Hyperbolic Invariant Manifolds Research Articles

3D Generating Surfaces in Hamiltonian Systems with Three Degrees of Freedom – II

Our paper is a continuation of a previous work referenced as [Katsanikas & Wiggins, 2024b ]. In this new paper, we present a second method for computing three-dimensional generating surfaces in Hamiltonian systems with three degrees of freedom. These 3D generating surfaces are distinct from the Normally Hyperbolic Invariant Manifold (NHIM) and have the unique property of producing dividing surfaces with no-recrossing characteristics, as explained in our previous work [Katsanikas & Wiggins, 2024b ]. This second method for computing 3D generating surfaces is valuable, especially in cases where the first method is unable to achieve the desired results. This research aims to provide alternative techniques and solutions for addressing specific challenges in Hamiltonian systems with three degrees of freedom and improving the accuracy and reliability of generating surfaces. This research may find applications in the broader field of dynamical systems and attract the attention of researchers and scholars interested in these areas.

Fully numerical computation of heteroclinic connection families in the spatial three-body problem

Heteroclinic connections in the spatial circular restricted three-body problem play an important role in astrodynamics. This paper presents a fully numerical methodology for the computation of the two-parameter families of connections that exist between families of invariant tori in the three-body problem. The computation of connections is presented as a two-point boundary value problem in which the initial and final states belong to unstable and stable manifolds of two respective normally hyperbolic invariant manifolds. A flow map torus computation procedure is modified to compute particular trajectories within these spaces, and a robust two-parameter continuation scheme is paired with a shooting technique to compute the entire connection family. Results are provided in the Sun–Earth and Earth–Moon representations of the spatial three-body problem. The Sun–Earth connection family works to verify the methodology with previously developed semi-analytical connection computation techniques. The Earth–Moon example demonstrates the methodology in a setting where fully numerical techniques are required to study the connection family.

The Classical Action as a Tool to Visualise the Phase Space of Hamiltonian Systems

In this paper, we analyse the classical action as a tool to reveal the phase space structure of Hamiltonian systems simply and intuitively. We construct a scalar field using the values of the action along the trajectories to analyse the phase space. The different behaviours of the trajectories around important geometrical objects like normally hyperbolic invariant manifolds, their stable and unstable manifolds, and KAM structures generate characteristic patterns in the scalar field generated by the action. Also, we present a simple argument based on the conservation of energy and the behaviour of the trajectories to understand the origin of the patterns in this scalar field. As examples, we study the phase space of open Hamiltonian systems with two and three degrees of freedom.

The vicinity of Earth–Moon [formula omitted] and [formula omitted] in the Hill restricted 4-body problem

The Hill restricted 4-body problem (HR4BP) is a periodic time-dependent Hamiltonian dynamical system that is both a generalization of the circular restricted 3-body problem (CR3BP) and Hill’s problem. In this work, the HR4BP is used to model the motion of an infinitesimal particle under the mutual gravitation of the Sun, Earth, and Moon. In the HR4BP, the equilibria L1 and L2 of the CR3BP are replaced by π-periodic orbits, the so-called EM L1,2 dynamical equivalents. We compute center manifold reductions about EM L1,2 to obtain a local overview of the dynamics in their vicinity. In this way, we show the existence of families of 2- and 3-dimensional quasi-periodic invariant tori around EM L1,2. Additionally, we exploit the first integral constructed in the center manifold reduction to compute hyperbolic invariant manifolds. We draw comparisons to the bicircular and quasi-bicircular restricted 4-body problems.

Orthogonal Approximation of Invariant Manifolds in the Circular Restricted Three-Body Problem

Methods to parameterize and approximate the hyperbolic invariant manifolds of particular solutions in the circular restricted three-body problem (CR3BP) are presented in this paper. Analytical representations obtained from these manifold approximations are instrumental in the synthesis of optimal trajectories for cislunar transport. A multivariate Chebyshev series is used to approximate the surfaces, thereby serving as tractable parametric representations of the complex properties of motion. It is demonstrated that the continuum of ballistic capture trajectories and their associated sensitivities on the manifold can be realized in functional form using simple algebraic operations. Two applications making use of the Chebyshev manifold approximations as a terminal constraint surface are presented. The first is a low-thrust trajectory optimization problem formulated such that the optimal free final state lying on the manifold is determined as an additional set of design parameters. The second is a guidance law designed to target the manifold in the vicinity of the nominal patch point. Each of these methods takes advantage of the Chebyshev approximations to provide additional flexibility for mission design in multibody dynamic environments. These applications offer tremendous optimism about the utility of function approximation methods in arriving at a formal representation for the invariant manifolds in the three-body problem for efficient generation of optimal trajectories.

