We establish a sharp rigidity–chaos dichotomy for planar two-centre billiards, motivated by a natural analogue of the Birkhoff–Poritsky conjecture: the only tables integrable at every energy should be ellipses confocal with the two centres. Let Ω be a bounded domain with 𝒞^1 boundary containing the segment joining the centres. At every fixed energy h≥ 0, if is not a confocal ellipse, we construct billiard trajectories that shadow the stable and unstable manifolds of the collision–reflection orbit and realise arbitrarily prescribed sequences of sufficiently large winding numbers around the segment. This yields an invariant set semiconjugate to the full shift on a countable alphabet, periodic trajectories with prescribed finite itineraries, and compact invariant subsystems with arbitrarily large topological entropy. If, in addition, is real-analytic, every real-analytic function on the fixed-energy phase space M_h that is invariant under the billiard map is constant. This establishes the real-analytic form of the two-centre Birkhoff–Poritsky conjecture throughout the non-negative-energy regime.

We investigate expansive solutions of the N-body problem in ℝ^d (d≥2) driven by homogeneous Newtonian potentials of degree -α. We establish the existence of half-entire expansive motions with prescribed initial configuration and asymptotic direction for a wide range of homogeneity exponents α. Our approach is variational and relies on the minimization of a suitably renormalized Lagrangian action, allowing us to treat in a unified framework the hyperbolic, parabolic, and hyperbolic-parabolic regimes in the sense of Chazy's classification. Beyond existence, we derive refined asymptotic expansions for all classes of expansive solutions, identifying higher-order correction terms and improving previously known growth estimates, including the classical Newtonian case α=1. In particular, for hyperbolic-parabolic solutions, we provide a detailed description of the interplay between linear escape of cluster centers and internal parabolic dynamics, extending the cluster scattering picture to general homogeneous potentials. Finally, we interpret these solutions within the geometric framework of the Jacobi-Maupertuis metric and the weak KAM theory. In this perspective, expansive motions correspond to geodesic rays and calibrating curves for the associated Hamilton-Jacobi equation, yielding a dynamical characterization of the boundary at infinity and a refined description of global viscosity solutions.

We investigate positive solutions for three-component competition-diffusion systems within B_1⊂ℝ^N : {[ -Δ u_1=μ u_1(1-u_1)-β u_1(u_2+ u_3), in B_1,; -Δ u_2=μ u_2(1-u_2)-β u_2(u_1 + u_3), in B_1,; -Δ u_3=γ u_3(1-u_3)-β u_3(u_1+u_2), in B_1,; ∂ u_1/∂ n=∂ u_2/∂ n=∂ u_3/∂ n=0, on∂ B_1,; ]. where N≥ 2 , 2μ>γ>μ >0 , β >μγ/2μ -γ . We obtain 2 bifurcation curves of unstable positive radial solutions by conducting a detailed local bifurcation analysis near the positive constant solutions, with β as the bifurcation parameter. Our analysis reveals two distinct bifurcation directions and helps us understand the instability of these solutions. Furthermore, we explore global bifurcation behavior by applying the index formula and Rabinowitz’s theorem. This establishes the existence, non-uniqueness, and instability of positive solutions.

We prove that certain renormalized value functions associated with the d-dimensional ( d≧ 2 ) N-body problem corresponding to different limiting shapes of expanding solutions, under the assumption that the center of mass is at the origin, are viscosity solutions of the associated Hamilton–Jacobi equation. We analyze their singularities, defined as the initial configurations for which the minimizer of the associated variational problem is not unique. Moreover, we estimate the size of the closure of the singular set by proving its ℋ^d(N-1)-1 -rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points.

In this paper we present SymOrb.jl (2024), a software which combines group representation theory and variational methods to provide numerical solutions of singular dynamical systems of paramount relevance in Celestial Mechanics and other interacting particles models. Among all, it prepares for large-scale search of symmetric periodic orbits for the classical n-body problem and their classification, paving the way towards a computational validation of Poincar & eacute; conjecture about the density of periodic orbits. Through the accessible language of Julia, Symorb.jl offers a unified implementation of an earlier version (dlfer-symorb, 2017). This paper provides theoretical and practical guidelines for the specific approach we adopt, complemented with examples.

