Math Insight

Consider the dynamical system \begin{align*} X_{ t+1} - X_t = \frac{ b X_t +c }{ k }, \quad \text{for $t=0,1,2,3, \ldots$} \end{align*} where $b$, $c$, and $k$ are parameters, and $k \ne 0$. Find all equilibria.

Consider the discrete dynamical system \[ \left\{ \begin{array}{r c l} w_{ t+1} & = & f(w_t)\\ w_0 & = & 5 \end{array} \right. \] where the function $f$ is graphed along with the diagonal $w_{ t+1} = w_t$, below.

1. Find the equilibria of the system. Label points corresponding to the equilibria on the graph above and give estimates of their numerical values.

2. Use cobwebbing to determine the stability of each equilibrium.
3. For any stable equilibrium, determine the interval around the equilibrium for which initial conditions $w_0$ lead to solutions $w_{ t }$ that approach the equilibrium as the time $t$ increases. In other words, find the largest interval around the equilibrium that is part of the basin of attraction.

Solve the dynamical system \begin{align*} r_{ t+1} - r_t &= 0.9 r_t\\ r_0 &= 72.6. \end{align*}

What is an equilibrium of a dynamical system? What does it mean for an equilibrium to be stable? To be unstable?

Consider the dynamical system \begin{align*} P_{ t +1}-P_t = \frac{(P_t -\lambda)(P_t-\alpha)(P_t-\mu)}{ \xi }, \quad \text{for $t=0,1,2,3, \ldots$ } \end{align*} where $\lambda$, $\alpha$, $\mu$, and $\xi$ are parameters, with $\xi \ne 0$. Find all equilibria.

Consider the dynamical system \begin{align*} P_{ n+1} - P_n &= \frac{ P_n(P_n -2)}{ 4 } \quad \text{for $n=0,1,2,3, \ldots$ .} \end{align*}

1. Find the equilibria.
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   Hint

   Pick a variable for the equilibria, substitute it for all values of the state variables, and solve for that variable. (We often use $E$ for the equilibrium variable, but you can choose a different symbol if you like.)

   Helpful pages

   • Equilibria in discrete dynamical systems

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2. 1. If $P_0 = -0.01$, calculate $P_1$, $P_2$, $P_3$, and $P_4$.
   2. If $P_0= 0.01$, calculate $P_1$, $P_2$, $P_3$, and $P_4$.
   3. Based on the above calculations, what can you infer about the stability of one of the equilibria?
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   Hint

   1. Plug in $P_0=-0.01$ into the dynamical system with $n=0$ to get $P_1$. Repeat with $n=1,2,3$ to get $P_2$, $P_3$, and $P_4$. A calculator would be nice for this.
   2. Do the same thing, only starting with $P_0=0.01$.
   3. Do both trajectories (that sequences of values calculated above) approach an equilibrium? If so, that is evidence that the equilibrium may be stable. If they move away from an equilibrium, it is evidence that that equilibrium may be unstable.

   Helpful pages

   • An introduction to discrete dynamical systems
   • The idea of stability of equilibria for discrete dynamical systems

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3. 1. If $P_0 = 1.99$, calculate $P_1$, $P_2$, $P_3$, and $P_4$.
   2. If $P_0= 2.01$, calculate $P_1$, $P_2$, $P_3$, and $P_4$.
   3. Based on the above calculations, what can you infer about the stability of one of the equilibria?
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   Hint

   1. Plug in $P_0=1.99$ into the dynamical system with $n=0$ to get $P_1$. Repeat with $n=1,2,3$ to get $P_2$, $P_3$, and $P_4$. A calculator would be nice for this.
   2. Do the same thing, only starting with $P_0=2.01$.
   3. Do both trajectories (that sequences of values calculated above) approach an equilibrium? If so, that is evidence that the equilibrium may be stable. If they move away from an equilibrium, it is evidence that the equilibrium may be unstable.

   Helpful pages

   • An introduction to discrete dynamical systems
   • The idea of stability of equilibria for discrete dynamical systems

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Consider the discrete dynamical system \[ \left\{ \begin{array}{r c l} x_{ t+1} & = & x_t\left(2.0-x_t\right) \\ x_0 & = & 1.2\\ \end{array} \right. \]

1. Calculate the equilibria analytically.

   $E=$________

2. Compute the next three points of the solution.

   $x_0=$________

   $x_1=$________

   $x_2=$________

   $x_3=$________

3. On the graph below, label the equilibria and cobweb the dynamical system for four steps.

Consider the dynamical system \begin{align*} W_{ t+1} -W_t &= \delta W_t \quad \text{for $t=0,1,2,3, \ldots$ } \end{align*} where $\delta$ is a parameter, and $\delta \ge -1$.

1. Solve the dynamical system to give a formula for $W_t$ in terms of the initial condition $W_0$, the parameter $\delta$, and the number $t$.
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   Hint

   To solve the dynamical system, first put it in “function iteration” form by adding $W_t$ to both sides. Then, the update equation will be in the form $W_{ t+1}$ equals something times $W_t$. This means, in each time step, we need to multiply by that something. Therefore, to go from the initial condition $W_0$ to the value $W_t$ in time step $t$, we just need to multiply that that something $t$ times. The solution is therefore an exponential function (either exponential growth or decay, depending on that something, but you get to worry about that in part c).

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   • Exponential growth and decay modeled by discrete dynamical systems

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2. Find the equilibria. Hint: treat the case $\delta=0$ separately from the case $\delta \ne 0$.
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   Hint

   Plug in some variable for the $W$'s and solve. There is one subtlety here. We are not told that $\delta \ne 0$, so we have to think about what happens in the special case when $\delta=0$.

   Helpful pages

   • Equilibria in discrete dynamical systems

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3. For each equilibrium, determine for which values of $\delta$ the equilibrium is stable. Hint: treat the case $\delta=0$ separately from the case $\delta \ne 0$.
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   Hint

   Your solution should be in the form of $W_t = (\mathit{something})^t W_0$, where the something is a function of the parameter $\delta$.

   Can you find an interval of $\delta$ where $(\mathit{something})^t$ gets smaller and smaller as $t$ increases? In that case, the solution will approach 0. (Is 0 an equilibrium?) That case would seem like exponential decay. Would you call the equilibrium stable or unstable in this case?

   Can you find an interval of $\delta$ where $(\mathit{something})^t$ gets larger and larger as $t$ increases? In that case, the solution will blow up if you start with any non-zero initial condition. That case would seem like exponential growth. If 0 happens to be an equilibrium, would you call it stable or unstable in this case?

   Is there a particular value of $\delta$ where $(\mathit{something})^t=1$ for all $t$? In this case, it would seem like the system is pretty boring with nothing changing. According to our definition of stability, would the equilibrium that you found be stable or unstable in this case?

   Helpful pages

   • The idea of stability of equilibria for discrete dynamical systems
   • Exponential growth and decay modeled by discrete dynamical systems

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