differential equations – Tennis Racket theorem


Torque-free Euler equations experiment seen in low gravity of Russian spacecraft is modelled here with a view to see its tumbling motion around the intermediate axis $omega_2$ rotation. However its reversal is not observed here. The boundary conditions do play a role, varying them did not much change the sine behaviour towards interfering periodic flips.

Due to easy demonstration possibility here I posted this hopefully interesting problem although strictly it is a physics problem.

{I1, I2, I3} = {8, 4, 0.4};
Dzhanibekov = {I1 TH1''(t) == (I2 - I3) TH2'(t) TH3'(t), 
   I2 TH2''(t) == (I3 - I1) TH3'(t) TH1'(t), 
   I3 TH3''(t) == (I1 - I2) TH1'(t) TH2'(t), TH1'(0) == -0.4, 
   TH2'(0) == 0.08, TH3'(0) == 0.65, TH1(0) == 0.75, TH2(0) == -0.85, 
   TH3(0) == 0.2};
NDSolve(Dzhanibekov, {TH1, TH2, TH3}, {t, 0, 15.});
{th1(u_), th2(u_), th3(u_)} = {TH1(u), TH2(u), TH3(u)} /. First(%);
Plot(Tooltip({th1'(t), th2'(t), th3'(t)}), {t, 0, 15}, 
 GridLines -> Automatic)

Please help choose better initial conditions for getting a jump around $theta_2$ axis. Thanks in advance.

Wing Nut Flips

Wiki Ref

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