List of abstracts

Abstract: The number of potential collision alerts is continuously increasing. Satellites equipped with low-thrust propulsion need to maneuver for a finite, possibly large, time to reduce the risk of collision to a safe threshold. This study offers a characterization of minimal requirements for such time in the context of the fast encounter between a low-thrust actuated satellite and a non-cooperative object. We first introduce a state vectorconsisting of the time of closest approach and the orbital elements of the primary at this time, and we deduce its equations of motion. Necessary conditions of optimality are then disclosed, and an approximate solution is proposed, which does not require to solve a shooting problem, but it is obtained by means of a backward integration from the safe set for any initial state in the proximity of the advertised collision. A thorough analysis of the optimal synthesis of this solution is proposed with special focus on the computation of the cut and conjugate loci.

Abstract: In the talk I discuss a new regularization technique discovered by Barutello, Ortega and Verzini, which in contrast to previous regularization techniques does not blow up the energy hypersurface but the loop space. In particular, this regularization technique does not need the energy hypersurface and is therefore suitable as well for non-autonomous systems where energy is not preserved. As I will explain in the talk due to the nonlocal nature of this regularization technique the regularized solutions will not satisfy an ODE but a delay equation.

Abstract: In recent years, a few researchers from both communities have initiated interdisciplinary collaborations that complement each other’s expertise. Consequently, tools from symplectic geometry have been successfully applied to trajectory design in mission planning and to bifurcation analysis in the three-body problem. This talk will present these applications and invite discussion on further collaborative opportunities. 

Abstract: In this talk, I will discuss the (circular, restricted) three-body problem from a symplectic perspective, focusing on geometric and topological aspects. Topics to be covered include:

-open books,

-fixed-point theorems,

-holomorphic foliations,

-cone structures.

Abstract: The local motion about a trajectory in a Hamiltonian Dynamical System (HDS) can be expressed as a Taylor series expanded about the nominal solution. The properties of these local solutions are controlled by the symplectic structure of the HDS, and provide fundamental connections between the stochastic evolution of spacecraft trajectories and the topological constraints of symplectic flows. The main properties are controlled by the state transition matrix (STM) evaluated in the neighborhood of the trajectory, which is a linear solution to a partial differential equation, and thus is a smooth map in phase space in the vicinity of a trajectory. The STM is also a symplectic map and thus conserves all of the fundamental structures found in a symplectic dynamical system. By analyzing this solution one can obtain insights into the local stability and instability of motion, can identify constraints on the volumetric propagation of solutions, and can expose other key insights.

In this talk we will introduce and discuss some of the fundamental structures that can be found in the state transition matrix and how they can be connected to fundamental results in symplectic topology. In addition, we will introduce a novel approach to analyzing the flow about an arbitrary trajectory. In particular, we introduce a recently derived ortho-symplectic transformation that can be applied to separate the basic flow components associated with the local motion (Boodram and Scheeres, 2025). This enables the definition of a local Poincaré-like map that can be defined between any two points on a trajectory, and which devolves to the classical Poincaré map and associated monodromy matrix when evaluated over periodic or quasi-periodic motion. Implications and applications of this transformation will be discussed.

Mini-course on elementary symplectic geometry, to catch up the audience on necessary concepts and definitions for the rest of the workshop.

Abstract: The exploitation of Dynamical Systems Theory to the study of the motion of objects in space has led to innovative mission designs and substantial propellant savings. Recent advancements in numerical continuation techniques have enabled a new class of dynamical structures known as quasi-periodic invariant tori (QPT) to emerge as an alternative option to periodic orbits for designing and operating spacecraft missions in chaotic dynamical environments such as the Circular Restricted Three-Body Problem. QPT have the potential to significantly increase the design domain of spacecraft missions as well as unlock new, fuel-efficient transfer opportunities in the vicinity of the secondary mass, e.g., the Moon. To demonstrate these advantages, we will showcase how heteroclinic connections between quasi-periodic invariant tori in the Earth-Moon system can be charted systematically by means of knot theory and the linking number, a topological property of curves in three-dimensional spaces. Heteroclinic connections are then continued throughout different members of is-energetic quasi-periodic families by means of two-parameter continuation tools.

