Dia: Dimecres, 4 d'octubre de 2023
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Filippo Giuliani (Politecnico de Milano)
Títol: Sobolev instability for the cubic NLS on irrational tori.
Resum: In the last two decades the study of instability in Sobolev spaces for nonlinear Hamiltonian partial differential equations on compact manifolds has drawn lots of attention in the mathematical community. A breaktrough result in this sense is due to Colliander-Keel-Staffilani-Takaoka-Tao (Invent. Math 2010), who showed the existence of solutions to the defocusing cubic NLS on the 2-dimensional square torus with arbitrarily small initial data and arbitrarily large Sobolev norms at later times. The mechanism to construct such unstable solutions is based on the study of the resonant dynamics of NLS and it has inspired several other works. However, Staffilani noticed that the same strategy would not applied for the NLS equation on 2-dimensional irrational tori, where the resonant structure is less rich.In this talk we discuss how we overcame this problem to prove Sobolev instability for the cubic NLS on irrational tori. Moroever, we present a recent result of this type where we take into account also the presence of smooth convolution potentials.
Dia: Dimecres, 25 d'octubre de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Román Moreno (UPC)
Títol: Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
Resum: In this talk we will discuss the splitting distance of a rapidly perturbed pendulum
Dia: Dimecres, 29 de novembre de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Mar Giralt (IMCCE de l'Observatoire de Paris)
Títol: An Arnold diffusion mechanism for the Galileo satellites
Resum: Among the various actions that are being taken to preserve the circumterrestrial environment, end-of-life disposal solutions play a key role. In this regard, innovative strategies should be conceived not only by means of novel technologies, but also following an advanced theoretical understanding.
A growing effort is devoted to exploit natural perturbations to lead the satellites towards an atmospheric reentry. In the case of the Medium Earth Orbit region, home of the navigation satellite Galileo, the main driver is the gravitational perturbation due to the Moon, that can increase the eccentricity in the long term. In this way, the pericenter altitude gets into the atmospheric drag domain and the satellite eventually reenters.
This is a joint work with Elisa Maria Alessi (IMATI), Inma Baldomá (UPC), Marcel Guardia (UB) and Alexandre Pousse (IMATI).
Dia: Dimecres, 20 de desembre de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Marcel Guàrdia, Universitat de Barcelona
Títol: Hyperbolic dynamics and oscillatory motions in the 3 Body Problem
Resum: Consider the three body problem with positive masses. In 1922 Chazy classified the possible final motions the three bodies may possess, that is the behaviors the bodies may have when time tends to infinity. One of them is what is known as oscillatory motions, that is, solutions of the three body problem such that the liminf (as time tends to infinity) of the relative positions between bodies is finite whereas the limsup is infinite. That is, solutions for which the bodies keep oscillating between an increasingly large separation and getting closer together. The first result of existence of oscillatory motions was provided by Sitnikov for a Restricted Three Body Problem, called nowadays Sitnikov model. His result has been extended to several Celestial Mechanics models, but always with rather strong assumptions on the values of the masses. In this talk I will explain how to construct oscillatory motions for the three body problem for any value of the masses (except for the case of three equal masses). The proof relies on the construction of hyperbolic invariant sets whose dynamics is conjugated to that of the shift of infinite symbols (i.e. symbolic dynamics). That is, we construct invariant sets for the three body problem with chaotic dynamics, which moreover contain oscillatory motions.
This is a joint work with Pau Martin, Jaime Paradela and Tere M. Seara.
Beer seminar de final de quadrimestre
Dia: Dimecres, 14 de febrer de 2024
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Anna Gierzkiewicz-Pieniążek (Jagiellonian University, Cracòvia)
Títol: The Sharkovskii Theorem for multidimensional maps with attracting periodic orbits
Resum: I would like to present a computer-assisted method for proving rigorously the existence of a wide variety of periodic orbits for multidimensional maps with an attracting n-periodic orbit, based on the proof of Sharkovskii Theorem for interval maps. The method [A. G., P. Zgliczyński, J Differ Equ, 314(2022),733-751] is a simple example of using CAPD library for C++.
Dia: Dimecres, 21 de febrer de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Rohil Prasad, University of Berkeley
Títol: On the dense existence of compact invariant set
Resum: This is joint work with Dan Cristofaro-Gardiner. We show that for any monotone-area preserving diffeomorphism of a closed surface or for any Reeb flow on a closed contact 3-manifold with torsion Chern class, there exist infinitely many distinct proper compact invariant subsets whose union is dense in the manifold. No genericity assumptions are required. The former class of systems includes all Hamiltonian diffeomorphisms of closed surfaces and the latter class of systems includes the geodesic flow of any Finsler metric on a closed surface. In particular, our methods can also show that any Finsler metric on a closed surface has infinitely many non-dense geodesics, with pairwise distinct closures, whose union is dense in the surface.
