Elementary discrete dynamical system problems

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Elementary discrete dynamical system problems

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What is an equilibrium of a dynamical system? What does it mean for an equilibrium to be stable? To be unstable?
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- Equilibria in discrete dynamical systems
- The idea of stability of equilibria for discrete dynamical systems
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Consider the dynamical system
where δ is a parameter, and δ≥−1.
1. Solve the dynamical system to give a formula for Wt in terms of the initial condition W0, the parameter δ, and the number t.
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  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 δ=0 separately from the case δ≠0.
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  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 $\delta=0$ .
  
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  Equilibria in discrete dynamical systems
  
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3. For each equilibrium, determine for which values of δ the equilibrium is stable. Hint: treat the case δ=0 separately from the case δ≠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?
  
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  The idea of stability of equilibria for discrete dynamical systems
  
  Exponential growth and decay modeled by discrete dynamical systems
  
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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.

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Hint

Don't try to multiply out the cubic. Just set $P_{ t +1}$ and $P_t$ equal to some variable that stands for the equilibrium value and solve for that variable. Solving for the values of that equilibrium variable should be easy when you leave the cubic in factored form.

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Solve the dynamical system
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Hint

Hint 1: Solving the dynamical system means writing $r_t$ , the state variable at time $t$ , just in terms of the time $t$ and the initial condition $r_0$ .

Hint 2: Since the dynamical system is linear, we can easily solve it. But, the first step is to put it into function iteration form, i.e., in the form $r_{ t+1} =$ some function of the previous value $r_t$ . So, you'll want to add $r_t$ to both sides.

Hint 3: Once you do that, you will have $r_{ t+1}$ equals $r_t$ times some number. In other words, to increment the time step, you need to multiply by that number. To go from the initial condition $r_0$ to $r_t$ , how many times do you need to multiply by that number?

If we start with a linear dynamical system, the solution will always be an exponential.

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- Exponential growth and decay modeled by discrete dynamical systems
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- Error in expression: stablelist_all
  Invalid format for expression: ([unstable,stable][Abs(slope1) < 1],[unstable,stable][Abs(slope2) < 1],[unstable,stable][Abs(slope3) < 1],[unstable,stable][Abs(slope4) < 1])
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Consider the discrete dynamical system
where the function f is graphed along with the diagonal wt+1=wt, 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.
Consider the dynamical system
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
  
  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.
  
  Do the same thing, only starting with $P_0=0.01$ .
  
  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
  
  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.
  
  Do the same thing, only starting with $P_0=2.01$ .
  
  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
  
  Hide help
Consider the discrete dynamical system
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*} 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.

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Hint

If $E$ is an equilibrium, this means then setting a state variable to $E$ will keep the state variable at that value forever. In particular, this means that if $X_t=E$ then $X_{ t+1}$ must also be $E$ . Just replace all values of the state variable with $E$ and solve for $E$ .

If you are stuck, you can first try to do this with substituting numbers in for the parameters $b$ , $c$ , and $k$ and finding the equilibria for this simpler case. Then, try changing the numbers and see what is different. Lastly, try to repeat your steps but with leaving the parameters just as $b$ , $c$ , and $k$ .

For the main point of the problem, you can assume that $b \ne 0$ . But, if you want to get every last detail, then you should also check what happens when $b=0$ .

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Elementary discrete dynamical system problems

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Math 201, Spring 2015