Invariant Sets Research Articles

An Analytical Framework for In-Plane Two-Impulse Transfers Between Lissajous Orbits

AbstractThis paper presents an analytical framework for in-plane two-impulse transfers between two Lissajous orbits around a single collinear libration point in the linearized circular restricted three-body problem. Based on the categorization of orbits near a collinear libration point in the linearized dynamics, we formulate two-impulse transfers as a two-point boundary value problem with a fixed time of flight (ToF). We explicitly derive and numerically verify the solution, along with its subsets corresponding to transfers involving asymptotic invariant manifolds. Additionally, we derive some general properties of solutions, including the inherent constraints arising from the limited distribution of invariant manifolds are identified and compared to their single-impulse counterparts. We further investigate the relation between the performance and the constraints of two-impulse transfers by numerically evaluating the cost of maneuvers for amplitude-decreasing transfers. Our main results demonstrate improved feasibility compared to the single-impulse transfers with minimum additional cost for a relatively large ToF, regardless of the size of the amplitude size, as well as the existence of a lower bound on the cost similar to the one found in the single-impulse case.

On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets

On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets

On the Definition of the Spin Charge in Asymptotically-Flat Spacetimes

Abstract We propose a solution to a classic problem in gravitational physics consisting of defining the spin associated with asymptotically-flat spacetimes. We advocate that the correct asymptotic symmetry algebra to approach this problem is the generalized‒BMS algebra gbms instead of the BMS algebra used hitherto in the literature for which a notion of spin is generically unavailable. We approach the problem of defining the spin charges from the perspective of coadjoint orbits of gbms and construct the complete set of Casimir invariants that determine gbms coadjoint orbits, using the notion of vorticity for gbms. This allows us to introduce spin charges for gbms as the generators of area-preserving diffeomorphisms forming its isotropy subalgebra. To elucidate the parallelism between our analysis and the Poincaré case, we clarify several features of the Poincaré embedding in gbms and reveal the presence of condensate fields associated with the symmetry breaking from gbms to Poincaré. We also introduce the notion of a rest frame available only for this extended algebra. This allows us to construct, from the spin generator, the gravitational analog of the Pauli‒Lubański pseudo-vector. Finally, we obtain the gbms moment map, which we use to construct the gravitational spin-charges and gravitational Casimirs from their dual algebra counterparts.

Anomalous Floquet Phases. A resonance phenomena

Floquet topological phases emerge when systems are periodically driven out-of-equilibrium. They gained attention due to their external control, which allows to simulate a wide variety of static systems by just tuning the external field in the high frequency regime. However, it was soon clear that their relevance goes beyond that, as for lower frequencies, anomalous phases without a static counterpart are present and the bulk-to-boundary correspondence can fail. In this work we discuss the important role of resonances in Floquet phases. For that, we present a method to find analytical solutions when the frequency of the drive matches the band gap, extending the well-known high frequency analysis of Floquet systems. With this formalism, we show that the topology of Floquet phases with resonances can be accurately captured in analytical terms. We also find a bulk-to-boundary correspondence between the number of edge states in finite systems and a set of topological invariants in different frames of reference, which crucially do not explicitly involve the micromotion. To illustrate our results, we periodically drive a SSH chain and a π-flux lattice, showing that our findings remain valid in various two-band systems and in different dimensions. In addition, we notice that the competition between rotating and counter-rotating terms must be carefully treated when the undriven system is a semi-metal. To conclude, we discuss the implications to experimental setups, including the direct detection of anomalous topological phases and the measurement of their invariants.

Renormalization in Lorenz maps - completely invariant sets and periodic orbits

The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincarè maps in geometric models of well-known Lorenz attractor. Using both analytical and symbolic approaches, we study connections between periodic points, completely invariant sets and renormalizations. We show that some renormalizations may be connected with completely invariant sets while some others don't. We provide an algorithm to detect the renormalizations that can be recovered from completely invariant sets. Furthermore, we discuss the importance of distinguish one-side and double-side preimage. This way we provide a better insight into the structure of renormalizations in Lorenz maps. These relations remained unnoticed in the literature, therefore we are correcting some gaps existing in the literature, improving and completing to some extent the description of possible dynamics in this important field of study.

Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons

Recent years have seen the emergence of a new understanding of scattering amplitudes in the simplest theory of colored scalar particles — the Tr(ϕ3) theory — based on combinatorial and geometric ideas in the kinematic space of scattering data. In this paper we report a surprise: far from the toy model it appears to be, the “stringy” Tr(ϕ3) amplitudes secretly contains the scattering amplitudes for pions, as well as non-supersymmetric gluons, in any number of dimensions. The amplitudes for the different theories are given by one and the same function, related by a simple shift of the kinematics. This discovery was spurred by another fundamental observation: the tree-level Tr(ϕ3) field theory amplitudes have a hidden pattern of zeros when a special set of non-planar Mandelstam invariants is set to zero. These zeros are not manifest in Feynman diagrams but are made obvious by the connection of these amplitudes to the new understanding of associahedra arising from “causal diamonds” in kinematic space. Furthermore, near these zeros, the amplitudes simplify, by factoring into a non-trivial product of smaller amplitudes. Remarkably the amplitudes for pions and gluons are observed to also vanish in the same kinematical locus. These properties for Tr(ϕ3) amplitudes hold and further generalize to the “stringy” Tr(ϕ3) amplitudes. The “kinematic causal diamond” picture suggests a unique shift of the kinematic data that preserves the zeros, and this shift is precisely the one that unifies colored scalars, pions, and gluons into a single object. We will focus in this paper on explaining the hidden zeros and factorization properties and the connection between all the colored theories, working for simplicity at tree level. Subsequent works will describe this new formulation for the Non-linear Sigma Model and non-supersymmetric Yang-Mills theory, at all loop orders.

Leveraging the Interplanetary Superhighway for Propellant–Optimal Orbit Insertion into Saturn–Titan System

This paper presents an innovative approach using Dynamical Systems Theory (DST) for interplanetary orbit insertion into Saturn−Titan three−body orbits. By leveraging DST, this study identifies invariant manifolds guiding a spacecraft into Titan−centered Distant Retrograde Orbits (DROs), strategically selected for their scientific significance. Subsequently, Particle Swarm Optimization (PSO) is employed to fine−tune the insertion parameters, thereby minimizing ΔV. The results demonstrate that the proposed method allows for a reduction in ΔV of over 70% compared to conventional approaches like patched conics−based flybys (2.68 km/s vs. 9.23 km/s), albeit with an extended time of flight, which remains notably faster than weak stability boundary transfers. This paper serves as an interplanetary mission planning methodology to optimize spacecraft trajectories for the exploration of the Saturn−Titan system.

Tangent Space Generators of Matrix Product States and Exact Floquet Quantum Scars

The advancement of quantum simulators motivates the development of a theoretical framework to assist with efficient state preparation in quantum many-body systems. Generally, preparing a target entangled state via unitary evolution with time-dependent couplings is a challenging task and very little is known about the existence of solutions and their properties. In this work we develop a constructive approach for preparing matrix product states (MPS) via continuous unitary evolution. We provide an explicit construction of the operator that exactly implements the evolution of a given MPS along a specified direction in its tangent space. This operator can be written as a sum of local terms of finite range, yet it is in general non-Hermitian. Relying on the explicit construction of the non-Hermitian generator of the dynamics, we demonstrate the existence of a Hermitian sequence of operators that implements the desired MPS evolution with an error that decreases exponentially with the operator range. The construction is benchmarked on an explicit periodic trajectory in a translationally invariant MPS manifold. We demonstrate that the Floquet unitary generating the dynamics over one period of the trajectory features an approximate MPS-like eigenstate embedded among a sea of thermalizing eigenstates. These results show that our construction is not only useful for state preparation and control of many-body systems, but also provides a generic route towards Floquet scars—periodically driven models with quasilocal generators of dynamics that have exact MPS eigenstates in their spectrum. Published by the American Physical Society 2024

Stochastic collapse: how gradient noise attracts SGD dynamics towards simpler subnetworks*

Abstract In this work, we reveal a strong implicit bias of stochastic gradient descent (SGD) that drives overly expressive networks to much simpler subnetworks, thereby dramatically reducing the number of independent parameters, and improving generalization. To reveal this bias, we identify invariant sets, or subsets of parameter space that remain unmodified by SGD. We focus on two classes of invariant sets that correspond to simpler (sparse or low-rank) subnetworks and commonly appear in modern architectures. Our analysis uncovers that SGD exhibits a property of stochastic attractivity towards these simpler invariant sets. We establish a sufficient condition for stochastic attractivity based on a competition between the loss landscape’s curvature around the invariant set and the noise introduced by stochastic gradients. Remarkably, we find that an increased level of noise strengthens attractivity, leading to the emergence of attractive invariant sets associated with saddle-points or local maxima of the train loss. We observe empirically the existence of attractive invariant sets in trained deep neural networks, implying that SGD dynamics often collapses to simple subnetworks with either vanishing or redundant neurons. We further demonstrate how this simplifying process of stochastic collapse benefits generalization in a linear teacher-student framework. Finally, through this analysis, we mechanistically explain why early training with large learning rates for extended periods benefits subsequent generalization.

