# Lab 9: :( # Name: Kelly # I worked on this code with: # Please do all of your work for this week's lab in this worksheet. If # you wish to create other worksheets for scratch work, you can, but # this is the one that will be graded. You do not need to do anything # to turn in your lab. It will be collected by your TA at the beginning # of (or right before) next week’s lab. # Be sure to clearly label which question you are answering as you go and to # use enough comments that you and the grader can understand your code.

#1 Exercise 1. By hand, find the equilibria of the system R0 = aR + bJ, J0 = cR + dJ. #(0,0) #Exercise 2. Declare a, b, c, d, R and J as symbolic variables. a = 0.25 b=0.5 c=0.65 d=0.75 var("R,J") Rprime(R)= a*R +b*J Jprime(J)= c*R + d*J

(R, J)

#Exercise 3. Make a list of eight points, two in each quadrant of the R−J plane, to serve as initial conditions for simulations. Recall that points are defined using parentheses: (R1, J1) is a point. Rpoints=[-3,-1,1,3] Jpoints=[-2,4,2,-4] coordinates = zip(Rpoints,Jpoints) coordinates

[(-3, -2), (-1, 4), (1, 2), (3, -4)]

#Exercise 4. Pick a positive value for a and d. (For simplicity, make them equal to each other.) Plot the system’s vector field, letting both R and J range from -10 to 10. Is the equilibrium stable or unstable? a=0.25 d=0.25 Rprime(R)= a*R Jprime(J)= d*J plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10))

#IT IS UNSTABLE #Exercise 5. Now, make a and d negative and plot the vector field. Is the equilibrium stable or unstable? a=-0.50 d=-0.50 Rprime(R)= a*R Jprime(J)= d*J plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10))

#it is stable #Exercise 6. Interpret the results of the previous two exercises in terms of the relationship between Romeo and Juliet. HINT: If you have trouble doing this from just the vector field, try running some simulations for various initial conditions using desolve_odeint. # when a & d are both positive, as romeo's love for juliet increases, juliet's love for romeo increases as well . When a & d are negative, as romeo's love for juliet decreases, juliets love for him decreases as well #Exercise 7. Keeping the absolute value of a and d the same, make one ofthem positive and the other negative. Plot the vector field as before. Does the equilibrium appear stable or unstable a=-0.50 d=0.50 Rprime(R)= a*R Jprime(J)= d*J plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10))

#It is semi stable, a saddle point #Exercise 8. Overlay a large red point at the equilibrium on top of the vector field (use the point command to plot points) and assign this plot to a variable. Then, use a for loop to simulate the dynamics of the system for each initial condition in the list you defined earlier, make a plot of the resulting trajectory and add it to the existing vector field. View the result. HINT: If your trajectories go outside the vector field, display the plot using show and specify xmin, xmax, ymin and ymax, as necessary. p=plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10)) + point([0,0], size =80, color = "red") for i in coordinates : traj = desolve_odeint([Rprime,Jprime],ics=i,times=srange(0,50,0.1),dvars=[R,J]) p += list_plot(zip(traj[:,0],traj[:,1]), color = "purple", thickness = 2, plotjoined= True) show(p,xmin=-10,xmax=10, ymin = -10, ymax=10)

#Exercise 9. Describe what you see in purely visual and dynamical terms, without referring to the meaning of the model. Is the equilibrium point stable or unstable? #the trajectory moves away from the origin going up and down and towards it going left and right. that makes the equillibrium point unstable #Exercise 10. Run simulations and plot time series for a couple of different initial conditions. Draw on the vector field, trajectories and time series to describe what is happening to Romeo and Juliet’s relationship in this model. p=plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10)) + point([0,0], size =80, color = "red") for i in coordinates : traj = desolve_odeint([Rprime,Jprime],ics=i,times=srange(0,50,0.1),dvars=[R,J]) p += list_plot(zip(traj[:,0],traj[:,1]), color = "purple", thickness = 2, plotjoined= True) show(p,xmin=-10,xmax=10, ymin = -10, ymax=10)

#Exercise 11. Pick a positive value for b and c, keeping them equal to each other. Plot the vector field. What kind of equilibrium point appears? var("R","J") b = 0.5 c = 0.5 Rprime = b*J Jprime = c*R

(R, J)

plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10))

#a saddle node bifurcation #Exercise 12. What happens if b and c are both negative? b = -0.5 c = -0.5 Rprime = b*J Jprime = c*R plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10))

