Math Insight

Single autonomous differential equation problems

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  1. For the dynamical system \begin{align*} \diff{ z }{t} = f(z, \gamma), \end{align*} where the function $f$ of $z$ also depends on a parameter $\gamma$, a bifurcation diagram with respect to the parameter $\gamma$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.
    1. For the following three values of $\gamma$, 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. $\gamma= -3$
      2. $\gamma= 4$
      3. $\gamma= 10$

    2. Identify any bifurcation points.

  2. Consider the dynamical system \begin{align*} \diff{ w }{t} &= -4 \left(w + 1\right) \left(w - 3\right) \left(w - 4\right)\\ w(0) & = w_0, \end{align*} where $w_0$ is an initial condition. Find all equilibria and analytically determine their stability. Use this information to draw a phase line diagram, including equilibria (using closed circles for stable equilibria and open circles for unstable equilibria) and vector field. Then, sketch a graph of the solutions corresponding to the equilibria (using solid lines for stable equilibria and dashed lines for unstable equilibria) and solutions for initial conditions $w_0 = -6.4, 2.2, 3.3, 6.4$.

  3. For the dynamical system $ \diff{ w }{t} = f(w,b),$ the function $f$ of $w$ depends on a parameter $b$, as shown in the graphs for $b=-3, 1, 5, 17$, 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 $-3 \le b \le 17$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $b=-3$

    $b=5$

    $b=1$

    $b=17$

  4. Estimate the solution of the differential equation \begin{align*} \diff{ z }{t} &= - 4 e^{- 1.8 z} + 7\\ z(0) &= 0.2, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.2$ to estimate $z(0.6)$.

  5. Consider the dynamical system \begin{align*} \diff{y}{t} = 3(y-2)(y-1)(y+1). \end{align*}
    1. Find the equilibria and use the stability theorem to calculate their stability.
    2. Sketch the vector field on the phase line illustrating the rate of change $\diff{y}{t}$.
    3. Graph the solution $y(t)$
      1. for the initial conditions $y(0)=1.5$.
      2. for the initial conditions $y(0)=2$.
      3. for the initial conditions $y(0)=2.5$.

  6. 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 $q(0)=0$.

  7. Consider the differential equation \begin{align*} \diff{ x }{ t } &= 2.9 x. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $x(0) = -0.4$?

  8. 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?

  9. 40
    Consider the dynamical system \begin{align*} \diff{ v }{t} &= f(v)\\ v(0) & = v_0, \end{align*} where the function $f$ is graphed to the right and $v_0$ is an initial condition. For each of the following initial conditions, sketch the graph of the solution $v(t)$.
    1. $v_0 = -7.9$
    2. $v_0 = -1.8$
    3. $v_0 = 3.4$
    4. $v_0=8.2$

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