Notes on files example?.mov and HopfPoint.mov

Theory

Fix the system parameters r and m.  Then, given α, the linearization around the relative equilibrium is given by \(TraditionalForm
      \`M\ \(x\& .. \) + S\ x\& .  + Λ == 0\), where \(TraditionalForm
      \`x = {δr\_1, δr\_2, δφ}\) and

M=(({{((𝓂 (Subscript[𝓂, 2]))/(1 - (λ^2))), (Subscript[𝓂, 2]) (-((α 
                                    (λ^2))/((Sqrt[1 - (λ^2)]) 
                                    (Sqrt[(r^2) - 
                                        (α^2) (λ^2)]))) - 1)
                          , 0}, {(Subscript[𝓂, 2]) (-((α 
                                    (λ^2))/((Sqrt[1 - (λ^2)]) 
                                    (Sqrt[(r^2) - 
                                        (α^2) (λ^2)]))) - 1)
                          , (((Subscript[𝓂, 2]) (r^2))/((r^2) - (α^2) (λ^2))), 0}, {0, 0, ((((((Subscript[ℓ, 1])))^2) (𝓂-1) (Subscript[𝓂, 2]) (α^2) (λ^2))/(𝓂+α (α + 2)))}}))

S=(({{0, 0, ((2 (𝓂-1) (α^2) μ)/((Subscript[ℓ, 1]) (((𝓂+α (α + 2)))^2) λ))}, {0, 0, ((2 (𝓂-1) α μ)/((Subscript[ℓ, 1]) (((𝓂+α (α + 2)))^2) λ))}, {-((2 (𝓂-1) (α^2) μ)/((Subscript[ℓ, 1]) (((𝓂+α (α + 2)))^2) λ)), -((2 (𝓂-1) α μ)/((Subscript[ℓ, 1]) (((𝓂+α (α + 2)))^2) λ)), 0}}))

Λ=(({{(((3 (𝓂^2)-(α - 6) α 𝓂+4 (α^2)) (μ^2))/(((((Subscript[ℓ, 1])))^4) (Subscript[𝓂, 2]) (((𝓂+α (α + 2)))^3) (λ^4)))+((ℊ 𝓂 (Subscript[𝓂, 2]))/((Subscript[ℓ, 1]) (((1 - (λ^2)))^(3/2)))), -(((α (3 α + 2)+𝓂 (4 α + 3)) (μ^2))/(((((Subscript[ℓ, 1])))^4) (Subscript[𝓂, 2]) (((𝓂+α (α + 2)))^3) (λ^4))), 0}, {-(((α (3 α + 2)+𝓂 (4 α + 3)) (μ^2))/(((((Subscript[ℓ, 1])))^4) (Subscript[𝓂, 2]) (((𝓂+α (α + 2)))^3) (λ^4))), (Subscript[𝓂, 2]) (((ℊ ((((Subscript[ℓ, 1])))^2) (r^2))/(((((((Subscript[ℓ, 1])))^2) ((r^2) - (α^2) (λ^2))))^(3/2)))-(((-3 (α^2)-6 α+𝓂-4) (μ^2))/(((((Subscript[ℓ, 1])))^4) ((((Subscript[𝓂, 2])))^2) (((𝓂+α (α + 2)))^3) (λ^4)))), 0}, {0, 0, ((α (μ^2))/(((((Subscript[ℓ, 1])))^2) (Subscript[𝓂, 2]) (((𝓂+α (α + 2)))^2) (λ^2)))}}))

(Notice that (Subscript[l, 1]) and (Subscript[m, 2]) still appear in the expressions above; in all my computations I fix (Subscript[l, 1]) = (Subscript[m, 2]) = 1.)  Here λ depends on α according to \(TraditionalForm
      \`λ\^2 = 
        \(L\^2 - r\^2\)\/\(L\^2 - α\^2\) /. 
          L -> 
            \((1 + α\/m)\) \((α\/\(1 + α\))\)\).

(Also, we take the gravitational acceleration g = 1.)

According to [Marsden&Scheurle] we have that the eigenvalues of the linearized system are the roots of the polynomial \(TraditionalForm
      \`p[γ] := 
        Det[γ\^2\ M + γ\ S + Λ]\).

In the included, I fix the system parameters r, m and plot the eigenvalues corresponding to a choice of α.

Comments

example1.mov

parameters {r->1,m->2}.

An animation of the Hamiltoninan-Hopf bifurcation at aprox. α == -1.67 for system parameters {r -> 1, m -> 2}.  

Remark:  Also we see passing of eigenvalues at α ≈ -(Sqrt[2]), i.e. where the branch converges to the straight down position; what can this mean, given that we have a discontinuity in the branch?
(Afterthought: a better observation seems to suggest that the passing occurs at an α<-(Sqrt[2]) .)

example2.mov

parameters {r->2,m->2}.

example3.mov

parameters {r -> 4/3, m -> 2}.

In this animation, proceding differently as we did in the previous ones, we keep α fixed and vary λ.  (Remember that for these particular system parameters the graph of λ vs .  α contains a vertical line at α = 4/3.)
Observe that we don't seem to recover any Hamiltonian-Hopf bifurcation in the limit λ -> 0.
Remarks: 1. What happens for values close to  .99 ≈ 1?

HopfPoint.mov

After figuring out that I could easily plot the movement of eigenvalues, I made an algorithm that detects for which value of α = (Subscript[α, hopf]) where splitting occurs (Hamiltonian Hopf bifurcation).  Given a pair of system parameters r, m, the "Hopf point" {(Subscript[α, hopf, ]) (Subscript[λ, hopf])} is thus determined, (where (Subscript[λ, hopf]) = λ[(Subscript[α, hopf])]).

In the animation the "Hopf point" is plotted in red.  m is taken to be equal to 2.