We prove $\textit{a priori}$ and $\textit{a posteriori}$ H\"older bounds and Schauder $C^{1,\alpha}$ estimates for continuous solutions to singular/degenerate equations with variable coefficients of type $$ \mathrm{div}\left(|u|^a A\nabla w\right)=0\qquad\mathrm{in \ }\Omega\subset\mathbb{R}^n, $$ where the weight $u$ solves an elliptic equation of type $\mathrm{div}\left(A\nabla u\right)=0$ with a Lipschitz-continuous and uniformly elliptic matrix $A$ and has a nontrivial, possibly singular, nodal set. Such estimates are uniform with respect to $u$ in a class of normalized solutions having bounded Almgren's frequency. More precisely, we provide $\textit{a priori}$ Hölder bounds in any space dimension, and Schauder estimates when $n=2$. When $a=2$, the results apply to the ratios of two solutions to the same PDE sharing their zero sets. Then, one can infer higher order boundary Harnack principles on nodal domains by applying the Schauder estimates for solutions to the auxiliary degenerate equation. The results are based upon a fine blow-up argument, Liouville theorems and quasiconformal maps.

We prove a priori Hölder bounds for continuous solutions to degenerate equations with variable coefficients of type div(u^2 A∇ w)=0 in Ω⊂ℝ^n, div(A∇ u)=0, where A is a Lipschitz continuous, uniformly elliptic matrix (possibly u has non-trivial singular nodal set). Such estimates are uniform with respect to u in a class of normalized solutions that have a bounded Almgren frequency. As a consequence, a boundary Harnack principle holds for the quotient of two solutions vanishing on a common set. This analysis relies on a detailed study of the associated weighted Sobolev spaces, including integrability of the weight, capacitary properties of the nodal set, and uniform Sobolev inequalities yielding local boundedness of solutions.

We give a complete characterization of the boundary traces $\varphi_i$ ($i=1,\dots,K$) supporting spiraling waves, rotating with a given angular speed $\omega$, which appear as singular limits of competition-diffusion systems of the type \[ \frac{\partial}{\partial t} u_i -\Delta u_i = \mu u_i -\beta u_i \sum_{j \neq i} a_{ij} u_j \text{ in } \Omega \times\mathbb{R}^+, \qquad u_i = \varphi_i \text{ on $\partial\Omega\times\mathbb{R}^+$}, \qquad u_i(\mathbf{x},0) = u_{i,0}(\mathbf{x}) \text{ for $\mathbf{x} \in \Omega$} \] as $\beta\to +\infty$. Here $\Omega$ is a rotationally invariant planar set and $a_{ij}>0$ for every $i$ and $j$. We tackle also the homogeneous Dirichlet and Neumann boundary conditions, as well as entire solutions in the plane. As a byproduct of our analysis we detect explicit families of eternal, entire solutions of the pure heat equation, parameterized by $\omega\in\mathbb{R}$, which reduce to homogeneous harmonic polynomials for $\omega=0$.

Advances in the variational approach to the n-body problem have led to significant progress in celestial mechanics, uncovering new types of possible orbits. In this paper, critical points of the Lagrangian action associated with the n-body problem are analysed using evolutionary algorithms to identify periodic and symmetrical solutions of the discretised system. A key objective is to locate minimum points of the action functional, as these correspond to feasible periodic solutions that satisfy the system’s differential equations. By employing both stochastic and deterministic algorithms, we explore the solution space and obtain numerical representations of these orbits. Next, we examine the stability of these orbits by treating them as critical points. One approach is to compute their discrete Morse index to distinguish between minimum points and saddle points. Another is to classify them based on their action levels. Finally, analysing the boundaries of their attraction basins allows us to identify non-minimal critical points via the Ambrosetti–Rabinowitz Mountain Pass Theorem. This leads to an updated version of the algorithm that provides a constructive proof of the theorem, yielding new orbits in specific cases. This paper builds upon and extends the results presented in Introna et al. (75th International Astronautical Conference, Milan, Italy), providing a more detailed theoretical framework and deeper insights into the formulation. Additionally, we present new numerical results and an extended analysis of the critical points found, further enhancing the findings of the previous study.