A càrrec de: Mikhail Hlushchanka, University of Amsterdam
Títol: Invariant graphs for rational maps: construction, application, and open problems
Resum: Invariant graphs provide nice combinatorial models for dynamical systems under consideration. As such, they appear naturally in various aspects of complex dynamics and have multiple applications. For instance, "Hubbard trees" were used to classify all postcritically-finite polynomials in the 80s. I will present the main approaches for the construction of invariant graphs, overview some of their applications, and discuss several open combinatorial problems in this area.
Dia: Dimecres, 28 de febrer de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Jared Todd Blanchard, Standford University
Títol: Analysis of Invariant Funnels in the Circular Restricted Three-Body Problem
Resum: Scientists have identified the ocean worlds Europa and Enceladus as the most promising targets in the search for life in the solar system. Designing trajectories to such deep space targets requires balancing fuel and time constraints. Study of the Circular Restricted Three-Body Problem (CR3BP) has allowed mission designers to find very efficient trajectories that arrive in realistic time frames. Mission design work relies on using the periodic and quasi-periodic orbits that add structure to the chaotic dynamics of the CR3BP, as well as their hyperbolic invariant manifolds to intelligently sample the state space. In this presentation, we review our novel method for computing invariant funnels around non-periodic trajectories that approach the secondary body. Invariant funnels are sets of trajectories that converge in position space to a target point and can often be generated in backward time by imposing a parallel velocity condition. We review some of their applications to mission design for the ocean worlds.
Dia: Dimecres, 6 de març de 2024
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Amadeu Delshams, UPC
Títol: Shadowing of non-transversal heteroclinic chains in lattices
Resum: We study the global instability of dynamical systems on complex lattices by shadowing chains of nontransversal heteroclinic connections between several periodic orbits. The systems we consider are inspired by the so-called toy model systems (TMS) used to show the existence of energy transfer from low to high frequencies in the nonlinear cubic Schrödinger equation (NLSE) or generalizations. Using the geometric properties of the complex projective space as a base space, we generate in a natural way collections of such systems containing these chains, both in the Hamiltonian and non-Hamiltonian setting. On the other hand, we characterize the property of "block diagonal dynamics along the heteroclinic connections" that allows these chains to be shadowed, a property which in general only holds for transversal heteroclinic connections. Due to the lack of transversality, only finite chains are shadowed, as there is a dropping dimensions mechanism in the evolution of any disk close to them. The main shadowing tools used in our work are the covering relations as introduced by one of the authors.
This is a joint work with Piotr Zgliczynski, Jagiellonian University, Krakow.
Dia: Dimecres, 13 de març de 2024
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Shoya Motonaga, Ritsumeikan University
Títol: Nonintegrability and related dynamics of the hydrogen atom in a circularly polarized microwave field
Resum: We consider the hydrogen atom with a circularly polarized microwave field (the CP problem). The model is given by a periodically perturbed Hamiltonian system with two degrees of freedom whose unperturbed system is the Kepler problem. We prove that the corresponding system in a rotating coordinate is not analytically Liouville integrable such that first integrals depend on the perturbation parameter analytically. Moreover, the persistence of infinitely many periodic orbits corresponding to elliptic orbits in the Kepler problem is provided. In our approach, the subharmonic Melnikov functions for periodic orbits play key roles. We also consider the case that the period of the periodic orbit goes to infinity.
Dia: Dimecres, 20 de març de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Edmond Koudjinan, Institute of Science and Technology Austria (ISTA)
Títol: On the Birkhoff conjecture for nearly centrally symmetric domains
Resum: In this talk, I will discuss a recent advance on Birkhoff conjecture, namely a proof that: an integrable, nearly centrally symmetric Birkhoff billiard table is necessarily an ellipse. This is done by combining recent breakthroughs by Bialy-Mironov (who prove the conjecture for centrally symmetric Birkhoff billiard tables) and by Kaloshin-Sorrentino (who prove the conjecture for Birkhoff billiard tables close to ellipses). In particular, we shall discuss the use of a nonstandard generating function discovered by Bialy-Mironov.
