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1. Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

   https://doi.org/10.1007/s00332-023-09891-4  

   Burby, J_W; Hirvijoki, E.; Leok, M. (February 2023, Journal of Nonlinear Science)

   Abstract M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formalU(1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether’s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal’s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along aU(1)-action. When the limiting rotation is non-resonant, these maps admit formalU(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formalU(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether’s theorem. When the unperturbedU(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds. 

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2. Combining finite element space-discretizations with symplectic time-marching schemes for linear Hamiltonian systems

   https://doi.org/10.3389/fams.2023.1165371  

   Cockburn, Bernardo; Du, Shukai; Sánchez, Manuel A. (April 2023, Frontiers in Applied Mathematics and Statistics)

   We provide a short introduction to the devising of a special type of methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. We restrict our attention to linear Hamiltonian systems, as the main results can be obtained easily and directly, and are applicable to many Hamiltonian systems of practical interest including acoustics, elastodynamics, and electromagnetism. After a brief description of the Hamiltonian systems of our interest, we provide a brief introduction to symplectic time-marching methods for linear systems of ODEs which does not require any background on the subject. We describe then the case in which finite-difference space-discretizations are used and focus on the popular Yee scheme (1966) for electromagnetism. Finally, we consider the case of finite-element space discretizations. The emphasis is placed on the conservation properties of the fully discrete schemes. We end by describing ongoing work. 

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3. Holomorphic Floer theory and Donaldson-Thomas invariants

   Bousseau, Pierrick (April 2024, Proceedings of symposia in pure mathematics)

   We present several expected properties of the holomorphic Floer theory of a holomorphic symplectic manifold. In particular, we propose a conjecture relating holomorphic Floer theory of Hitchin integrable systems and Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds. More generally, we conjecture that the BPS spectrum of a 4-dimensional N = 2 quantum field theory can be recovered from the holomorphic Floer theory of the corresponding Seiberg-Witten integrable system. 

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4. Title: String-Math 2022 Article: Holomorphic Floer theory and Donaldson-Thomas invariants

   Bousseau, Pierrick (April 2024, AMS)

   Donag, Ron; Langer, Adrian; Sułkowski, Piotr; Wendland, Katrin (Ed.)

   We present several expected properties of the holomorphic Floer theory of a holomorphic symplectic manifold. In particular, we propose a conjecture relating holomorphic Floer theory of Hitchin integrable systems and Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds. More generally, we conjecture that the BPS spectrum of a 4-dimensional N = 2 quantum field theory can be recovered from the holomorphic Floer theory of the corresponding Seiberg-Witten integrable system. 

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5. Qualitative inverse problems: mapping data to the features of trajectories and parameter values of an ODE model

   https://doi.org/10.1088/1361-6420/acd414  

   Duan, X.; Rubin, J. E.; Swigon, D. (May 2023, Inverse Problems)

   Abstract Our recent work on linear and affine dynamical systems has laid out a general framework for inferring the parameters of a differential equation model from a discrete set of data points collected from a system being modeled. It introduced a new class of inverse problems where qualitative information about the parameters and the associated dynamics of the system is determined for regions of the data space, rather than just for isolated experiments. Rigorous mathematical results have justified this approach and have identified common features that arise for certain classes of integrable models. In this work we present a thorough numerical investigation that shows that several of these core features extend to a paradigmatic linear-in-parameters model, the Lotka–Volterra (LV) system, which we consider in the conservative case as well as under the addition of terms that perturb the system away from this regime. A central construct for this analysis is a concise representation of parameter and dynamical features in the data space that we call theP_n-diagram, which is particularly useful for visualization of the qualitative dependence of the system dynamics on data for low-dimensional (smalln) systems. Our work also exposes some new properties related to non-uniqueness that arise for these LV systems, with non-uniqueness manifesting as a multi-layered structure in the associatedP₂-diagrams. 

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