#a saddle node bifurcation in the other direction #Exercise 13. Keeping the absolute value of b and c the same, make one of them positive and the other negative. Plot the vector field as before. b = -0.5 c = 0.5 Rprime = b*J Jprime = c*R plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10))

#it's a neutral center #Exercise 14. Overlay a large red point at the equilibrium on top of the vector field (use the point command to plot points) and assign this plot to a variable. Then, use a for loop to simulate the dynamics of the system for each initial condition in the list you defined earlier, make a plot of the resulting trajectory and add it to the existing vector field. View the result. p=plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10)) + point([0,0], size =80, color = "red") for i in coordinates : traj = desolve_odeint([Rprime,Jprime],ics=i,times=srange(0,50,0.1),dvars=[R,J]) p += list_plot(zip(traj[:,0],traj[:,1]), color = "purple", thickness = 2, plotjoined= True) show(p,xmin=-10,xmax=10, ymin = -10, ymax=10)

#Exercise 15. Describe what you see in purely visual and dynamical terms, without referring to the meaning of the model. Is the equilibrium point stable or unstable? #stable since all trajectories spiral towards the origin #Exercise 16. Run simulations and plot time series for a couple of different initial conditions. Draw on the vector field, trajectories and time series to describe what is happening to Romeo and Juliet’s relationship in this model. p=plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10)) + point([0,0], size =80, color = "red") for i in coordinates : traj = desolve_odeint([Rprime,Jprime],ics=i,times=srange(0,50,0.1),dvars=[R,J]) p += list_plot(zip(traj[:,0],traj[:,1]), color = "purple", thickness = 2, plotjoined= True) show(p,xmin=-10,xmax=10, ymin = -10, ymax=10)

#Exercise 17. Plot the vector field for this system. Since it will be hard to tell exactly what’s going on from the vector field alone, pick an initial condition, run a simulation and overlay the trajectory on top of the vector field. var("R","J") Rprime(R)= J Jprime(J)=-R-0.05*J p=plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10)) + point([0,0], size =80, color = "red") traj = desolve_odeint([Rprime,Jprime],ics=coordinates[0],times=srange(0,50,0.1),dvars=[R,J]) p += list_plot(zip(traj[:,0],traj[:,1]), color = "purple", thickness = 2, plotjoined= True) show(p,xmin=-10,xmax=10, ymin = -10, ymax=10)

(R, J)

#Exercise 18. Plot time series for R and J. Describe what is happening in terms of Romeo and Juliet’s relationship for i in coordinates : traj = desolve_odeint([Rprime,Jprime],ics=i,times=srange(0,50,0.1),dvars=[R,J]) p += list_plot(zip(traj[:,0],traj[:,1]), color = "purple", thickness = 2, plotjoined= True) show(p,xmin=-10,xmax=10, ymin = -10, ymax=10)

#Exercise 19. This kind of equilibrium is called a stable spiral. Why is the word “stable” appropriate? #because the trajectory is spiraling towards the eq point #Exercise 20. Plot the vector field for this system and overlay a trajectory ontop of it. var("R","J") Rprime(R)= J Jprime(J)=-R+0.05*J p=plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10)) + point([0,0], size =80, color = "red") traj = desolve_odeint([Rprime,Jprime],ics=coordinates[2],times=srange(0,50,0.1),dvars=[R,J]) p += list_plot(zip(traj[:,0],traj[:,1]), color = "purple", thickness = 2, plotjoined= True) show(p,xmin=-10,xmax=10, ymin = -10, ymax=10)

(R, J)

#Exercise 21. Plot time series for R and J. Describe what is happening in terms of Romeo and Juliet’s relationship. var("R","J") Rprime(R)= J Jprime(J)=-R+0.05*J p=plot_vector_field((Rprime,Jprime),(R,-10,10),(J,-10,10)) + point([0,0], size =80, color = "red") for i in coordinates : traj = desolve_odeint([Rprime,Jprime],ics=i,times=srange(0,50,0.1),dvars=[R,J]) p += list_plot(zip(traj[:,0],traj[:,1]), color = "purple", thickness = 2, plotjoined= True) show(p,xmin=-10,xmax=10, ymin = -10, ymax=10)

(R, J)

#As romeo's love for Juliet increases, Juliet's love for him decreases with some increase in proportion to her own love #Exercise 22. What might this kind of equilibrium be called? #unstalbe spiral, a source