Poster presentations
22nd School on Interactions between Dynamical Systems and Partial Differential Equations (JISD 2026)
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Adrian Arriaga Heteroclinic connections between Lyapunov orbits in the Sun–Earth solar sail circular restricted three-body problem Universidad Nacional Autónoma de México (UNAM)
The stable and unstable invariant manifolds associated with periodic orbits around the collinear libration points of the circular restricted three-body problem (CR3BP) play a fundamental role in organising transport in phase space. Building on earlier work (Conley, McGehee, Llibre–Martínez–Simó, Koon–Lo–Marsden–Ross) on heteroclinic connections between Lyapunov orbits around L1 and L2, this work extends the analysis to the CR3BP with solar radiation pressure, where a solar sail modifies the effective potential and displaces the equilibrium points (Farrés, Jorba).
We consider the Hamiltonian case in which the sail is oriented perpendicular to the Sun–sail line. Under this assumption, the solar radiation pressure reduces the Sun's gravitational attraction by a factor (1 − β), where β is the sail lightness number. The system preserves its Hamiltonian structure, but the collinear equilibrium points SL1 and SL2 are displaced toward the Sun and the Jacobi constant is modified. We compute these families numerically using a shooting method and numerical continuation, and analyse stability via Floquet theory. The invariant manifolds are globalised by perturbing along the eigenvectors and integrating forward or backward in time. To detect heteroclinic connections, we introduce a Poincaré section through the Earth and represent the manifold intersections as curves in a reduced phase space.
With this algorithm we identify a critical value of the Jacobi constant at which the stable and unstable manifolds of the Lyapunov orbits around SL1 and SL2 first become tangent. This critical value acts as a threshold beyond which transversal intersections appear and heteroclinic transport becomes possible at the first return. Understanding these connections informs the study of low-energy transfers to and from strategic locations near the Earth.
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Brayan Guerra Slow dynamics in coupled neural oscillators: a geometric approach via invariant manifolds Universidad Nacional Autónoma de México (UNAM)
We investigate the emergence of slow dynamics in a pair of weakly coupled neural oscillators described by the Wilson–Cowan model. In the uncoupled regime, two identical autonomous oscillators, each with an asymptotically stable periodic orbit, define a product system in R⁴ with an invariant torus given by the Cartesian product of the individual limit cycles. This torus is normally hyperbolic: its tangent dynamics has two neutral directions associated with the independent phase shifts, while the transverse directions contract.
We then introduce a weak unidirectional coupling by perturbing one oscillator through a term of order ε depending on the second oscillator. For sufficiently small coupling, Fenichel's persistence theorem ensures the torus continues as a nearby normally hyperbolic manifold. The coupling breaks the degeneracy of the neutral spectrum: one Floquet multiplier stays equal to one, reflecting time-translation invariance, while the second moves strictly inside the unit circle. This splitting creates a weakly contracting direction inside the stable bundle, generating a slow manifold embedded in the stable manifold of the perturbed periodic orbit.
The periodic orbit is computed numerically by a shooting method, and numerical continuation in the perturbation parameter yields a nearby periodic orbit for the coupled system. To approximate the invariant manifolds, we apply the parameterisation method for periodic orbits, formulating the homological equations in normal Floquet coordinates and solving them in Fourier space. As the main numerical result, we compute an approximation of the slow stable manifold of the coupled system, giving an explicit geometric representation of the weakly attracting direction generated by the coupling. These results provide a geometric mechanism through which weak interactions induce time-scale separation in Wilson–Cowan oscillators, with potential relevance for synchronisation, phase coordination, and transient neural responses.
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Edhin Mamani On the maximizing measure for closed visibility manifolds Federal University of Minas Gerais, Brazil
The geodesic flow of a compact Riemannian manifold of negative curvature is a classical example of an Anosov flow of geometric origin, and many of its dynamical and ergodic properties are well understood. The existence and uniqueness of the maximizing measure was proved by Margulis and Bowen in the early 1980s, and later extended to broader classes: Knieper (1998) to compact rank-1 manifolds of non-positive curvature via the Patterson–Sullivan measure, and Gelfert–Ruggiero (2018) to compact higher-genus surfaces without focal points via an expansive factor flow; the first author extended this to higher-genus surfaces without conjugate points. In the present work we generalise the Gelfert–Ruggiero approach to n-dimensional compact manifolds without conjugate points, assuming a gap-entropy condition and a special global geometry property called the visibility condition. This is joint work with Rafael Ruggiero.