Dia: Dimecres, 3 d'abril de 2024
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Andrew Clarke, UPC
Títol: Chaotic properties of billiards in circular polygons
Resum: Circular polygons are closed plane curves formed by concatenating a finite number of circular arcs so that, at the points where two arcs meet, their tangents agree. These curves are strictly convex and 𝒞1, but not 𝒞2. We study the billiard dynamics in domains bounded by circular polygons. We prove that there is a set accumulating on the boundary of the domain in which the return dynamics is semiconjugate to a transitive shift on infinitely many symbols. Consequently the return dynamics has infinite topological entropy. In addition we give an exponential lower bound on the number of periodic orbits of large period, and we prove the existence of trajectories along which the angle of reflection tends to zero with optimal linear speed. These results are based on joint work with Rafael Ramírez-Ros.
Dia: Dimecres, 17 d'abril de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Corentin Fierobe, Institute of Science and Technology Austria (ISTA)
Títol: On the Birkhoff conjecture for nearly centrally symmetric domains
Resum: Perturbations of integrable systems have been intensively studied, mostly from a KAM point of view. There the question is to understand if invariant curves persist after a small perturbation, and which one. Numerous results show that it is true for curves on which the dynamics is conjugated to a rotation of diophantine rotation number. More recent results about so-called integrable billiards and Birkhoffs conjecture underline the importance of invariant curves on which the dynamic is conjugated to a rational rotation when one studies rigidity phenomena of perturbed systems. In the talk we will present a result on invariant curves in an analytic family of twist maps which extends a result by Arnaud, Massetti and Sorrentino in the two-dimensional case, and which can notably be applied to different billiard models. It is the fruit of a joint work with A. Sorrentino.
Dia: Dimecres, 24 d'abril de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Maximilian Engel, Freie Universität Berlin.
Títol: Shear-induced chaos via stochastic forcing: a tale of finding positive Lyapunov exponents.
Resum: We discuss the phenomenon of shear-induced chaos, coined by Wang and Young about twenty years ago and referring to chaotic behavior as a result of shear being magnified by some forcing, in the context of stochastic perturbations. As a latest result, we show the positivity of Lyapunov exponents for the normal form of a Hopf bifurcation, perturbed by additive white noise, under sufficiently strong shear strength. This completes a series of related results for simplified situations which we can exploit by studying suitable limits of the shear and noise parameters. Some general ideas concerning conditioned random dynamics, computer-assisted proofs and continuity of Lyapunov exponents will be highlighted along the way.
A càrrec de: Francisco J. Beron Vera, University of Miami.
Títol: Inestabilidades térmicas en el océano hamiltoniano
Resum: Se estudia la estabilidad de una corriente zonal baroclínica en el plano β en un modelo cuasigeostrófico de gravedad reducida con variación lateral de flotabilidad. Este tipo de modelo, llamado modelo de Ripa, provee una descripción sencilla para las circulaciones de pequeña escala, la llamada submesoescala, comúnmente observadas en imágenes de color de la superficie del océano, las cuales es de sperarse que proliferen a medida de que el océano absorba calor de la troposfera en continuo calentamiento y los gradientes de temperatura se tornen más empinados. Esto hace de gran interés entender el funcionamiento del modelo.
En ausencia de viscosidad y forzamiento, el modelo de Ripa es hamiltoniano, pero en el sentido generalizado de Lie-Poisson. Además conservar energía y momento zonal, como consecuencia de simetría ante translaciones temporales y zonales via el teorma de Noether, respectivamente, el modelo conserva una familia infinita de funcionales que forman el núcleo del paréntesis de Lie-Poisson, llamadas casimires.
El uso de estas integrales de movimiento en el método de Arnold permite demostrar estabilidad de Lyapunov para ciertos parámetros que determinan la corriente de arriba, que representa un equilibrio condicionado del sistema. La existencia de equilibrios llamadas método de Shepherd. Los resultados se acompañan de simulaciones numéricas directas.
Es de mi interés particular promover la colaboración en el último punto, cuyo desarrollo riguroso requiere de la utilización de métodos de integración que preserven la estructura geométrica del modelo de Ripa.
Dia: Dimecres, 15 de maig de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Fenfen Wang, Universitat de Barcelona
Títol: Response solution to ill-posed Boussinesq equation with quasi-periodic forcing of Liouvillean frequency.
Resum: We focus on the existence of response solution (i.e.,quasi-periodic solution with the same frequency as the forcing) for the quasi-periodically forced generalized ill-posed Boussinesq equation, where the forcing frequency is Liouvillean (beyond Diophantine or Brjuno frequency). The proof is based on a modified Kolmogorov-Arnold-Moser (KAM) iterative scheme. We need to, at every step of KAM iteration, construct a symplectic transformation in a such way that the composition of these transformations reduce the original system to a new system which possesses zero as equilibrium. Note that the model under consideration is ill-posed and has complicated Hamiltonian structure. This makes homological equations appearing in KAM iteration to be different from the ones in the classical infinite-dimensional KAM theory. We strengthen the existing results in the literature where the system is well-posed or the forcing frequency is assumed to be Diophantine.