Abstract: Arnold diffusion is a phenomenon in Hamiltonian dynamics in which small perturbations of nearly integrable systems can induce slow but unbounded changes in the action variables. First described by V.I. Arnold (1964), it allows certain trajectories to drift across phase space, despite the perturbations being arbitrarily small.

In this talk, we will introduce the Arnold diffusion mechanism and illustrate an application to orbital mechanics. Focusing on the Medium Earth Orbit (MEO) region, home to navigation satellites such as GPS and Galileo, we show how natural perturbations (in our case, third-body effects) can be exploited to guide satellites toward atmospheric reentry at the end of their operational life.

Using Galileo as a case study, we analyze a hierarchy of Hamiltonian models incorporating the Earth’s oblateness and the Moon’s gravitational attraction. We demonstrate how Arnold diffusion can trigger eccentricity growth along certain trajectories, lowering the satellite into the atmospheric drag domain.

This is a joint work with E.M Alessi, I. Baldomá and M. Guardia.

Abstract: Accurate knowledge of the state of Resident Space Objects (RSO) is essential for Space Situational Awareness (SSA) applications. Uncertainty distributions rapidly degrade under nonlinear dynamics, such as high-eccentricity orbits or multi-body environments, and become non-Gaussian. Regularisation techniques in astrodynamics are used to remove singularities of the system and improve numerical integration but have not really been explored for efficient uncertainty propagation. Regularisation typically involves transformation of coordinates and time, introducing a fictitious time as the new independent variable. This work proposes a framework for linear covariance propagation in time-stabilised dynamics, in which state and covariance are propagated in fictitious time using a Sundman transformation. Propagation in fictitious time improves uncertainty realism, as evaluated through the Cramér-von Mises test, demonstrating greater statistical consistency with the true distribution than physical time propagation, at the cost of breaking the Hamiltonian (and thus symplectic) structure.

A novel analytical synchronisation method is introduced to recover the covariance in physical time at an arbitrary epoch following fictitious time propagation. Covariance synchronisation is validated on eccentric orbits in Keplerian two-body dynamics, showing accurate recovery of both the physical covariance and associated uncertainty realism. By combining uncertainty propagation in time-stabilised dynamics with synchronisation in physical time, this framework provides a computationally efficient method for uncertainty propagation in nonlinear dynamics relative to physical time evolution. While this study focuses on time stabilisations that do not preserve the Hamiltonian structure, potential future work can extend this framework to Hamiltonian regularised formulations, in which the symplectic form is preserved. Results find application in long-term state and uncertainty propagation in two-body and three-body dynamics for object tracking and trajectory prediction.

Abstract: Arnold's J⁺ invariant tracks direct tangencies which do not occur in families of periodic orbits in Hamiltonian Systems. Its extension into Stark-Zeeman Settings - which model specific three body systems - and to links of families of periodic orbits can be used to study pairs of families of periodic orbits in Stark-Zeeman Systems. As an example we will discuss a construction of pairs of periodic orbits in the Euler Problem. 

Abstract: Suppose we have a set of observations of a dynamical system that sample a smooth manifold. Examples include isolated stable periodic or quasiperiodic motions, or a family of stable fixed points or smooth attractors as a parameter changes (a bifurcation diagram).

We will show how to build an atlas of charts that represent the point cloud as a manifold. The approach uses a continuation method with local representations of the manifold near a point in the cloud found by collecting points in a neighborhood and fitting them with a Taylor series. A new point is chosen on the boundary of the union of the local approximations, which ensures that the continuation extends the manifold at each step. The approach is not dependent on the topology of the manifold. It produces a restricted Laguerre-Voronoi diagram on the manifold, whose dual is a simplicial mesh.

We will give examples of a bifurcation diagram of a periodic orbit, a quasiperiodic torus, and a torus containing a stable and unstable periodic orbit.

https://doi.org/10.1137/24M1678647

Abstract: Hamiltonian Monte Carlo (HMC) generates proposals by simulating Hamiltonian dynamics, but its efficiency depends strongly on the choice of coordinates: complex geometries can lead to stiff trajectories and poor exploration. From a geometric perspective, this reflects performing Hamiltonian flow in non-adapted canonical coordinates.

