A classical ODE system can be described by three odes. The equations are xX=0(y-x) y=rx=y=xz I=xy=-bz Here a, r, b are all positive parameters. Consider the scenario that o=10, b=8/3, r=28. 1 Try to solve this ode with the initial condition to be (0,1,0) with I to be long enough, Then a. Plotyvs t. Look at the figure and try to describe what you observed i. Hint: do you see a patterny il. Hint: think again, is it really a pattern? Maybe try to run it longer? b. Plot zvs x. Look at the figure and try to describe what you observed, I, Hint: Do you see a patterny i, Hint: think again, . Plot x vs yvs 2 in 3D. Describe what you see. 2. Try to solve this with a slightly different initial condition (0.01,1,0). a. Plot yvs t. Plot this in the same figure with the plot from 1.a. Using different color to distinguish these two plots. Describe what you see and try to explain/speculate the reason. 3. One obvious fixed point of Lorenz equation is (0,0,0). There are two ather fixed points, try to solve for them. Plot these three fixed points in the plot in the above sections as well, See where they are? 4, For the two non-zero fixed points, show that the characteristic equation for the gigenvalues of the Jacobian matnx s B r{o+b+ 1A+ (r+a)bd+2ba(r=1)=0 5. There is a bifurcation point where _ a+b+3 W=t =1 Our choice of r=28 make it bigger than r_H. Try to repeat 1,2,3 with a r value smaller than r_H and make another plot, see what the trajectories look like and explain what you see