Compute the coefficient matrix of the linearization of the system of differential equations at each stationary point of the system.
\(x' = x + y^2 \hspace{0.4 in} y' = x + y\)
\(x'=x-xy, \hspace{0.4in} y' = y+2xy\)
\(x' = (1+x)\sin(y) \hspace{0.4 in} y' = 2 - x - \cos(y)\)
\(x' = x-y^2 \hspace{0.4 in} y' = y - x^2 \)
\(x' = 1 - xy \hspace{0.4 in} y' = x - y^2 \)
\(x' = y \hspace{0.4 in} y' = x + 2x^3 \)
For 7-18, determine the stability and classify each stationary point of the system
\(x' = x, \hspace{.4in} y' = -2y + x^3\)
\(x' = 2 - y, \hspace{.4in} y' = 3 - x^2\)
\(x' = -(2+y)y, \hspace{.3in} y' = x(1-y) \)
\(x' = 1-y \hspace{0.4 in} y' = x^2 - y^2 \)
\(x' = -x + 2xy \hspace{0.4 in} y' = y - x^2 - y^2 \)
\(x' = 4x-y \hspace{0.4 in} y'=5x-x^2\)
\(x' = -x-x^2, \hspace{.4in} y'= y + 2xy \)
\(x' = (1+x)(y-x) \hspace{0.4 in} y' = (2 - x)(y+x)\)
\(x' = 1+2y, \hspace{.3in} y'= 1-3x^2 \)
\(x' = -x + y + x^2, \hspace{.3in} y' = y - 2xy \)
\(x' = (12 - 2x - 3y)x, \hspace{.3in} y' = (5x-15)y\)
\(x' = x-x^2-xy, \hspace{.3in} y' = \frac{1}{2}y - \frac{1}{4}y^2 - \frac{3}{4}xy\)
Derive and explain the linearization of a system of differential equations,
\(x' = f(x,y)\) and \(y'=g(x,y)\). That is, show how
\[\frac{d}{dt} \begin{pmatrix} \tilde{x} \\ \tilde{y} \end{pmatrix} = \begin{pmatrix} \partial_xf(x_o,y_o) & \partial_yf(x_o,y_o) \\ \partial_xg(x_o,y_o) & \partial_yg(x_o,y_o) \end{pmatrix} \begin{pmatrix} \tilde{x} \\ \tilde{y} \end{pmatrix}\]
Sketch the phase portrait of \(x' = y\), \(y'= \sin(x)\) using the methods of this chapter. Compare it the the phase portrait of the methods used in the previous chapter, “Integral Methods to Autonomous Planar Systems".
Consider the nonlinear planar system \[\begin{split} x^\prime &= 1-y\\ y^\prime &= x^2 - y^2 \end{split}\] Determine the stability and classify each stationary point of the system if possible. Sketch a phase portrait.
Consider the system \[x^\prime = -y + x^3 + xy^2,\quad y^\prime = x + y^3 + yx^2.\]
Linearize the system about \((0,0)\) and classify this stationary points. What can you say about the stability of \((0,0)\)?
Show that \(V(x(t),y(t)) = x(t)^2 + y(t)^2\) is strictly increasing in \(t\) away from \((x,y) = (0,0)\). What can you deduce about the stability of \((0,0)\)?
A damped pendulum of mass \(m\) and lenght \(\ell\) moving in a plane can be described by the equation \[m\ell\theta^{\prime\prime} = -\gamma\theta^\prime - ma\sin{(\theta)}.\] where \(\gamma >0\) is the damping coefficient of the pendulum. This can be written as a system \[x^\prime = y, \quad y^\prime = -2\mu y - \omega^2\sin(x),\] where \(x = \theta\), \(y=\theta^\prime\), \(\mu = \gamma/(2m\ell)\) and \(\omega^2 = a/\ell\). Answer the following
Find all stationary points and write the Jacobian matrix at each of the stationary points.
Determine the type and stability of each stationary point when \(\mu^2 >\omega^2\).
Determine the type and stability of each stationary point when \(\mu^2 < \omega^2\).
Determine the type and stability of each stationary point when \(\mu^2 = \omega^2\).
The following questions are related to a special type of differential system of the form \[\begin{split}x^\prime &= -\partial_xU(x,y)\\y^\prime &= -\partial_yU(x,y)\end{split}\]referred to as a gradient system.
Show that a planar system \[x^\prime = f(x,y), \quad y^\prime = g(x,y)\] is a gradient system if and only if \[\partial_yf(x,y) = \partial_xg(x,y).\]
Show that a linear planar system \(\xBld^\prime = \ABld\xBld\) is a gradient system if and only if \(\ABld = \ABld^T\).
Suppose that \((x_0,y_0)\) is a stationary solution of the gradient system \(x^\prime = -\partial_xU(x,y), \quad y^\prime = -\partial_yU(x,y)\), then clearly \((x_0, y_0)\) is also a critical point of \(U(x,y)\). Show the following
If \((x_0,y_0)\) is a strict local minimum of \(U(x,y)\), then \((x_0,y_0)\) is either a node or radial sink.
If \((x_0,y_0)\) is a strict local maximum of \(U(x,y)\), then \((x_0,y_0)\) is a node or radial source.
If \((x_0,y_0)\) is a saddle point of \(U(x,y)\), then \((x_0,y_0)\) corresponds to a saddle.
Can a gradient system have critical points that correspond to spirals or centers? Give an example or show why not.
If \((x(t),y(t))\) is a solution to a gradient system. Show that \(U(x(t),y(t))\) is always stricly decreasing in \(t\) except at stationary points.
Show that if \[x^\prime = f(x,y), \quad y^\prime = g(x,y)\] is Hamiltonian, then \[x^\prime = g(x,y), \quad y^\prime = -f(x,y)\] is a gradient system.