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(10 points) In this exercise, you examine a simple model for the elliptic burster (ellip.ode,ellip.m) : xyz=x(1z+rr2)y=y(1z+rr2)+z=(rr0cz) where r=x2+y2. Simulate the burster for 1000 time units and plot x(t) over that period with the parameters =0.01,r0=0.2,c=0.1. Now plot r versus z. (In MatLab, columm 4 and column 5, in XPP, z and r; click on Window Fit to get the correct window). It is just a simple limit cycle. Now we will analyze it. Let x=rcos,y=rsin as a change of variables and show that your get: r=r(1z+rr2)z=(rr0cz)=1. So, r,z do not depend on and you now have a planar system. Let's set c=0. Sketch the phase-plane with z on the horizontal axis and r on the vertical include the nullclines. Find the equilibria assuming r00. Determine the stability as a function of r0. For what range of values of r0 do you expect to find limit cycles in the (z,r)-system. Limit cycles in (z,r) mean elliptic bursting in x,y,z !! Can this elliptic burster system have chaos? (Can a 2D system of ODEs have chaos?)