Math Insight

Single autonomous differential equation problems

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  1. For the dynamical system $ \diff{ s }{t} = f(s,b),$ the function $f$ of $s$ depends on a parameter $b$, as shown in the graphs for $b=2, 5, 6, 7$, below. For values of $b$ in between those shown, $f$ changes smoothly, so its graph will be somewhere in between the snapshots shown. Sketch a bifurcation diagram with respect to the parameter $b$, for $2 \le b \le 7$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $b=2$

    $b=6$

    $b=5$

    $b=7$

  2. For the differential equation \begin{align*} \diff{x}{t} = f(x), \end{align*} the function $f(x)$ is graphed below. Autonomous differential equation example function 1
    1. Sketch the vector field illustrating the rate of change $\diff{x}{t}$.
    2. Find the equilibria and calculate their stability.
    3. Graph the solution $x(t)$
      1. for initial condition $x(0)=3.8$.
      2. for initial condition $x(0)=4.2$.

  3. Consider the differential equation \begin{align*} \diff{ z }{ t } &= 2.5 z. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $z(0) = 6.2$?

  4. For the differential equation \begin{align*} \diff{q}{t} = e^{-q^2-3q+1} \end{align*} find the equilibria and use the stability theorem to calculate their stability. Graph the solution for the initial condition $q(0)=0$.

  5. 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?

  6. For the dynamical system $ z'(t) = h(z, \beta)$, where the function $h$ of $z$ also depends on a parameter $\beta$, a bifurcation diagram with respect to the parameter $\beta$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.
    1. For the following three values of $\beta$, determine the number of equilibria, their values, rounded to the nearest integer, and their stability. For each case, sketch the phase line, including equilibria and vector field. Use a solid circle for stable equilibria and an open circle for unstable equilibria.
      1. $\beta= -8$
      2. $\beta= -1$
      3. $\beta= 8$
    2. Identify any bifurcation points.

  7. Estimate the solution of the differential equation \begin{align*} \diff{ w }{t} &= 8 e^{- 1.4 w^{2}}\\ w(0) &= 0.7, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.1$ to estimate $w(0.3)$.

  8. Consider the dynamical system \begin{align*} \diff{ y }{t} &= -2 \left(y + 1\right) \left(y - 5\right) \left(y - 7\right)\\ y(0) & = y_0, \end{align*} where $y_0$ is an initial condition. The graph of the function $f(y) = -2 \left(y + 1\right) \left(y - 5\right) \left(y - 7\right)$ is shown below. Use the graph to sketch the solution $y(t)$ for each of the following initial conditions: $y_0 = -1.6$, $y_0 = -1$, $y_0 = 0.2$, $y_0 = 5$, $y_0 = 6.4$, $y_0 = 7$, and $y_0=7.6$.

  9. For the dynamical system \begin{align*} \diff{ z }{t} &= \left(- 8 z^{2} + 2 z\right) e^{- z}, \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 1241, Fall 2020