Classifying Equilibria_ HelpNotes.ipynb

Kernel: SageMath 9.1

Hi everyone. Here are the help notes to clarify some points of lab-8.

#FORGET ABOUT THE subs function. We will not be using it for this lab.
#Instead, we shall find the expression for the derivative (using the diff function)
#However, we shall store that expression as a mathematical function of some dummy variable, say P. Let's see.

Nprime(N) = 0.2 * N * (1 - N/200) * (N/100 - 1)
#Let us see what diff function does:
print(diff(Nprime))

N |--> -0.0000100000000000000*(N - 100)*(N - 200) - 0.0000100000000000000*(N - 100)*N - 0.0000100000000000000*(N - 200)*N

We see that the diff function simply differentiated Nprime and gave an expression. I now want to store this expression as a mathematical function of some dummy variable,P

derNprime(P) = diff(Nprime(P))#THE MOST IMPORTANT LINE. Converts the EXPRESSION FOR THE DERIVATIVE OF NPRIME INTO A MATHEMATICAL FUNCTION named derNprime.
print(derNprime)

P |--> -0.0000100000000000000*(P - 100)*(P - 200) - 0.0000100000000000000*(P - 100)*P - 0.0000100000000000000*(P - 200)*P

#derNprime is the mathematical function that represents the derivative of Nprime.
#Since derNprime is a mathematical function, I can compute it for a specific input simply by:
print(derNprime(100))

0.100000000000000

#13. We can therefore do exercise 13 as:
Nprime(N) = 0.2 * N * (1 - N/200) * (N/100 - 1)
derNprime(P) = diff(Nprime(P))
print(derNprime)

P |--> -0.0000100000000000000*(P - 100)*(P - 200) - 0.0000100000000000000*(P - 100)*P - 0.0000100000000000000*(P - 200)*P

#16.
def stability (chng_eqn, eql_list):
    
    derXprime(P) = diff(chng_eqn(P)) #Converts the derivative of chng_eqn into a mathematical function named derXprime.
    
    for eq in eql_list:
        val = derXprime(eq)
        if val < 0:
            print("STABLE")
        elif val > 0:
            print("UNSTABLE")
        else:
            print("TEST FAILS")

my_eqn(N) = 0.2 * N * (1 - N/200) * (N/100 - 1);
eq_pts = [0, 100, 200]
stability(my_eqn, eq_pts)

STABLE
UNSTABLE
STABLE