Computing Natural Transitions Between Tori Near Resonances in the Earth–Moon System

Natural transitions between bounded motions near mean-motion resonances occur throughout our solar system and are valuable in trajectory design. Such phenomena have been examined for natural transitions between periodic orbits near resonances within multi-body systems. However, families of quasi-periodic trajectories, tracing the surface of invariant 2-tori, significantly expand the solution space of bounded motions near resonances. Yet, identifying natural transitions between spatial 2-tori has previously been cumbersome due to the high dimensionality of the associated solution space. This paper approaches the challenge in constructing these natural transfers by using a combination of Poincaré mapping, a well-known technique from dynamical systems theory, and manifold learning, a technique for dimension reduction. The presented approach involves projecting a higher-dimensional dataset of intersections recorded from the hyperbolic invariant manifolds of two 2-tori onto a lower-dimensional embedding, enabling rapid identification of initial guesses for natural transfers. These initial guesses are then corrected and input to a continuation scheme to recover families of geometrically similar transfers connecting families of invariant 2-tori. This approach is demonstrated by constructing families of natural transitions between tori near distinct resonances in the Earth–Moon circular restricted three-body problem.

Optimal Stabilization of Periodic Orbits

In this contribution the optimal stabilization problem of periodic orbits is studied in a symplectic geometry setting. For this, the stable manifold theory for the point stabilization case is generalized to the case of periodic orbit stabilization. Sufficient conditions for the existence of a normally hyperbolic invariant manifold (NHIM) of the Hamiltonian system are derived. It is shown that the optimal control problem has a solution if the related periodic Riccati equation has a unique periodic solution. For the analysis of the stable and unstable manifold a coordinate transformation is used which is moving along the orbit. As an example, an optimal control problem is considered for a spring-mass oscillator system, which should be stabilized at a certain energy level.

Partial barriers to chaotic transport in 4D symplectic maps.

Chaotic transport in Hamiltonian systems is often restricted due to the presence of partial barriers, leading to a limited flux between different regions in phase space. Typically, the most restrictive partial barrier in a 2D symplectic map is based on a cantorus, the Cantor set remnants of a broken 1D torus. For a 4D symplectic map, we establish a partial barrier based on what we call a cantorus-NHIM-a normally hyperbolic invariant manifold with the structure of a cantorus. Using a flux formula, we determine the global 4D flux across a partial barrier based on a cantorus-NHIM by approximating it with high-order periodic NHIMs. In addition, we introduce a local 3D flux depending on the position along a resonance channel, which is relevant in the presence of slow Arnold diffusion. Moreover, for a partial barrier composed of stable and unstable manifolds of a NHIM, we utilize periodic NHIMs to quantify the corresponding flux.

Phase Space Reaction Dynamics Associated with an Index-2 Saddle Point for Time-Dependent Hamiltonian Systems

We analyze benchmark models for reaction dynamics associated with a time-dependent index-2 saddle point. The influence of index-2 saddle points on chemical reaction dynamics has received a great deal of attention in recent years and we extend this work in a new and important direction. Our model allows us to incorporate time dependence of a general form, subject to an exponential growth restriction. Under these conditions, we analytically compute the time-dependent normally hyperbolic invariant manifold and its time-dependent stable and unstable manifolds. These time-dependent phase space structures allow us to quantitatively determine the effect of the time dependence on reacting and nonreacting trajectories.

Ridesharing Options from Geosynchronous Transfer Orbits to the Sun–Earth Region

Ridesharing options for secondary payloads to be delivered to regions beyond geostationary altitude are increasingly available with propulsive evolved expendable launch vehicle secondary payload adapter rings. The preliminary mission design for secondary payloads must consider the variability of the dropoff orbit orientation: a factor that is a function of the primary mission. In this analysis, assume that the dropoff orbit is an Earth-centered geosynchronous transfer orbit (GTO). Then, propellant-efficient transfers to periodic and quasi-periodic orbits near the Sun–Earth Lagrange point are constructed from a range of GTO orientations. Dynamical structures (such as hyperbolic invariant manifolds) associated with orbits near Sun–Earth Lagrange point are leveraged to identify and summarize various types of transfer opportunities.