In integrable systems, action–angle variables provide an optimal representation in which the dynamics reduce to uniform motion on invariant tori. Motivated by this structure, we introduce Action–Angle Hamiltonian Monte Carlo (AAHMC), a framework that learns approximate canonical transformations mapping the phase space of HMC into coordinates resembling action–angle variables.

The transformations are constructed using normalising flows which regularise the dynamics. In the transformed coordinates, trajectories become closer to integrable, enabling more efficient exploration of complex target distributions.

This perspective suggests a connection between probabilistic inference and classical techniques from symplectic geometry and celestial mechanics, where constructing near-integrable coordinates plays a central role.

Abstract: To be announced

Abstract: We study the bifurcation structure of highly inclined near halo orbits with close approaches to the light primary, in the circular restricted three-body problem (CR3BP). Using a Hamiltonian formulation together with Moser regularization, we develop a numerical framework for the continuation of periodic orbits and the computation of their Floquet multipliers which remains effective near collision. We describe vertical collision orbits and families emerging from its pitchfork, period-doubling, and period-tripling bifurcations in the limiting Hill's problem, including the halo and butterfly families. We continue these into the CR3BP using a perturbative framework via a symplectic scaling, and construct bifurcation graphs for representative systems (Saturn-Enceladus, Earth-Moon, Copenhagen) to identify common dynamical features. Conley-Zehnder indices are computed to classify the resulting families. Together, these results provide a coherent global picture of polar orbit architecture near the light primary, offering groundwork for future mission design, such as Enceladus plume sampling missions. This is joint work with Dayung Koh and Otto van Koert. 

Abstract: Stability analysis of periodic orbits rely on linear analysis tools such as Floquet Multipliers or the Broucke stability diagram, typically aided by Monte Carlo simulations. However, these techniques fail to potentially capture the nonlinear behaviour of the system near bifurcations of linearly stable orbits. A novel approach is proposed to expand upon continuation methods to automatically gain insight into the nonlinear stability of periodic orbits and identify bifurcation via Differential Algebra and Lyapunov–Schmidt Reduction. Differential algebra enables representation of functions as Taylor polynomial series with mathematical operations defined on them enabling a locally dense semi-analytical solution. The Lyapunov–Schmidt method can give a bifurcation equation, coefficients of which can be used to approximate the map’s flow thus giving insights into nonlinear stability.The motivating example are the planar Quasi Satellite Orbits of the Mars Moon eXploer mission which exhibited linear stability but were nonlinearly unstable near the 1:3 subharmonic bifurcation. Results show that the Differential Algebra is capable of easily creating parametrised Poincare maps which can be used to get a bifurcation normal form via the Lyapunov-Schmidt Reduction. Applying, approximate flow following Kuznetsov’s work in Element of Applied Bifurcation Theory, the stability of different orbits can be shown. 

Abstract: A tutorial for Floer homology in celestial mechanics: In this talk we will give an overview of Floer homology and some other tools from symplectic dynamics. In particular, we will explain how to compute this homology in practice in celestial mechanics. We discuss how to set up the numerics and how to interpret this homology. Finally, we also discuss how and what orbits can be found using this theory.

Abstract: In this talk I will present a contact geometric approach to the study of attractors and other invariant sets of random dynamical systems with bounded noise, as well as in systems modelling the dynamical propagation of uncertainty. I will show in particular that to any such system there is naturally associated a contact dynamical system on the unit cotangent bundle of the state space --the so-called boundary system-- with the following characteristic weak cocycle property: boundaries of invariant sets of the random dynamical system lift in a unique way to backward invariant Legendrian submanifolds (possibly with singularities) of the boundary system. Using this correspondence principle, I will present recent results on the stability and bifurcations of attractors of random dynamical systems, in relation to the associated Legendrian manifold and singularity/bifurcation theory of the boundary flow. Finally, I will indicate further challenges and future perspectives towards the deeper interconnections between the theory of random dynamical systems and the contact geometry and dynamics of their boundary cocycles.  