This paper deals with the so-called Boltzmann billiard, that is, a billiard subjected to a central force of the type V(r)=-α/r-β/r^2, α and β being positive constants, and with a straight reflection table. In the particular case of α and β positive, we prove the presence of a symbolic dynamics, and hence of positive topological entropy, at positive energy and for β sufficiently small.

We investigate the integrability of Kepler billiards-mechanical billiard systems in which a particle moves under the influence of a Keplerian potential and reflects elastically at the boundary of a strictly convex planar domain. Our main result establishes that, except possibly for one location of the gravitational center, analytic integrability at high energies occurs only when the domain is an ellipse and the center is placed at one of its foci. This provides a partial affirmative answer to a Keplerian analogue of the classical Birkhoff-Poritsky Conjecture. Our approach is based on the construction of symbolic dynamics arising from chaotic subsystems that emerge in the high-energy regime. Depending on the geometric configuration of the boundary and the location of the attraction center, we construct three types of symbolic dynamics by shadowing chains of punctured Birkhoff-type trajectories. These constructions yield subsystems conjugated to Bernoulli shifts, implying positive topological entropy and precluding analytic integrability. We further analyze the notion of focal points of the second kind, showing that in real-analytic, non-elliptic domains there can be at most one such point, while ellipses are the only domains admitting two-coinciding with their classical foci. Finally, we demonstrate the existence of an infinite-dimensional family of non-elliptic domains possessing a focal point of the second kind, and conclude with numerical simulations illustrating chaotic behavior in such cases.

We analyze the bifurcation phenomenon for the following two-component competition system: -Δ u_1=μ u_1(1-u_1)-βα u_1u_2, in B_1⊂ℝ^N, -Δ u_2=σ u_2(1-u_2)-βγ u_1u_2, in B_1⊂ℝ^N, ∂ u_1/∂ n= ∂ u_2/∂ n =0, on ∂ B_1, where N≥ 2, α>γ>0, σ≥μ>0 and β>σ/γ. More precisely, treating β as the bifurcation parameter, we initially perform a local bifurcation analysis around the positive constant solutions, obtaining precise information of where bifurcation could occur, and determine the direction of bifurcation. As a byproduct, the instability of the constant solution is provided. Furthermore, we extend our exploration to the global bifurcation analysis. Lastly, under the condition σ=μ, we demonstrate the limiting configuration on each bifurcation branch as the competition rate β→+∞.

We investigate the existence of rotating spirals for three-component competition-diffusion systems in B-1 C R-2: {partial derivative(t)u(1) - Delta u(1) = f(u(1)) - beta alpha u(1)u(2) - beta gamma u(1)u(3), in B-1 x R+, partial derivative(t)u(2) - Delta u(2) = f (u(2)) - beta gamma u(1)u(2) - beta alpha u(2)u(3), in B-1 x R+, partial derivative(t)u(3) - Delta u(3) = f(u(3)) - beta alpha u(1)u(3) - beta gamma u(2)u(3), in B-1 x R+, u(i)(x, 0) = u(i,0)(x), i = 1, 2, 3, in B-1, with Neumann or Dirichlet boundary conditions, where f (s) = mu s(1 - s) with mu, beta, alpha, gamma > 0 and alpha not equal gamma. For the Neumann problem, we establish the existence of rotating spirals by applying the multi-parameter bifurcation theorem. As a byproduct, the instability of the constant positive solution is proved. In addition, for the non-homogeneous Dirichlet problem, the Rothe fixed point theorem is employed to prove the existence of rotating spirals. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

In this note, we present the theoretical and computational approach our research group is developing to detect chaotic behaviours in the classical N-body problem of Celestial Mechanics. Inspired by the famous Poincaré conjecture, we numerically determine large sets of symmetric periodic orbits and we encode their simple motions by revealing the existence of a symbolic dynamics. Finally, we explore a new idea for combining periodic orbits to create new solutions with increasing complexity, using AI algorithms. We also revisit Conley’s ideas on low-energy transport orbits.