Dia: Dimecres, 22 de maig de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Lasse Rempe, University of Liverpool
Títol: A counterexample to Eremenko's conjecture.
Resum: Let f be a transcendental entire self-map of the complex plane. The escaping set of f consists of those points that tend to infinity under iteration of f. (For example, all real numbers belong to the escaping set of the exponential map, since they tend to infinity under repeated exponentiation.) In 1989, Eremenko conjectured that every connected component of the escaping set is unbounded.
Eremenko's conjecture has been a central problem in transcendental dynamics in the past decade. A number of stronger versions of the conjecture have been disproved, while weaker ones has been established, and the conjecture has also been shown to hold for a number of classes of functions. I will describe joint work with David Martí-Pete and James Waterman in which we construct a counterexample to Eremenko's conjecture.
The talk should be accessible to a general mathematical audience, including postgraduate students.
Dia: Dimecres, 29 de maig de 2024
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.A càrrec de: Pierre Houzelstein, École Normale Supérieure, París
Títol: Transfer operator approach to stochastic oscillators
Resum: Oscillatory behaviour is an ubiquitous phenomenon in natural systems, such as those modelled by neuroscience and climate sciences. Phase reduction is a powerful way of dealing with deterministic systems exhibiting oscillatory behavior: the N-dimensional state of the system is encoded into a one dimensional phase variable via projection onto the closed limit cycle attractor. However, real-world oscillations are often noisy and modelled using stochastic differential equations (SDEs). In this case, the deterministic phase reduction approach is ill-defined; finding an analogous method for stochastic oscillators is an open question. In this talk, we will present a novel stochastic phase reduction framework, designed with the deterministic approach in mind [1]. It relies on the spectral decomposition of the Kolmogorov backwards operator (aka the generator of the stochastic Koopman operator). We note that this object is the source of much current interest due to the development of modern data analysis methods aiming to reconstruct transfer operators from time series, such as Dynamic Mode Decomposition (DMD) [2]. We will start by introducing the phase reduction approach for determin- istic systems, and show its limitations when noise is present. We will then discuss about transfer operators and how their spectral properties may be used to infer relevant statistical properties of the relevant system, with the particular case of oscillators in mind. Finally, we will introduce numerical and data-driven methods to construct those operators, and apply them to case examples.
References: [1] Pierre Houzelstein, Peter J. Thomas, Benjamin Lindner, Boris S. Gutkin and Alberto Pérez-Cervera, Generalized dynamical phase reduction for stochastic oscillators, arXiv:2402.02856 (2024). [2] Matthew O. Williams, Ioannis G. Kevrekidis and Clarence W. Rowley, A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition, Journal of Nonlinear Science (2015).A càrrec de: Renato Calleja, UNAM
Títol: From the Lagrange Triangle to the Figure-Eight Choreography: About a conjecture of C. Marchal.
Resum: In the context of the three-body problem with equal masses, Marchal conjectured in 1999 that the most symmetric continuation class of Lagrange’s equilateral triangle solution, known as the P12 family of Marchal, includes the remarkable figure-eight choreography discovered by Moore in 1993 and proven to exist by Chenciner and Montgomery in 2000. In this talk, I will present a framework for verifying the existence of the P12 family as a zero-finding problem. Additionally, we will explore the relation between this verification and Marchal's conjecture. This work is a collaboration with Carlos García-Azpeitia, Olivier Hénot, Jean-Philippe Lessard, and Jason Mireles James.
Dia: Dimecres, 5 de juny de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.A càrrec de: Max Kreider, Case Western Reserve University
Títol: Arnold tongues for the synchronization of stochastic oscillator
Resum: Large-scale brain oscillations may reflect the synchronous behavior of neuron populations. However, the mechanisms underlying collective neuron dynamics are not well understood. One approach is to model neuron populations as systems of oscillators: ordinary differential equations with stable limit-cycle solutions. Arnold tongues characterize synchronization regimes of coupled oscillators as a function of frequency difference and coupling strength. However, analysis is often complicated by high dimensionality and nonlinear dynamics.
Phase reduction is effective for studying systems of coupled deterministic oscillators by describing the synchronized dynamics in terms of a one-dimensional phase (timing) variable for each oscillator. However, neural activity is noisy; stochastic phase concepts can extend phase reduction to noisy oscillators. Of particular utility is the asymptotic stochastic phase, derived from the Q function: the slowest decaying mode of the stochastic Koopman operator. The Q function simplifies system dynamics for single oscillators [Perez et al 2023 PNAS], yet a characterization of the synchronization of coupled stochastic oscillators remains an open question.
