Automatic Extraction and Characterization of Structures in Area-preserving Maps
1. Background and motivation
Many natural phenomena found in various areas, such as orbital mechanics, fluid dynamics or quantum mechanics, can be described in terms of Hamiltonian dynamics. Multi-dimensional Hamiltonian systems often exhibit chaotic behaviour, which makes their analysis difficult. The use of surfaces of sections, such as a Poincaré return map, is commonly employed to get insight into the phase space of dynamical systems, which present both, regular and chaotic behaviour.
1.1 Surfaces of section for the study of dynamical systems
In the last decade, maps have been used extensively in a wide range of scientific and
engineering problems to understand the dynamical structure of complex systems. A
Poincaré surface of section, as illustrated in Figure 1, can be interpreted as a discrete
dynamical system with a state space that is one dimension smaller than the original
continuous dynamical system. On the map, a periodic trajectory becomes a point,
while a non-periodic trajectory is represented by a set of points.
One of the advantages of the Poincaré map lies in its power as a visualization tool.
Such a map reduces the order of the problem, condensing quantities of information
into a lower-dimensional image. Poincare maps reveal, at a glance, regions of wellordered
behaviour, despite the chaotic nature of the underlying problem. An example
of a Poincaré map is represented in Figure 2. This map is generated in the Planar
Circular Restricted Three-Body Problem (PCR3BP), so that the system is initially
four-dimensional. To create the map, a grid of initial conditions is selected and
integrated forward in time. The intersections of each trajectory with the surface of
section create the Poincaré map. Three types of behaviours are easily identified on the
map represented in Figure 2: periodic orbits, quasi-periodic motion, and chaotic
Fig. 2 Example of Poincaré map (courtesy of Haapala )
Generating maps provides a global picture of the phase space for complex dynamical
systems and offers features that might be difficult to identify otherwise. However,
despite how valuable these maps are, their implementation is not practical. Two main
limitations prevent their utilisation:
An effective analysis of maps remains a difficult task. The complexity of surfaces of
sections often makes the identification of the topology challenging. In computer
visualization, discrete dynamical systems and area-preserving maps are not typical
research topics. Helman and Hesselink introduce some topological approaches to
reproduce a topologic skeleton . Peikert and Sadlo employ a Poincaré map
approach to the visualization of vortex rings [3,4]. Recently, Tricoche, Garth and
Sanderson presented a method to automatically extract and characterise structures on
area-preserving maps . Figure 3 illustrates the algorithm of Tricoche et al., which
captures very subtle structures in the individual islands of the map.
Fig. 3 Structures of a Poincaré map captured via visualization algorithm (courtesy of Tricoche )
1.3 The Keplerian map for resonant transfers between Jovian's moons
Different maps can be generated for different applications. Besides the traditional Poincaré map, other types of maps are the Periapsis Poincaré maps or Keplerian maps, which display the semi-major axis of each trajectory as it evolves over time as a function of the initial periapsis angle ? . A Keplerian map is represented in Figure 4, where the semi-major axis is denoted ‘a’ on the y-axis of the map.
Keplerian maps offer some advantages compared to the traditional Poincaré maps:
Some authors consider Keplerian maps to determine the long-time evolution of nearly parabolic comets [8,9]. In this investigation, Keplerian maps are employed to identify resonant transfer trajectories applicable to spacecraft in a planet-moon system [6,10]. A very challenging part in the design of a planetary moon tour, such as a multi-moon orbiter in the Jupiter system, is the orbital transfer from one planetary moon to another for low-energy transfers. Multiple gravity assists by moons could be used in conjunction with ballistic capture to drastically decrease fuel usage. In planetary systems, the strong dependence on the three-body regimes of motion precludes the use of a patched conic approach. Instead, some recent approaches employ patched three-body models to enable multiple “resonant-hopping” gravity assists. An example of a low-energy inter-moon transfer via resonant gravity assists is represented in Figure 5.
Fig. 5 Inter-moon transfer via resonant gravity assists in the Jupiter system (courtesy Ross et al. )
In Figure 5(a), the spacecraft lowers its perijove by a sequence of successive resonant orbits with the outer moon M1. Once the spacecraft’s orbit comes close to grazing the orbit of the inner moon M2, the inner moon takes “control.” The spacecraft orbit where this occurs is denoted E. In Figure 5(b), the spacecraft now receives gravity assists from the inner moon at perijove and decreases its apojove by following a sequence of successive resonant orbits. Then, the spacecraft gets ballistically captured by the inner moon M2.
2 Study Objective
The goal of the Ariadna project is the implementation of a fast and accurate visualization algorithm to characterize and extract structures directly in areapreserving maps. In particular, the project is based on the features of Keplerian maps and intends to apply this algorithm to extract sequences of resonant orbits to generate low-energy inter-moon transfers.3 Proposed Methodology
The following methodology is proposed for this study, and should be discussed in the
proposal, though argued alternatives are welcome as long as they promise to achieve
the project goals.
Fig. 5 Figure 2: Various features on the same Keplerian map (courtesy Ross and Scheeres ); (a) Unstable resonant orbits; (b) Unstable/stable manifolds corresponding to a 1:2 resonant orbit; (c) Exit zone for ballistic capture.- Automatic extraction of initial conditions directly from the map
An accurate way to extract data from the map should be introduced. In particular, the algorithm should extract accurate sequences of resonant orbits that can be employed to generate low-energy inter-moon transfers.
The efficiency of the detection and precision of the extraction are closely related to how well and how fast the Keplerian map is generated. The more iterations of the maps, the more accurate the extraction is but the slower it gets. Therefore, trade-offs need to be investigated between the process that generates the map and the visual algorithm that detects and extracts the information from the map.
This Ariadna project proposal is addressed at research groups with expertise in any of the following domains: computer science and vision, dynamical systems and chaotic motion, applied mathematics, orbital mechanics, astrodynamics and mission. design.
4 ACT Contributions
The project will be conducted in close scientific collaboration with ACT-researchers. In particular, ACT-researchers will provide expertise in orbital mechanics and dynamical systems and will provide knowledge of Keplerian maps.
 Haapala, A., ‘Trajectory Design using Periapse Maps and Invariant Manifolds,’ MS. Thesis, Purdue University, West Lafayette, Indiana, 2010.
Tricoche, X., Garth, C., Sanderson, A., ‘Visualizing Invariant Manifolds in Area-
Preserving Maps,’ Topology-Based Methods in Data Analysis and Visualization IV,
Peikert, R., Hauser, H., Carr, H. (eds), Springer, to appear 2011.