Topological shadowing methods in arnold diffusion: weak torsion and multiple time scales

Consider a map which possesses a normally hyperbolic invariant manifold of any even dimension with transverse homoclinic channels. We develop a topological shadowing argument to prove the existence of Arnold diffusion along the invariant manifold, shadowing some iterations of the inner dynamics carried by the invariant manifold and the outer dynamics induced by the stable and unstable foliations. In doing so, we generalise an idea of Gidea and de la Llave (2006 Discrete Contin. Dyn. Syst. 14 295), based on the method of correctly aligned windows and a so-called transversality-torsion argument. Our proof permits that the dynamics on the invariant manifold satisfy only a non-uniform twist condition, and, most importantly for applications, that the splitting of separatrices be small in certain directions and thus the associated drift in actions very slow; diffusion occurs in the directions of the manifold having non-small splitting. Furthermore we provide estimates for the diffusion time.

The deterministic core of stochastically perturbed nonlinear mechanical systems

Invariant manifolds, such as normally hyperbolic invariant manifolds and spectral submanifolds, are the key to understanding the dynamical behaviour of many nonlinear mechanical systems and serve as natural candidates for model-order reduction. While numerous related invariant manifold results are available for unforced and periodically forced nonlinear mechanical systems, their applicability to random external forcing remains to be established. Here, we clarify the continued role of deterministic invariant manifolds, more specifically normally hyperbolic invariant manifolds and spectral submanifolds under the addition of small white noise excitation. Subsequently, we demonstrate our results on several mechanical systems.

Constructing motion primitive sets to summarize periodic orbit families and hyperbolic invariant manifolds in a multi-body system

Constructing motion primitive sets to summarize periodic orbit families and hyperbolic invariant manifolds in a multi-body system

A deep dive into the $$2g+h$$ resonance: separatrices, manifolds and phase space structure of navigation satellites

Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the $2g+h$ lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity space for fixed semi-major axis. This allows to compute the change in the width of the $2g+h$ resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, $i_{L}$. Since $i_L$ is comparable to the resonance's separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination $i_{p}$, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold $\mathcal{C}$. Using $i_p$ as a label, we compute its range of values for which $\mathcal{C}$ becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in $\mathcal{C}$ allows to explain the role of the initial phase $h$ noticed in several works. (iv) Through Fast Lyapunov Indicator (FLI) cartography, we portray the stable and unstable manifolds of the NHIM as the altitude increases. Manifold oscillations dominate in phase space between $a=24,000$ km and $a=30,000$ km as a result of the sweeping of the $2g+h$ resonance by the $h-\Omega_{\rm{Moon}}$ and $2h-\Omega_{\rm{Moon}}$ resonances. The noticeable effects of the latter are explained as a consequence of the relative inclination of the Moon's orbit with respect to the ecliptic. The role of the phases $(h,\Omega_{\rm{Moon}})$ in the structures observed in the FLI maps is also clarified. Finally,(v) we discuss how the understanding of the manifold dynamics could inspire end-of-life disposal strategies.

Some Notes on Two Tests for Stability in Lossy Power Systems

Analysis of the small-signal stability of power system is commonly based on the swing equation model. Due to the special structure of the power grid swing equations, an equilibrium set corresponding to a so-called frequency equilibrium has to be analyzed. We present two new and short proofs for two tests by Skar concerning both the nonuniformly and the uniformly damped system. Based on the latter we derive a simple stability test for uniformly damped systems, which does not rely on eigenvalue computations. The nonhyperbolicity of the equilibria in original coordinates is tackled by the concept of normally hyperbolic invariant manifolds. The derivations are completely based on the theory of quadratic pencils.

Phase space geometry of isolated to condensed chemical reactions.

