions as they evolve over time.

In drawing a phase portrait, we focus on critical points and trajectory patterns that indicate how the system behaves in their proximity. These patterns help confirm the conclusions drawn from eigenvalues and provide additional insights when linear analysis alone is insufficient.
  • Phase portraits reveal whether critical points are stable or unstable.
  • In the exercise, critical points like \((-1, \pi n)\) show as centers or saddle points depending on eigenvalues.
  • The phase portrait can be used to check solutions for scenarios where eigenvalue analysis provides ambiguous or inconclusive information.

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Most popular questions from this chapter

Can be interpreted as describing the interaction of two species with population densities \(x\) and \(y .\) In each of these problems carry out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, or unstable. (d) Sketch the trajectories in the neighborhood of each critical point. (e) Draw a phase portrait for the system. (f) Determine the limiting behavior of \(x\) and \(y\) as \(t \rightarrow \infty\) and interpret the results in terms of the populations of the two species. $$ \begin{array}{l}{d x / d t=x(1.5-0.5 y)} \\ {d y / d t=y(-0.5+x)}\end{array} $$

(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. $$ d x / d t=2 x^{2} y-3 x^{2}-4 y, \quad d y / d t=-2 x y^{2}+6 x y $$

The equation $$ u^{\prime \prime}-\mu\left(1-\frac{1}{3} u^{\prime 2}\right) u^{\prime}+u=0 $$ is often called the Rayleigh equation. (a) Write the Rayleigh equation as a system of two first order equations. (b) Show that the origin is the only critical point of this system. Determine its type and whether it is stable or unstable. (c) Let \(\mu=1 .\) Choose initial conditions and compute the corresponding solution of the system on an interval such as \(0 \leq t \leq 20\) or longer. Plot \(u\) versus \(t\) and also plot the trajectory in the phase plane. Observe that the trajectory approaches a closed curve (limit cycle). Estimate the amplitude \(A\) and the period \(T\) of the limit cycle. (d) Repeat part (c) for other values of \(\mu,\) such as \(\mu=0.2,0.5,2,\) and \(5 .\) In each case estimate the amplitude \(A\) and the period \(T\). (e) Describe how the limit cycle changes as \(\mu\) increases. For example, make a table of values and/or plot \(A\) and \(T\) as functions of \(\mu .\)

We will prove part of Theorem 9.3 .2 : If the critical point \((0,0)\) of the almost linear system $$ d x / d t=a_{11} x+a_{12} y+F_{1}(x, y), \quad d y / d t=a_{21} x+a_{22} y+G_{1}(x, y) $$ is an asymptotically stable critical point of the corresponding linear system $$ d x / d t=a_{11} x+a_{12} y, \quad d y / d t=a_{21} x+a_{22} y $$ then it is an asymptotically stable critical point of the almost linear system (i). Problem 12 deals with the corresponding result for instability. Consider the linear system (ii). (a) Since \((0,0)\) is an asymptotically stable critical point, show that \(a_{11}+a_{22}<0\) and \(\left.a_{11} a_{22}-a_{12} a_{21}>0 . \text { (See Problem } 21 \text { of Section } 9.1 .\right)\) (b) Construct a Liapunov function \(V(x, y)=A x^{2}+B x y+C y^{2}\) such that \(V\) is positive definite and \(\hat{V}\) is negative definite. One way to ensure that \(\hat{V}\) is negative definite is to choose \(A, B,\) and \(C\) so that \(\hat{V}(x, y)=-x^{2}-y^{2} .\) Show that this leads to the result $$ \begin{array}{l}{A=-\frac{a_{21}^{2}+a_{22}^{2}+\left(a_{11} a_{22}-a_{12} a_{21}\right)}{2 \Delta}, \quad B=\frac{a_{12} a_{22}+a_{11} a_{21}}{\Delta}} \\\ {C=-\frac{a_{11}^{2}+a_{12}^{2}+\left(a_{11} a_{22}-a_{12} a_{21}\right)}{2 \Delta}}\end{array} $$ where \(\Delta=\left(a_{11}+a_{22}\right)\left(a_{11} a_{22}-a_{12} a_{21}\right)\) (c) Using the result of part (a) show that \(A>0\) and then show (several steps of algebra are required) that $$ 4 A C-B^{2}=\frac{\left(a_{11}^{2}+a_{12}^{2}+a_{21}^{2}+a_{22}^{2}\right)\left(a_{11} a_{22}-a_{12} a_{21}\right)+2\left(a_{11} a_{22}-a_{12} a_{21}\right)^{2}}{\Delta^{2}}>0 $$ Thus by Theorem 9.6.4, \(V\) is positive definite.

Consider the system of equations $$ d x / d t=y-x f(x, y), \quad d y / d t=-x-y f(x, y) $$ where \(f\) is continuous and has continuous first partial derivatives. Show that if \(f(x, y)>0\) in some neighborhood of the origin, then the origin is an asymptotically stable critical point, and if \(f(x, y)<0\) in some neighborhood of the origin, then the origin is an unstable critical point. Hint: Construct a Liapunov function of the form \(c\left(x^{2}+y^{2}\right) .\)
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