We investigate periodic trajectories in a classical system consisting of multiple mutually repelling electrons constrained to move along a half-line, with an attractive nucleus fixed at the origin. Adopting a variational framework, we seek critical points of the associated Lagrangian action functional using a modified Lusternik-Schnirelmann theory for manifolds with boundary. Furthermore, in the limit where the electron charges vanish, we demonstrate that frozen planet orbits converge to segments of a brake orbit in a Kepler-type problem, thereby drawing a strong analogy with Schubart orbits in the gravitational N-body problem.

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: { ∂ t u ¯ − y − a ∇ ⋅ ( y a ∇ u ¯ ) = 0 a m p ; in B 1 + × ( − 1 , 0 ) − ∂ y a u ¯ = q ( x , t ) u a m p ; on B 1 × { 0 } × ( − 1 , 0 ) , \begin{equation*} \begin {cases} \partial _t \overline {u} - y^{-a} \nabla \cdot (y^a \nabla \overline {u}) = 0 \quad &\text { in } \mathbb {B}_1^+ \times (-1,0) \\ -\partial _y^a \overline {u} = q(x,t)u \quad &\text { on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*} where a ∈ ( − 1 , 1 ) a\in (-1,1) is a fixed parameter, B 1 + ⊂ R N + 1 \mathbb {B}_1^+\subset \mathbb {R}^{N+1} is the upper unit half ball and B 1 B_1 is the unit ball in R N \mathbb {R}^N . Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator H s u ( x , t ) = 1 | Γ ( − s ) | ∫ − ∞ t ∫ R N [ u ( x , t ) − u ( z , τ ) ] G N ( x − z , t − τ ) ( t − τ ) 1 + s d z d τ . \begin{equation*} H^su(x,t) = \frac {1}{|\Gamma (-s)|} \int _{-\infty }^t \int _{\mathbb {R}^N} \left [u(x,t) - u(z,\tau )\right ] \frac {G_N(x-z,t-\tau )}{(t-\tau )^{1+s}} dzd\tau . \end{equation*} We characterise the possible blow-ups and we examine the structure of the nodal set of solutions vanishing with a finite order. More precisely, we prove that the nodal set has at least parabolic Hausdorff codimension one in R N × R \mathbb {R}^N\times \mathbb {R} , and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to an Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results including Federer’s reduction principle and the parabolic Whitney’s extension.

We prove uniform Hölder estimates in a class of singularly perturbed competition-diffusion elliptic systems, with the particular feature that the interactions between the components occur three by three (ternary interactions). These systems are associated to the minimization of Gross-Pitaevski energies modeling ternary mixture of ultracold gases and other multicomponent liquids and gases. We address the question whether this regularity holds uniformly throughout the approximation process up to the limiting profiles, answering positively. A very relevant feature of limiting profiles in this process is that they are only partially segregated, giving rise to new phenomena of geometric pattern formation and optimal regularity.

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(\rho^aA\nabla w\right)=\rho^af+\mathrm{div}\left(\rho^aF\right) \quad\textrm{in}\; \Omega \] for exponents $a>-1$, where the weight $\rho$ vanishes in a non degenerate manner on a regular hypersurface $\Gamma$ which can be either a part of the boundary of $\Omega$ or mostly contained in its interior. As an application, we extend such estimates to the ratio $v/u$ of two solutions to a second order elliptic equation in divergence form when the zero set of $v$ includes the zero set of $u$ which is not singular in the domain (in this case $\rho=u$, $a=2$ and $w=v/u$). We prove first $C^{k,\alpha}$-regularity of the ratio from one side of the regular part of the nodal set of $u$ in the spirit of the higher order boundary Harnack principle established by De Silva and Savin. Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension $n=2$, we provide local gradient estimates for the ratio which hold also across the singular set.

We seek frozen planet orbits for the helium atom through an application of the Mountain Pass Lemma to the Lagrangian action functional. Our method applies to a wide class of gravitational-like interaction potentials thus generalising the results in [7] (Cieliebak, Frauenfelder and Volkov - 2023). We also let the charge of the two electrons tend to zero and perform the asymptotic analysis to prove convergence to a limit trajectory having a collision-reflection singularity between the electrons.