Problem 18
Consider the nonlinear system $$ \begin{aligned} &x^{\prime}=x(1-x)-x y, \\ &y^{\prime}=y(2-y)+2 x y . \end{aligned} $$ Show that \((-1 / 3,4 / 3)\) is an equilibrium point of the system. a) Without the use of technology, calculate the Jacobian of the system at the equilibrium point \((-1 / 3\), \(4 / 3)\). What is the equation of the linearization at this equilibrium point? Use [v,e]=eig( \(\mathrm{J})\) to find the eigenvalues and eigenvectors of this Jacobian. b) Enter the system in pplane6. Find the equilibrium point at \((-1 / 3,4 / 3)\). Does the data in the Equilibrium point data window agree with your findings in part (a)? Note: The eigenvalues of the Jacobian predict classification of the equilibrium point. In this case, the point \((-1 / 3,4 / 3)\) is a saddle because the eigenvalues are real and opposite in sign. c) Display the linearization. Does the equation of the linearization agree with your findings in part (a)?
Problem 20
Most of the time, the linearization accurately predicts the behavior of a nonlinear system at an equilibrium point. There are exceptions, most notably when the the matrix \(A\) has purely imaginary eigenvalues, or when one of the eigenvalues is zero. For example, consider the system $$ \begin{aligned} &x^{\prime}=-y+x\left(x^{2}+y^{2}\right) \\ &y^{\prime}=x+y\left(x^{2}+y^{2}\right) \end{aligned} $$ Enter the system in pplane6, set the display window so that \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\), select Arrows for the direction field, then click Proceed. In the PPLANE6 Display window, select Solutions \(\rightarrow\) Show nullclines to overlay the nullclines on the vector field and note the presence of an equilibrium point at \((0,0)\). a) Use Keyboard input to start a solution trajectory at \((0.5,0)\) and note that the origin behaves as a spiral source. If the solution takes too long to stop on its own use the Stop button. b) Select Solutions \(\rightarrow\) Find an equilibrium point and find the equilibrium point at \((0,0)\). Note that the eigenvalues of the Jacobian are purely imaginary, indicating that the linearization has a center, not a spiral source, at the origin. Display the linearization and draw some solution trajectories.
Problem 21
In Exercises \(21-30\), perform steps i) and ii) for the linear system $$ \mathbf{x}^{\prime}=\mathbf{A x}, $$ where \(\mathbf{A}\) is the given matrix. i) Find the type of the equilibrium point at the origin. Do this without the aid of any technology or pplane6. You may, of course, check your answer using pplane. ii) Use pplane6 to plot several solution curves - enough to fully illustrate the behavior of the system. You will find it convenient to use the "linear system" choice from the Gallery menu. If the eigenvalues are real, use the "Keyboard input" option to include straight line solutions starting at \(\pm 10\) times the eigenvectors. If the equilibrium point is a saddle point, compute and plot the separatrices. $$ \left[\begin{array}{rr} 1 & 1 \\ -18 & 10 \end{array}\right] $$
Problem 23
In Exercises \(21-30\), perform steps i) and ii) for the linear system $$ \mathbf{x}^{\prime}=\mathbf{A x}, $$ where \(\mathbf{A}\) is the given matrix. i) Find the type of the equilibrium point at the origin. Do this without the aid of any technology or pplane6. You may, of course, check your answer using pplane. ii) Use pplane6 to plot several solution curves - enough to fully illustrate the behavior of the system. You will find it convenient to use the "linear system" choice from the Gallery menu. If the eigenvalues are real, use the "Keyboard input" option to include straight line solutions starting at \(\pm 10\) times the eigenvectors. If the equilibrium point is a saddle point, compute and plot the separatrices. $$ \left[\begin{array}{rr} -1 & 0 \\ 3 & -3 \end{array}\right] $$
Problem 24
In Exercises \(21-30\), perform steps i) and ii) for the linear system $$ \mathbf{x}^{\prime}=\mathbf{A x}, $$ where \(\mathbf{A}\) is the given matrix. i) Find the type of the equilibrium point at the origin. Do this without the aid of any technology or pplane6. You may, of course, check your answer using pplane. ii) Use pplane6 to plot several solution curves - enough to fully illustrate the behavior of the system. You will find it convenient to use the "linear system" choice from the Gallery menu. If the eigenvalues are real, use the "Keyboard input" option to include straight line solutions starting at \(\pm 10\) times the eigenvectors. If the equilibrium point is a saddle point, compute and plot the separatrices. $$ \left[\begin{array}{rr} 6 & 1 \\ -18 & 0 \end{array}\right] $$