Here, we compute the Q function of coupled stochastic oscillators for three qualitatively different systems: a two-dimensional ring model, a four-dimensional linear system for which the Q function and low-lying spectrum are known analytically, and a four-dimensional nonlinear system. We demonstrate that in each case the eigenvalues corresponding to the Q function exhibit a qualitatively similar bifurcation, and that the synchronization boundary as a function of frequency difference and coupling strength resembles an Arnold tongue. We argue that our stochastic phase-based approach can contribute to the modeling and analysis of large-scale electrophysiological recordings.
Dia: Dimecres, 12 de juny de 2024
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.A càrrec de: Pau Roldan, UPC
Títol: Shortest transition time orbits near L1 in the Spatial RTBP
Resum: We consider the spatial restricted three-body problem, as a model for the motion of spacecraft relative to the Sun-Earth system. We focus on the dynamics near the equilibrium point L1 located between the Sun and the Earth. We show that it is possible for the spacecraft to transition from an orbit that is nearly planar relative to the ecliptic, to an inclined orbit, at zero energy cost. (And in fact it can transition through any prescribed sequence of inclinations).
We provide several explicit constructions of such orbits, and also develop an algorithm to design orbits that achieve the *shortest transition time*.
Our main new tool is the `Standard Scattering Map' (SSM), a semi-analytical representation of the scattering map originally introduced by Delshams, de la Llave and Seara. The SSM can be used in many other situations, from Arnold diffusion problems to transport phenomena in applications.
In collaboration with Amadeu Delshams (UPC, Barcelona) and Marian Gidea (Yeshiva University, NY).
Dia: Dimecres, 3 de juliol de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.A càrrec de: Álvaro Fernández, UB
Títol: Invariant tori in Hamiltonian systems: rigorous results and algorithms
Resum: We consider the existence of partially hyperbolic invariant tori with Diophantine frequencies and their invariant manifolds in real analytic Hamiltonian systems.
First, we present an a-posteriori KAM theorem stating that if we have embeddings for the torus and its stable bundle satisfying that the error in their functional invariance equations is small enough, then there is an invariant torus with stable and unstable invariant bundles nearby. The method of proof is based on the parameterization method and consists on constructing an iterative procedure that defines sequences of embeddings that converge to solutions of their invariance equations in a complex strip of the torus. The results rely on the geometrical properties of the system and of invariant tori and do neither assume action-angle coordinates nor a perturbative setting. Additionally, the theorem is constructed such that it applies both to autonomous and quasi-periodic Hamiltonian systems. The iterative procedure is applied in the Elliptic Restricted Three Body Problem where we compute a large set of 3-dimensional invariant tori with their stable and unstable bundles.
We also provide algorithms to compute higher order expansions of the whiskers. We will introduce the methodology for autonomous systems where we compute tori, bundle, and higher order terms simultaneously with an iterative procedure. We will also show some results in the Circular Restricted Three Body Problem and provide some comments on the generalization to quasi-periodic Hamiltonians.
Dia: Dimecres, 17 de juliol de 2024
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.A càrrec de: Patricia Sanchez Martin, UPC
Títol: Exploring galactic dynamics from single-time data
Resum: The dynamical systems approach to studying galactic shape and dynamics involves analyzing barred galaxies as a dynamical system governed by a gravitational potential created by the building blocks of the galaxy. The associated invariant manifolds of the planar family of Lyapunov orbits around the unstable equilibrium points delineate the arms and rings of barred galaxies. Orbits within these manifolds facilitate the transfer of matter between the inner and outer regions of the galaxy, with the Lyapunov orbits serving as gateways.
We aim to use the recent Gaia DR3 data release, which provides highly accurate positions and velocities of stars in the Milky Way and nearby regions, such as the Large Magellanic Cloud, at a single moment in time. This dataset presents a unique opportunity to validate our dynamical systems approach to explaining galactic features. There is a pressing need for methods to identify key galactic features from a single snapshot. In this study, we introduce a novel approach to identify these features.
Once our method identifies the building blocks of the galaxy from a single snapshot, we will utilize the results to model the potentials of the components of the galaxy. Combined with the detected angular velocity of the bar, these potentials will be incorporated into the classical dynamical system model. Once validated through simulations, our methodology can be applied to data from the Large Magellanic Cloud, a barred galaxy known for its asymmetrical density distribution in the central region and arms.
This is joint work with J. Amorós (UPC) and J. Masdemont (UPC).
Last updated: Sun Jul 14 13:36:48 2024