Abstract: In this investigation, using numerical continuation, Kustaanheimo-Stiefel regularization, and a novel "symplectic toolkit", we carry out an extensive numerical study of periodic orbit families for the Earth-Moon CR3BP. Near the Moon we investigate prograde, retrograde, and Halo orbits, discovering previously-unknown orbit families linking them together through bifurcations and singularities - also confirming a 1968 conjecture of Broucke. Earth prograde and retrograde orbits are also studied, finding infinite chains linking 1:2N and 1:2N+1 resonant orbits. These connections provide insights into the global network structure of families of periodic orbits, identifying orbit families near others of interest for mission design.

Abstract: Bifurcations of periodic orbits are ubiquitous in Hamiltonian dynamics, and their generic types are fully classified by Meyer. While the invariance of Floer homology guarantees that this behavior is controlled on the homology level, its effect on the chain complex level has yet to be discovered.

In this ongoing joint work with Hong-kwon Cho, we investigate the structural changes of the local Floer chain complex across generic bifurcations. By employing localized toy Hamiltonians, we construct explicit Floer cylinders near the bifurcating orbits. This concrete chain-level construction aims to bridge symplectic topology and dynamical systems, providing a new framework to analyze orbital bifurcations using Floer-theoretic invariants.

Abstract: One standard way of proving existence of trajectories in the circular restricted three-body problem is through Poincaré-Birkhoff type fixed point theorems, all of which rely on some "twist condition" on the boundary dynamics of a Poincaré section.This talk will give a brief history of the evolution of this assumption, up to its latest version, the Weakened Twist Condition, as well as a conjecture on its veracity. 

Abstract:  To be announced

Abstract: The theory of Lagrangian Coherent Structures (LCS) can be used in astrodynamics to reveal the boundaries between qualitatively different dynamical behaviours, and identify structures analogous to invariant manifolds in systems with arbitrary time dependence. To advance its application to astrodynamics, this presentation introduces an automatic method for the objective computation of LCS in arbitrary coordinate systems, together with an improved LCS generation method offering higher resolution and reliability. These methods are used to compute the LCS associated with the ballistic capture mechanism in the Sun-Mars Elliptic Restricted Three-Body Problem, and the meaning of these surfaces in terms of the dynamical behaviours they separate is investigated in detail.

Abstract: We present a computational framework for constructing local orbital elements in the vicinity of periodic orbits of the circular restricted 3-body problem (CR3BP), motivated by the need for geometrically meaningful and numerically efficient coordinates for cislunar mission design. The approach builds on high-order jet transport and recent advances in the explicit numerical computation of normal forms for Poincaré maps to obtain nonlinear invariant coordinates that separate oscillatory and hyperbolic dynamics near a selected periodic orbit. These coordinates generalize classical action-angle descriptions and provide a practical representation of quasi-periodic motion, stable and unstable manifolds, and nearby trajectories within a single unified framework.

We detail the construction of the Poincaré map, its high-order Taylor representation, and the associated normal form transformations, including the recovery of the periodic-orbit phase and continuation along orbit families. The resulting local orbital elements admit direct numerical evaluation and enable fast propagation and geometric interpretation of motion near the reference orbit. Convergence proper ties, truncation error behavior, and optimal normalization order are analyzed to quantify the domain of validity and computational accuracy relative to direct numerical integration.

The methodology is demonstrated for representative cislunar periodic orbits. We discuss implications for mission design, highlighting how nonlinear invariant coordinates can support trajectory analysis, uncertainty representation, and maneuver design near periodic-orbit regimes. These results establish local orbital elements derived from normal form theory as a practical dynamical systems tool for astrodynamics analysis and education in multi-body environments.