Problem 25
In Exercises \(21-30\), perform steps i) and ii) for the linear system $$ \mathbf{x}^{\prime}=\mathbf{A x}, $$ where \(\mathbf{A}\) is the given matrix. i) Find the type of the equilibrium point at the origin. Do this without the aid of any technology or pplane6. You may, of course, check your answer using pplane. ii) Use pplane6 to plot several solution curves - enough to fully illustrate the behavior of the system. You will find it convenient to use the "linear system" choice from the Gallery menu. If the eigenvalues are real, use the "Keyboard input" option to include straight line solutions starting at \(\pm 10\) times the eigenvectors. If the equilibrium point is a saddle point, compute and plot the separatrices. $$ \left[\begin{array}{rr} 9 & -1 \\ 9 & 3 \end{array}\right] $$
Problem 26
In Exercises \(21-30\), perform steps i) and ii) for the linear system $$ \mathbf{x}^{\prime}=\mathbf{A x}, $$ where \(\mathbf{A}\) is the given matrix. i) Find the type of the equilibrium point at the origin. Do this without the aid of any technology or pplane6. You may, of course, check your answer using pplane. ii) Use pplane6 to plot several solution curves - enough to fully illustrate the behavior of the system. You will find it convenient to use the "linear system" choice from the Gallery menu. If the eigenvalues are real, use the "Keyboard input" option to include straight line solutions starting at \(\pm 10\) times the eigenvectors. If the equilibrium point is a saddle point, compute and plot the separatrices. $$ \left[\begin{array}{ll} 9 & 1 \\ 9 & 3 \end{array}\right] $$
Problem 27
In Exercises \(21-30\), perform steps i) and ii) for the linear system $$ \mathbf{x}^{\prime}=\mathbf{A x}, $$ where \(\mathbf{A}\) is the given matrix. i) Find the type of the equilibrium point at the origin. Do this without the aid of any technology or pplane6. You may, of course, check your answer using pplane. ii) Use pplane6 to plot several solution curves - enough to fully illustrate the behavior of the system. You will find it convenient to use the "linear system" choice from the Gallery menu. If the eigenvalues are real, use the "Keyboard input" option to include straight line solutions starting at \(\pm 10\) times the eigenvectors. If the equilibrium point is a saddle point, compute and plot the separatrices. $$ \left[\begin{array}{rr} 7 & 4 \\ -10 & -5 \end{array}\right] $$
Problem 28
In Exercises \(21-30\), perform steps i) and ii) for the linear system $$ \mathbf{x}^{\prime}=\mathbf{A x}, $$ where \(\mathbf{A}\) is the given matrix. i) Find the type of the equilibrium point at the origin. Do this without the aid of any technology or pplane6. You may, of course, check your answer using pplane. ii) Use pplane6 to plot several solution curves - enough to fully illustrate the behavior of the system. You will find it convenient to use the "linear system" choice from the Gallery menu. If the eigenvalues are real, use the "Keyboard input" option to include straight line solutions starting at \(\pm 10\) times the eigenvectors. If the equilibrium point is a saddle point, compute and plot the separatrices. $$ \left[\begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array}\right] $$
Problem 29
In Exercises \(21-30\), perform steps i) and ii) for the linear system $$ \mathbf{x}^{\prime}=\mathbf{A x}, $$ where \(\mathbf{A}\) is the given matrix. i) Find the type of the equilibrium point at the origin. Do this without the aid of any technology or pplane6. You may, of course, check your answer using pplane. ii) Use pplane6 to plot several solution curves - enough to fully illustrate the behavior of the system. You will find it convenient to use the "linear system" choice from the Gallery menu. If the eigenvalues are real, use the "Keyboard input" option to include straight line solutions starting at \(\pm 10\) times the eigenvectors. If the equilibrium point is a saddle point, compute and plot the separatrices. $$ \left[\begin{array}{rr} 6 & -4 \\ 18 & -6 \end{array}\right] $$
