Math Insight

Elementary discrete dynamical systems problems, part 2

Problem 1

Consider the dynamical system \begin{align*} x_{n+1} &= x_n^2 \quad \text{for $n=0,1,2,3, \ldots$} \end{align*}

1. Find all equilibria.
2. Determine the stability of the equilibria using calculus.
3. Graph the function and confirm the stability of the equilibria using cobwebbing.

Problem 2

Consider the dynamical system \begin{align*} y_{t+1} &= y_t^3 \quad \text{for $t=0,1,2,3, \ldots$} \end{align*}

1. Find all equilibria.
2. Determine the stability of the equilibria using calculus.
3. Graph the function and confirm the stability of the equilibria using cobwebbing.

Problem 3

Consider the dynamical system \begin{align*} z_{t+1} -z_t &= z_t(1-z_t) \quad \text{for $t=0,1,2,3, \ldots$} \end{align*}

1. Find all equilibria.
2. Determine the stability of the equilibria using calculus.
3. Graph the function and confirm the stability of the equilibria using cobwebbing.

Problem 4

Consider the dynamical system \begin{align*} w_{n+1} -w_n &= 3w_n(1-w_n/2) \quad \text{for $n=0,1,2,3, \ldots$} \end{align*}

1. Find all equilibria.
2. Determine the stability of the equilibria.

Problem 5

Consider the dynamical system \begin{align*} v_{n+1} -v_n &= 0.9v_n(2-v_n) \quad \text{for $n=0,1,2,3, \ldots$} \end{align*}

1. Find all equilibria.
2. Determine the stability of the equilibria.

Problem 6

Consider the dynamical system \begin{align*} u_{t+1} -u_t &= a u_t(1-u_t) \quad \text{for $t=0,1,2,3, \ldots$} \end{align*} where $a$ is a positive parameter.

1. Find all equilibria.
2. For each equilibrium, find the values of $a$ for which you can determine that the equilibrium is stable and the values of $a$ for which you can determine the equilibrium is unstable.

Problem 7

Consider the dynamical system \begin{align*} s_{t+1} -s_t &= 0.5s_t(1-s_t/b) \quad \text{for $t=0,1,2,3, \ldots$} \end{align*} where $b$ is a positive parameter.

1. Find all equilibria.
2. Determine the stability of the equilibria.
3. Does the stability of any of the equilibria depend on $b$?

One you have worked on a few problems, you can compare your solutions to the ones we came up with.