Parametric stability of a double pendulum with variable length and with its center of mass in an elliptic orbit

Figure 1.  Double pendulum in an elliptical orbit. Orbital plane

Figure 2.  Graphics of: (1) $ \beta^{(2)}(e) $, (2) $ \beta^{(3)}(e) $, (3) $ \beta^{(5}(e) $, (4) $ \beta^{(6)}(e) $, (5) $ \beta^{(7)}(e) $, (6) $ \beta^{(8)}(e) $ in the plane $ e\times \beta $

Figure 3.  Graphics of: (1) $ \beta^{(1)}(e) $ and (2) $ \beta^{(4)}(e) $ in the plane $ e\times \beta $

Figure 4.  Regions of stability and instability for the equilibrium point $ E_2 $. Graphics of (1): $ \beta^{(1)}(e) $, (2): $ \beta^{(2)}(e) $, (3): $ \beta^{(3)}(e) $, (4): $ \beta^{(4)}(e) $, (5): $ \beta^{(5)}(e) $, (6): $ \beta^{(6)}(e) $, (7): $ \beta^{(7)}(e) $, (8): $ \beta^{(8)}(e) $

Figure 5.  Regions of stability and instability for the equilibrium point $ E_3 $. Graphics of (1): $ \beta^{(1)}(e) $, (2): $ \beta^{(2)}(e) $, (3): $ \beta^{(3)}(e) $, (4): $ \beta^{(4)}(e) $, (5): $ \beta^{(5)}(e) $, (6): $ \beta^{(6)}(e) $, (7): $ \beta^{(7)}_+(e) $, (8): $ \beta^{(7}_-(e) $, (9): $ \beta^{(8)}(e) $, (10): $ \beta^{(9)}(e) $

Figure 6.  Regions of stability and instability for the equilibrium point $ E_3 $. Graphics of $ \beta^{(7)}_+(e) $ and $ \beta^{(7)}_-(e) $. The shaded region is where the equilibrium point is unstable