Abstract: We propose an optimization-free guidance method for passive-safe spacecraft rendezvous using Barrier States, an augmented-state framework that embeds safety information into Hill-frame relative dynamics. For linearized relative orbital motion modeled as a Hamiltonian system, the proposed transformation incorporates safety constraints directly into the guidance coordinates and yields closed-form feedback laws without requiring online optimization or quadratic programming. The method simultaneously addresses terminal rendezvous, keep-out-zone avoidance, and passive safety under actuator failure. Numerical simulations of Hill-frame proximity operations demonstrate accurate rendezvous performance together with improved safety margins and fewer collision cases in Monte Carlo analyses. 

Abstract: Hamiltonian mechanics provides the mathematical foundations for geometric insights and structure-preserving numerical methods in astrodynamics, yet explicit symplectic integrators have historically been limited to separable systems. This work presents interrelated contributions that extend explicit symplectic integration to nonseparable Hamiltonians, with applications to restricted three-body dynamics and trajectory optimization. We demonstrate that explicit symplectic integrators can be realized for the Circular and Elliptic Restricted Three-Body Problem, achieving superior long-term stability and computational efficiency over conventional non-symplectic methods. 

Building on this, we generalize the framework to adaptive time-stepping via the extended phase space method, successfully resolving the error divergence characteristics that are traditionally associated with variable time steps in symplectic integrators and systematically analyzing the influence of monitor and control functions and splitting strategies on accuracy and stability. These structure-preserving methods are further extended to the indirect optimal control setting, propagating the state-costate Hamiltonian system arising from Pontryagin's Minimum Principle with improved convergence over classical Runge-Kutta integration on a minimum-fuel station-keeping problem in the CR3BP.

Abstract: We study random dynamical systems with bounded noise via a deterministic set-valued formulation. Stationary measures of the random system correspond to minimal attractors of the set-valued system, and they may undergo discontinuous (topological) bifurcations. We introduce a finite-dimensional boundary map on the unit tangent space that preserves the (co-)normal bundle, making it a contactomorphism. The smooth structure of the boundary map enables bifurcation analysis of the invariant manifolds. In a specific Hénon map example, we demonstrate that bifurcations in the random system correspond to those in the boundary map.

Abstract: A multi-orbit cycler is a periodic solution of the planar circular restricted three-body problem (PCR3BP) that alternately undergoes temporary capture with both primaries. These trajectories arise within phase-space regions bounded by invariant manifold tubes associated with Lyapunov periodic orbits about the collinear Lagrange points. We examine families of symmetric multi-orbit cyclers possessing x-axis symmetry and investigate the bifurcation structure governing their existence. Numerical continuation of periodic solutions indicates that, for a given symmetric cycler family, at most two solutions exist at a fixed Jacobi constant. In addition, these families exhibit a saddle-center bifurcation at the minimum-energy configuration, producing a stable-unstable pair of periodic orbits, which guarantees that every symmetric cycler family will have at least one stable subfamily.

Abstract: Growing activity across cislunar space is bringing new operational, regulatory, and modelling challenges. Existing sensors, tools, and services for situational awareness are difficult or impossible to apply reliably to cislunar space, where operators instead depend on voluntary ephemeris sharing and mutual coordination. The strongly perturbed dynamical environment also presents challenges for long-term modelling, and the absence of universally accepted workflows and fidelities frustrates sustainability studies, impact analyses, and post-mission disposal planning.

This talk aims to introduce the open challenges in cislunar operations, and to identify them as a potentially fertile area of application for symplectic geometry. The talk will begin with a high-level overview of the operational, sustainability, and regulatory challenges, drawing on experience at the UK's spaceflight regulator conducting technical assessments of LEO, GEO and xGEO missions. Attention will then turn to outlining the methods and techniques that are being developed internally, before using doctoral and subsequent research as a basis to define areas where symplectic approaches may provide utility.

Abstract: Our research is motivated by the following practical question: can a rocket travel between any two points in the gravitational field of the Earth of the Moon, using its engines only at the beginning and at the end of the journey? This question, known as the two-boost problem, arises naturally in the context of space mission design. On the other hand, Floer homology is a robust algebraic tool which connects the dynamics of a Hamiltonian system to the geometry of the energy level sets. In my talk I will show how to use Floer theory to answer the two-boost problem in the setting of the restricted circular planar three-body problem.