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Efficient representation of invariant manifolds of periodic orbits in the CRTBP

  • * Corresponding author: R. Castelli

    * Corresponding author: R. Castelli
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  • This paper deals with a methodology for defining and computing an analytical Fourier-Taylor series parameterisation of local invariant manifolds associated to periodic orbits of polynomial vector fields. Following the Parameterisation Method, the functions involved in the series result by solving some linear non autonomous differential equations. Exploiting the Floquet normal form decomposition, the time dependency is removed and the differential problem is rephrased as an algebraic system to be solved for the Fourier coefficients of the unknown periodic functions. The procedure leads to an efficient and fast computational algorithm. Motivated by mission design purposes, the technique is applied in the framework of the Circular Restricted Three Body problem and the parameterisation of local invariant manifolds for several halo orbits is computed and discussed.

    Mathematics Subject Classification: Primary: 37C27, 70F15; Secondary: 37D10.


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  • Figure 1.  Schematic picture of the action of the conjugating map. The function $\mathbb{P}(\theta, \sigma)$ maps the cylinder in parameter space into the local manifold, in such a way that trajectories of the linear vector field (5) are mapped into trajectories on the manifold.

    Figure 2.  Image of the parameterisation of the local stable (left) and unstable (right) manifold associated to the periodic orbit of Example 1. The black line is the periodic orbit.

    Figure 3.  Image of the parameterisation of the manifolds associated to the Lyapunov orbit of Example 2. The figures at the bottom show the image of $\mathbb{P}(\theta, \sigma)$ for some fixed values of $\theta$ and of $\sigma$.

    Figure 4.  Unstable local invariant manifold for the Halo orbit of example 3.

    Figure 5.  Left: values of the $\log(|a_{\alpha, k}|)$ for $\alpha = 1$ (top line) up to $\alpha = 10$ (bottom line). Right: plot of $\|a_\alpha\|$ in logarithmic scale.

    Figure 6.  Image of the parameterisation of the stable and unstable local manifold associated to the orbit of Exemple 6.

    Figure 7.  Left: The blu-green figure is the image of the parameterisation for the local unstable manifold, the red lines are fibres on the manifold obtained by numerical integration. Right: Visualisation of the planes $\Pi_1$, $\Pi_2$.

    Figure 8.  Poincaré sections of the manifolds at the planes $\Pi_1$, $\Pi_2$. The blue line is the section of the manifold obtained with the parameterisation, the red dots refer to the manifold obtained by numerical integration.

    Figure 9.  Heteroclinic obit. The black curves are the periodic Lyapunov orbits, the red circles are the sections on the stable and unstable manifolds. The green line is the curve connecting the two manifolds, obtained by solving $F(\theta_1, \theta_2, T) = 0$. The blue lines are orbits on the stable and unstable manifolds.

    Table 1.  Computational time

    m N time (s)
    30 6 1.2
    30 10 8.6
    40 6 1.4
    40 10 9.1
    50 4 0.4
    50 6 1.9
    50 9 7.6
     | Show Table
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