The complexity of gas and condensed phase chemical reactions has generally been uncovered either approximately through transition state theories or exactly through (analytic or computational) integration of trajectories. These approaches can be improved by recognizing that the dynamics and associated geometric structures exist in phase space, ensuring that the propagator is symplectic as in velocity-Verlet integrators and by extending the space of dividing surfaces to optimize the rate variationally, respectively. The dividing surface can be analytically or variationally optimized in phase space, not just over configuration space, to obtain more accurate rates. Thus, a phase space perspective is of primary importance in creating a deeper understanding of the geometric structure of chemical reactions. A key contribution from dynamical systems theory is the generalization of the transition state (TS) in terms of the normally hyperbolic invariant manifold (NHIM) whose geometric phase-space structure persists under perturbation. The NHIM can be regarded as an anchor of a dividing surface in phase space and it gives rise to an exact non-recrossing TS theory rate in reactions that are dominated by a single bottleneck. Here, we review recent advances of phase space geometrical structures of particular relevance to chemical reactions in the condensed phase. We also provide conjectures on the promise of these techniques toward the designand control of chemical reactions.

Self-induced synchronization by large delay

We investigate the dynamics of a delay differential equation obtained by perturbing a vector field f:Rn→Rn, admitting a stable periodic orbit, by using a delayed feedback control term ηg(x,x(t−τ)) of same regularity, where η is small and τ large so that ητ be bounded but non small. We prove that trajectories starting in a neighborhood (of size independent of the parameters η,τ) of this original periodic orbit enter asymptotically a periodic regime, and that the number of such distinct periodic regimes increases (almost) linearly when ητ increases infinitely. Our result is based on the construction of an invariant manifold via a process inspired by the Lyapunov-Perron method for integral operators associated to ordinary differential equations, and on the persistence of normally hyperbolic invariant manifolds for semi-flows on Banach spaces. The statement we provide here complements already known results on periodic orbits of delay differential equations.

Computer assisted proof of drift orbits along normally hyperbolic manifolds

Normally hyperbolic invariant manifolds theory provides an efficient tool for proving diffusion in dynamical systems. In this paper we develop a methodology for computer assisted proofs of diffusion in a-priori chaotic systems based on this approach. We devise a method, which allows us to validate the needed conditions in a finite number of steps, which can be performed by a computer by means of rigorous-interval-arithmetic computations. We apply our method to the generalized standard map, obtaining diffusion over an explicit range of actions.

An algorithm for computing phase space structures in chemical reaction dynamics using Voronoi tessellation

An efficient sampling algorithm is proposed for computing reactive islands, which consist of intersections of reactive trajectories and a Poincaré surface of section (PSOS), and are characterized by the number of times the reactive trajectories intersect the PSOS before/after reactions take place. The boundaries of reactive islands are series of intersections of cylindrical stable and unstable invariant manifolds emanated from a normally hyperbolic invariant manifold located, typically, nearby a rank-one potential energy saddle, which forms “reactivity boundaries”. The reactive islands structures enable us to predict the fate of trajectories. The proposed algorithm estimates regions of reactive islands on the PSOS as a Voronoi diagram constructed from the set of labeled intersections of computed molecular dynamics trajectories. The algorithm iteratively refines the boundaries of the estimated reactive islands efficiently, by computing trajectories of which initial condition is sampled from the estimated boundaries of reactive islands in the Voronoi diagram of the previous iteration step. The efficiency of the proposed algorithm is scrutinized, compared with random sampling algorithm and using a two-degrees of freedom Hamiltonian system of a double well potential, and further possible extensions of the proposed algorithm are addressed.

The Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems with Three or More Degrees of Freedom – II

We develop a method for the construction of a dividing surface using periodic orbits in Hamiltonian systems with three or more degrees-of-freedom that is an alternative to the method presented in [ Katsanikas & Wiggins, 2021 ]. Similar to that method, for an [Formula: see text] degrees-of-freedom Hamiltonian system, we extend a one-dimensional object (the periodic orbit) to a [Formula: see text] dimensional geometrical object in the energy surface of a [Formula: see text] dimensional space that has the desired properties for a dividing surface. The advantage of this new method is that it avoids the computation of the normally hyperbolic invariant manifold (NHIM) (as the first method did) and it is easier to numerically implement than the first method of constructing periodic orbit dividing surfaces. Moreover, this method has less strict required conditions than the first method for constructing periodic orbit dividing surfaces. We apply the new method to a benchmark example of a Hamiltonian system with three degrees-of-freedom for which we are able to investigate the structure of the dividing surface in detail. We also compare the periodic orbit dividing surfaces constructed in this way with the dividing surfaces that are constructed starting with a NHIM. We show that these periodic orbit dividing surfaces are subsets of the dividing surfaces that are constructed from the NHIM.