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1. Arnold diffusion in the elliptic Hill four-body problem: geometric method and numerical verification

   Burgos-Garcia, Jaime; Gidea, Marian; Sierpe, Claudio (April 2024, Nonlinearity)

   We present a mechanism for Arnold diffusion in energy in a model of the elliptic Hill four-body problem. Our model is expressed as a small perturbation of the circular Hill four-body problem, with the small parameter being the eccentricity of the orbits of the primaries. The mechanism relies on the existence of two normally hyperbolic invariant manifolds (NHIM's), and on the corresponding homoclinic and heteroclinic connections. The dynamics along homoclinic/heteroclinic orbits is encoded via scattering maps, which we compute numerically. Having several scattering maps, at each point we select the scattering map that gives the largest gain in energy or the scattering map that gives the smallest loss in energy. Using Birkhoff's Ergodic Theorem we show that there are pseudo-orbits generated by the selected scattering maps along which, on average, the energy grows by an amount independent of the small parameter. A shadowing lemma yields the existence of diffusing orbits. 

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2. 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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3. Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability

   https://doi.org/10.1016/j.physd.2024.134058  

   Chekroun, Mickaël D; Liu, Honghu (April 2024, Physica D: Nonlinear Phenomena)

   Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics. To identify these invariant sets we adopt an approach combining Galerkin–Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle-node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems. These dynamical insights enable us in turn to design a stochastic model whose solutions---as the delay parameter drifts slowly through its critical values---produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping ‘‘points’’ beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena. 

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4. Relative cubulation of relative strict hyperbolization

   https://doi.org/10.1112/jlms.70093  

   Lafont, Jean‐François; Ruffoni, Lorenzo (April 2025, Journal of the London Mathematical Society)

   Abstract We prove that many relatively hyperbolic groups obtained by relative strict hyperbolization admit a cocompact action on a cubical complex. Under suitable assumptions on the peripheral subgroups, these groups are residually finite and even virtually special. We include some applications to the theory of manifolds, such as the construction of new non‐positively curved Riemannian manifolds with residually finite fundamental group, and the existence of non‐triangulable aspherical manifolds with virtually special fundamental group. 

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5. Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant

   https://doi.org/10.1090/cams/4  

   Hopkins, Michael; Lin, Jianfeng; Shi, XiaoLin Danny; Xu, Zhouli (February 2022, Communications of the American Mathematical Society)

   In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin ⁡ ( 2 ) \operatorname {Pin}(2) -equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the Pin ⁡ ( 2 ) \operatorname {Pin}(2) -equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from B Pin ⁡ ( 2 ) B\operatorname {Pin}(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the j j -based Atiyah–Hirzebruch spectral sequence. 

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