Mini-Lecture 5-2 - Discrete-time models of dynamical systems.ipynb

Kernel: SageMath 9.3

All of our models so far (differential equations) have assumed that time ($t$) is continuous.

This was why we needed to use Euler's method, with a “small” $\Delta t$, to approximate solutions to our differential equations. And this was always just an approximation: we can make it more accurate by using a smaller $\Delta t$, but we can never get the exact solution using Euler's method.

But sometimes, it makes sense to think of time as discrete. This means that the $t$ variable can only be a whole number.

This makes sense, for example, for models of populations that have a distinct mating season, or a very fixed life cycle that repeats each year (such as many insects).

Discrete time models

(Also known as difference equations)

f(X) = 2*X
X0 = 5

X1 = f(X0)
X1

10

X2 = f(X1)
X2

20

X3 = f(X2)
X3

40

X4 = f(X3)
X4

80

X5 = f(X4)
X5

160

f(X) = 2*X
X0 = 5

X_list = [X0]
for n in srange(0, 10):
    current_X = X_list[n]
    next_X = f(current_X)
    X_list.append(next_X)

X_list

[5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120]

list_plot(X_list, size=30, axes_labels=("$n$ (or $t$)", "$X$"))

f(X) = 4*X*(1 - X)
X0 = 0.4

X1 = f(X0)
X1

0.960000000000000

X2 = f(X1)
X2

0.153600000000000

X3 = f(X2)
X3

0.520028160000000

X4 = f(X3)
X4

0.998395491228058

X5 = f(X4)
X5

0.00640773729417265

f(X) = 4*X*(1 - X)

solution = [0.4]
t_list = srange(0, 10000)
for t in t_list:
    solution.append(f(solution[-1]))

solution = list(zip(t_list, solution))

p = list_plot(solution[:16], size=30)
p.show(axes_labels=("$t$", "$X$"))

p = list_plot(solution[:61], size=30)
p.show(axes_labels=("$t$", "$X$"))

p += list_plot(solution[:61], plotjoined=True, color="gray")
p.show(axes_labels=("$t$", "$X$"))

p = list_plot(solution[9300:9500], size=20)
p.show(axes_labels=("$t$", "$X$"))

p += list_plot(solution[9300:9500], plotjoined=True, color="gray")
p.show(axes_labels=("$t$", "$X$"))