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Rafael Martínez Vergara Existence of a two-periodic strange non-chaotic attractor (TSNA) in quasiperiodically forced systems Universitat de Barcelona
This work explores the emergence of two-periodic strange non-chaotic attractors (TSNAs) in quasiperiodically forced dynamical systems. We analyse a two-parameter family of quasiperiodically forced maps acting on the cylinder, built from a piecewise-linear map in the fibre direction and a smooth forcing, with an irrational rotation in the base. We establish the existence of a continuous bifurcation curve b*(a) along which the system undergoes a non-smooth period-doubling bifurcation, and show that at this critical value the closure of the associated attractor has positive two-dimensional Lebesgue measure. The resulting fractalisation is characterised by the divergence of the Lipschitz constant of the attracting invariant curve as b approaches b*(a) from below. The analysis builds on recent results on non-smooth bifurcations in families of one-dimensional piecewise-linear quasiperiodically forced maps.
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Victor Medeiros Typical continuity of fractal dimensions of dynamical Lagrange and Markov spectra for holomorphic horseshoes on complex surfaces IMPA, Brazil
The Markov and Lagrange spectra are fractal sets associated with the best constants of approximation in Diophantine analysis. They have an intricate structure and admit a dynamical characterisation in terms of symbolic dynamics and hyperbolic horseshoes, which lets one define dynamical Lagrange spectra associated with a real hyperbolic horseshoe. A striking property of the classical and real dynamical Lagrange spectra is that the Hausdorff dimension of the truncated spectrum is a continuous function of the truncation parameter, as proved by Moreira in a series of papers.
Moving to the complex setting, where one approximates a complex number by Gaussian integers, the spectrum admits a dynamical characterisation through a full-branched holomorphic expanding map. The main result generalises Moreira's theorem to the complex setting of horseshoes associated with holomorphic automorphisms: in a Baire-residual set of automorphisms of C², the Hausdorff dimension of the truncated dynamical spectra associated with complex horseshoes is a continuous function.
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Dmitrii Mints High-order homoclinic tangencies and universal dynamics for multidimensional diffeomorphisms Imperial College London, United Kingdom
This research studies the dynamics of smooth multidimensional diffeomorphisms from the Newhouse domain, the open regions in the space of maps where systems with homoclinic tangencies are dense. We prove that in the space of smooth and real-analytic multidimensional maps, in any neighbourhood of a map with a bi-focus periodic orbit whose invariant manifolds are tangent, there exist open regions (subdomains of the Newhouse domain) where maps with high-order homoclinic tangencies of corank 2 are dense, and maps with universal two-dimensional dynamics are residual. Here the invariant manifolds forming the tangency share a plane of common tangent vectors. This is joint work with D. Turaev.
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Elena Pilar Ochoa Ochoa Gevrey class for a locally three-phase-lag thermoelastic beam system Universidad del Bío-Bío, Chile
This work studies the behaviour of solutions for the three-phase-lag heat equation with localised dissipation on an Euler–Bernoulli beam model, where the dissipation influences the regularity of the solutions. We consider a beam composed of three components, two elastic and a third dissipative one following the three-phase-lag theory, with the dissipative coefficient vanishing outside the central interval.
The main result is that the associated semigroup belongs to Gevrey class 5 when a condition on the phase-lag coefficients holds (τ*α > k*τq). In the opposite case (τ*α < k*τq), the shifted operator Aγ = A − γI is dissipative and, following the same steps, the corresponding semigroup is likewise a Gevrey semigroup of class 5.
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Joana Pech Alberich Monotonicity of the first Dirichlet eigenvalue of regular polygons Universitat Politècnica de Catalunya (UPC)
We settle a conjecture of Antunes and Freitas from 2006: for regular N-sided polygons of fixed area π, the first Dirichlet eigenvalue λ1(N) and the quotients λ1(N)/λ1(N+1) are monotonically decreasing in N. The proof combines two approaches: sharp asymptotic expansions with rigorous error bounds (for N ≥ 64) and computer-assisted proofs using interval arithmetic (for N < 64). This is joint work with Joel Dahne and Javier Gómez-Serrano.
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