Flocking analysis and control of a nonlinear collective migration model

Abstract In this paper, we study a nonlinear collective migration model with the Cucker–Smale type weight, the nonlinear velocity coupling and the distributed network. Finite-time flocking tracking can be achieved by the alignment force gathering agents and the tracking force matching target. A trade-off existing between the two forces is established by the tracking strategy that can be viewed as a control. When the strategy is time-invariant, finite-time flocking tracking would occur for any initial state under the long-range weight, only for partial initial state under the short-range weight. An invariant set of the system is proposed and proved to be an attractive domain of the flocking state. Then two time-varying strategies, the average strategy and the maximum strategy, are designed to overcome the constraint of initial state under the short-range weight. The average strategy has to mobilize all agents simultaneously, but only causes once switch. The maximum strategy only mobilizes the agent with the largest velocity deviation, but produces more switches. Several numerical simulations are provided to observe the effects of the time-invariant and time-varying strategies.

Predictive Framework for Flow Reversals and Excursions in Turbulence.

We present a dynamical framework for intermittent reversals and excursions (R&Es) of large-scale circulations in turbulence. We show that R&Es can occur when turbulent trajectories in phase space shadow invariant manifolds of certain unstable periodic orbits (UPOs). Consequently, substantial flow reorganization and extreme fluctuations in flow metrics observed during R&Es can be reconstructed by splicing the unstable manifolds of such dynamically relevant UPOs. Using this geometrical framework, we predict imminent R&Es and preemptively avert these extreme events using closed-loop control.

Stochastic optimization of a mass-in-mass cell with piecewise hybrid nonlinear–linear restoring force

This article investigates the optimization under uncertainty of a mass-in-mass meta-cell for its potential use within a metamaterial. The specificity of the proposed mass-in-mass system stems from the hybrid nonlinear–linear stiffness at the inner level. It is well known that these systems are highly sensitivity to small perturbations in loading conditions or design parameters. In fact, the sensitivity is such that the system can exhibit discontinuous behaviors. Therefore the proposed optimization approach not only accounts for sources of uncertainties but also can handle discontinuous responses. The objective of the stochastic optimization is to find the stiffness properties of the mass-in-mass system which minimize the expected value of a specific efficiency metric. In order to better understand the system’s dynamic behavior and the origins of the discontinuities, slow invariant manifolds and frequency response curves are provided. The efficiency of the optimized system with hybrid stiffness is compared with that of a similar optimized system featuring pure cubic nonlinearity.

Research on energy transfer and vibration suppression of nonlinear energy sink with nonlinear damping

The nonlinear energy sink is a nonlinear shock absorber that provides targeted energy transfer and plays a crucial role in structural vibration suppression. In this paper, the energy transfer and dynamic characteristics of nonlinear damping nonlinear energy sink systems are analyzed. Firstly, the theoretical model of the nonlinear damping NES system is described, and secondly, the equations of the system in the free state are derived by using the complex variable average method and the multi-scale method to obtain the energy transfer efficiency equation of the system. Through the study of the slow invariant manifold and energy transfer efficiency of the system, the influence of each system parameter on the energy transfer efficiency is analyzed. Then the equations of the system under harmonic excitation are processed in the same method to obtain the boundary equations of the system, through which the saddle-node bifurcation diagram of the system is plotted to study the influence of each system parameter on the saddle-node bifurcation characteristics of the system. Finally, the slow-varying equation of the system under harmonic excitation is derived, and the influence of each system parameter on the frequency detuning parameter interval of the strongly modulated response is obtained using the phase trajectory and a one-dimensional mapping diagram of the system, and the time-mileage diagram and Poincare section are used to prove the damping effect of the strongly modulated response.

Integrated Motion Control Strategy Considering Parameter Robustness for Distributed Vehicle by Wire

<div>To address the issues of functional conflicts in execution subsystems and the deterioration of control performance due to model parameter uncertainties in the motion control of distributed vehicle by wire, this article proposes an integrated control strategy considering parameter robustness. This strategy aims to compensate for model mismatch, resolve functional conflicts, and achieve motion coordination. Based on the over-actuation characteristics of distributed vehicle by wire, this article constructs the dynamic model and utilizes the tire cornering properties along with phase portraits to delineate the working regions of the execution subsystems. To deal with model parameter uncertainties and mismatch, tube-based model predictive control (tube-based MPC) is applied to the control strategy design, which compensates for model deviations through state feedback and constructs a robust positively invariant set (RPI) to constrain the system state. Correspondingly, the weights of control inputs are adjusted adaptively, according to the working regions, to optimize the coordination logic of integrated control. In order to verify the effectiveness and feasibility of the strategy, extreme driving condition tests are executed on hardware-in-the-loop (HIL) and real vehicle test platforms. The test results indicate that the strategy proposed in this article is able to reduce the sideslip angle and tracking error of yaw rate, improve driving stability under extreme conditions through integrated control, and especially, it can still maintain precise stability control performance under severe model mismatch, exhibiting strong robustness facing parameter uncertainties.</div>

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.

