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Worksheet on going from discrete dynamical systems to continuous dynamical systems
Elementary dynamical systems
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Show a graph of the solution to the discrete model $x_{n+1}-x_n = 2.4 x_n(1-x_n)$ with initial conditions $x_0=0.01$. Be sure to carefully label what your values on your time axis mean.
Moving to continuous time
How did your solution to the
model of equation (3)
with $\Delta t$ compare with the solution to the original discrete model?
When changing $\Delta t$ to 8, what do you observe? Is the solution $x(t)$ calculated at intervals $\Delta t=8$ more or less the same as the solution calculated at intervals $\Delta t= 16$? If it looks any different, describe those difference.
Describe what changes you observe as you decrease $\Delta t$ from 16 down to 0.5. Do the changes become more or less dramatic as you continue to decrease $\Delta t$ to smaller values? Can you predict what might happen if you decreased $\Delta t$ to even smaller values?
Does the transformation from the difference equation to the differential equation seem reasonable? Or does it seem to be a bunch of nonsense? Why or why not?
From using the Forward Euler algorithm to estimate the solution to the continuous logistic equation with $r=0.1$, $1$, and $10$ and $M=10$ and $100$, answer the following questions.
What happens to $x(t)$ as $t$ gets large? How does this behavior depend on the initial condition $x(0)$? Do you get the same long term behavior for any value of the initial population size $x(0)$? Or do you get different behavior for different ranges of $x(0)$?
How long does it take $x(t)$ to get to this long term behavior. One way to answer this might be to examine, if you think that $x(t)$ should approach a value $c$, how long does it take $x(t)$ to get to that 90% of value if you start halfway there, i.e., with $x(0)=c/2$? How does this time depend on $r$ or $M$?
What value of $\Delta t$ did you need to choose so that it was small enough to give accurate results? Did this value depend on the parameters $r$ or $M$, or did it depend on the initial condition $x(0)$?
What happens to the estimate solution if you take too large of a time step $\Delta t$?
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Elementary dynamical systems
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From discrete dynamical systems to continuous dynamical systems
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Introduction to autonomous differential equations