7. Light and shadow continue their eternal dance. Shadow grows in the spaces that light has...

   

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7. Light and shadow continue their eternal dance. Shadow grows in the spaces that light has left while the inhabitants of this universe are wont to produce light where the shadow is darkest. L' = S S' = -L (8 pts) Sketch the vector field of this system using a minimum of eight vectors. Then sketch out two approximate trajectories for this system and name the 2D equilibrium point that is present. S -3 -2 0 3 sink center stable spiral saddle point 11 8. Bifurcation analysis Consider model M for system S which the state variable X(t). The change equation X'(t) utilizes parameter p. Consider the bifurcation diagram for parameter p below. 60 50 40 X(t) 30 20 10 0 1 2 3 5 P 7 8 9 10 a. (6 pts) Identify and label all bifurcation points on the above diagram. b. (4 pts) Draw a qualitative phase portrait for the system at p = 3. 12 9. You are provided the following code, which analyzes a population model. def test(Nprime, eqlist): dNprimedN(N)=diff(Nprime(N)) for eqpoint in eqlist: if dNprimedN(eqpoint)>0: print("1st kind of EP") elif dNprimedN(eqlist) <0: print ("2nd kind of EP") else: print(Test failed) model(X) = 0.2*X*(1-X/350)*(X/100-1) eqs = [a,b,c] test(model, eqs) (10) (11) (12) a. (3 pts) Explain what function Line 2 is performing in this code. b. (3 pts) Find and describe one coding error above. Mark the number for the line of code you picked in the box below. c. (2 pts) What type of equilibrium point is being described in Lines 4 and 5? stable unstable semistable d. (3 pts) For the equation in Line 10, what are the three equilibrium points? Write them in ascending order in the "eqs" list, replacing a, b, c in Line 11 13 7. Light and shadow continue their eternal dance. Shadow grows in the spaces that light has left while the inhabitants of this universe are wont to produce light where the shadow is darkest. L' = S S' = -L (8 pts) Sketch the vector field of this system using a minimum of eight vectors. Then sketch out two approximate trajectories for this system and name the 2D equilibrium point that is present. S -3 -2 0 3 sink center stable spiral saddle point 11 8. Bifurcation analysis Consider model M for system S which the state variable X(t). The change equation X'(t) utilizes parameter p. Consider the bifurcation diagram for parameter p below. 60 50 40 X(t) 30 20 10 0 1 2 3 5 P 7 8 9 10 a. (6 pts) Identify and label all bifurcation points on the above diagram. b. (4 pts) Draw a qualitative phase portrait for the system at p = 3. 12 9. You are provided the following code, which analyzes a population model. def test(Nprime, eqlist): dNprimedN(N)=diff(Nprime(N)) for eqpoint in eqlist: if dNprimedN(eqpoint)>0: print("1st kind of EP") elif dNprimedN(eqlist) <0: print ("2nd kind of EP") else: print(Test failed) model(X) = 0.2*X*(1-X/350)*(X/100-1) eqs = [a,b,c] test(model, eqs) (10) (11) (12) a. (3 pts) Explain what function Line 2 is performing in this code. b. (3 pts) Find and describe one coding error above. Mark the number for the line of code you picked in the box below. c. (2 pts) What type of equilibrium point is being described in Lines 4 and 5? stable unstable semistable d. (3 pts) For the equation in Line 10, what are the three equilibrium points? Write them in ascending order in the "eqs" list, replacing a, b, c in Line 11 13