The Balsa-Wood Glider: Equilibrium Point Analysis with Bifurcation

Computation of the Fundamental Matrix at the equilibrium point:

\(A = {{
          D[\(-Sin[a]\) - D\ v^2, v] /. {a -> \(-ArcTan[D]\), 
              v -> 1/\((D^2 + 1)\)^\((1/4)\)}, 
			
          D[\(-Sin[a]\) - D\ v^2, a] /. {a -> \(-ArcTan[D]\), 
              v -> 1/\((D^2 + 1)\)^\((1/4)\)}}, {
          D[v - Cos[a]/v, v] /. {a -> \(-ArcTan[D]\), 
              v -> 1/\((D^2 + 1)\)^\((1/4)\)}, 
			
          D[v - Cos[a]/v, a] /. {a -> \(-ArcTan[D]\), 
              v -> 1/\((D^2 + 1)\)^\((1/4)\)}}}\)

Its eigenvalues:

eig = Eigenvalues[A]

A graph of the real part of the eigenvalue versus the parameter D:

rep = 
      Plot[Evaluate[Re[eig]], {D, 0, 4}, 
        PlotStyle -> {{AbsoluteThickness[2], Red}}, 
        AxesLabel -> {"D", None}];

A graph of the imaginary part of the eigenvalue versus the parameter D:

imp = 
      Plot[Evaluate[Im[eig]], {D, 0, 4}, 
        PlotStyle -> {{AbsoluteThickness[2], Blue}}, 
        AxesLabel -> {"D", None}];

Show[rep, imp];

The bifurcation from spiral sink to sink occurs at D = 
      2 (Sqrt[ 2.])