Math Insight

Single autonomous differential equation problems

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  1. For the dynamical system $ \diff{ s }{t} = g(s,a),$ the function $g$ of $s$ depends on a parameter $a$, as shown in the graphs for $a=-10, -6, -2, 10$, below. For values of $a$ in between those shown, $g$ changes smoothly, so its graph will be somewhere in between the snapshots shown. Sketch a bifurcation diagram with respect to the parameter $a$, for $-10 \le a \le 10$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $a=-10$

    $a=-2$

    $a=-6$

    $a=10$

  2. Consider the dynamical system \begin{align*} \diff{ y }{t} &= 1 \left(y + 7\right) \left(y + 4\right) \left(y - 3\right)\\ y(0) & = y_0, \end{align*} where $y_0$ is an initial condition. The graph of the function $f(y) = 1 \left(y + 7\right) \left(y + 4\right) \left(y - 3\right)$ is shown below. Use the graph to sketch the solution $y(t)$ for each of the following initial conditions: $y_0 = -7.6$, $y_0 = -7$, $y_0 = -5.5$, $y_0 = -4$, $y_0 = 0.9$, $y_0 = 3$, and $y_0=4.0$.

  3. For the dynamical system \begin{align*} \diff{r}{t} = -re^{r} \end{align*} find the equilibria and use the stability theorem to calculate their stability. Graph the solution for the initial condition $r(0)=5$.

  4. Consider the dynamical system \begin{align*} \diff{u}{t} = u(2-u). \end{align*}
    1. Using any valid method, determine the equilibria of the dynamical system and their stability.
    2. Graph of the solution of the dynamical system with initial condition $u(0)=0.8$.
    3. Use the Forward Euler algorithm with time step $\Delta t=2$ to estimate the solution with initial condition $u(0)=0.8$. Take four time steps, estimating $u(2)$, $u(4)$, $u(6)$, and $u(8)$. Does the behavior of the solution estimated by Forward Euler match your previous results? If not, how is the behavior different?
    4. Use the Forward Euler algorithm with time step $\Delta t=1$ to estimate the solution with initial condition $u(0)=0.8$. Take four time steps, estimating $u(1)$, $u(2)$, $u(3)$ and $u(4)$. Does the behavior of the solution estimated by Forward Euler match your previous results? If not, how is the behavior different?
    5. Use the Forward Euler algorithm with time step $\Delta t=0.5$ to estimate the solution with initial condition $u(0)=0.8$. Take four time steps, estimating $u(1/2)$, $u(1)$, $u(3/2)$ and $u(2)$. Does the behavior of the solution estimated by Forward Euler match your previous results? If not, how is the behavior different?

  5. For the dynamical system \begin{align*} \diff{ v }{t} = h(v, b), \end{align*} where the function $h$ of $v$ also depends on a parameter $b$, a bifurcation diagram with respect to the parameter $b$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.
    1. For the following three values of $b$, determine the number of equilibria, their values, rounded to the nearest integer, and their stability. Sketch the phase line, including equilibria. Use a solid circle for stable equilibria and an open circle for unstable equilibria.
      1. $b= -8$
      2. $b= -3$
      3. $b= 7$

    2. Identify any bifurcation points.


  6. Consider the dynamical system \begin{align*} \diff{ x }{t} &= f(x)\\ x(0) & = x_0, \end{align*} where the function $f$ is graphed to the right and $x_0$ is an initial condition. For each of the following initial conditions, sketch the graph of the solution $x(t)$.
    1. $x_0 = -7.2$
    2. $x_0 = -1.8$
    3. $x_0 = 7.2$
    4. $x_0=9.8$

  7. Estimate the solution of the differential equation \begin{align*} \diff{ z }{t} &= - 7 \ln{\left (0.5 z^{2} \right )}\\ z(0) &= -1, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.1$ to estimate $z(0.3)$.

  8. Consider the differential equation \begin{align*} \diff{ y }{ t } &= 6.1 y. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $y(0) = 3.5$?

  9. For the dynamical system \begin{align*} \diff{ w }{t} &= \left(- 2 w^{2} + 10 w\right) e^{- w}, \end{align*} find all equilibria and analytically determine their stability (i.e., use the stability theorem). Using this information, draw a phase line diagram with equilibria and vector field. (Be sure to indicate the stability of the equilibria.)

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Math 201, Spring 22