Semi-analytical computation of bifurcation of orbits near collinear libration point in the restricted three-body problem

A unified semi-analytical solution is presented for constructing the phase space near collinear libration points in the Circular Restricted Three-body Problem (CRTBP), encompassing Lissajous orbits, quasihalo orbits, Axial orbits, and their invariant manifolds, as well as transit and non-transit orbits. Based on classical in-plane and out-of-plane frequency resonance mechanisms, the Lindstedt–Poincaré method could only derive separate analytical solutions for the invariant manifolds of Lissajous orbits and halo orbits, falling short for the invariant manifolds of quasihalo orbits. In this paper, by introducing a coupling coefficient η and a bifurcation equation, a unified series solution for these orbits is systematically developed using a coupling-induced bifurcation mechanism and Lindstedt–Poincaré method. Analyzing the bifurcation equation obtained from different coupling forms reveals bifurcation conditions for all kinds of orbits near collinear libration points. When η = 0, the series solution describes non-bifurcated orbits, while when η ≠ 0, the solution describes bifurcated orbits, including quasihalo orbits, Axial orbits, and their invariant manifolds, as well as newly bifurcated transit and non-transit orbits. This unified semi-analytical framework provides a more comprehensive understanding of the complex phase space structures near collinear libration points in the CRTBP.

The Wong-Zakai approximations of invariant manifolds for retarded partial differential equations with multiplicative white noise

We focus on the dynamics and Wong-Zakai approximation for a class of retarded partial differential equations subjected to multiplicative white noise. We show that when restricted to a local region and under certain conditions, there exists a unique global solution for the truncated system driven by either the white noise or the approximation noise. Such solution generates a random dynamical system, and the solutions of Wong-Zakai approximations are convergent to solutions of the stochastic retarded differential equation. We also show that there exist invariant manifolds for the truncated system driven by either the white noise or the approximation noise, which are then the local manifolds for the untruncated systems, and prove that such invariant manifolds of the Wong-Zakai approximations converge to those of the stochastic retarded differential equation.

Exactly conservative physics-informed neural networks and deep operator networks for dynamical systems

We introduce a method for training exactly conservative physics-informed neural networks and physics-informed deep operator networks for dynamical systems, that is, for ordinary differential equations. The method employs a projection-based technique that maps a candidate solution learned by the neural network solver for any given dynamical system possessing at least one first integral onto an invariant manifold. We illustrate that exactly conservative physics-informed neural network solvers and physics-informed deep operator networks for dynamical systems vastly outperform their non-conservative counterparts for several real-world problems from the mathematical sciences.

A note on invariant manifolds for stochastic partial differential equations in the framework of the variational approach

In this note we provide conditions for local invariance of finite dimensional submanifolds for solutions to stochastic partial differential equations (SPDEs) in the framework of the variational approach. For this purpose, we provide a connection to SPDEs in continuously embedded spaces.

Iterative learning robust security predictive tracking control for small delay batch processes: Deception attacks on unreliable network channel

During the batch process, the need for security control is becoming increasingly urgent with the gradual penetration of network control technology. For small time delay batch processes subject to deception attacks, an iterative learning robust security predictive tracking control approach is developed. Reviewing previous studies for iterative learning control, the issue that the repetitive character of the batch process would mask the characteristics of deception attacks was not considered. Moreover, the gains of iterative learning control law are utilized offline, which makes large deviations for the system state as the operating time increases. This will lead to “excessive-input” conditions for control inputs, which means more consumption of energy and production resources. To address these challenges, we introduce robust security invariant sets to improve the robustness of the system by constraining the system state to be in a safe range. Also, the design of the iterative learning controller incorporates deception attack data for historical batches, which is continuously improved and optimized to withstand deception attacks. Meanwhile, by calculating stability conditions online, the real-time control law gain is obtained, which could make the control inputs adjusted online based on the real-time system’s dynamic characteristics, avoiding excessive inputs and improving the efficiency of energy and resource utilization. Additionally, the stability analysis proves that the system is input to state stability. Ultimately, simulation verifies the effectiveness and feasibility of the developed method.