A survey on the blow-up method for fast-slow systems

In this document we review a geometric technique, called the blow-up method, as it has been used to analyze and understand the dynamics of fast-slow systems around non-hyperbolic points. The blow-up method, having its origins in algebraic geometry, was introduced to the study of fast-slow systems in the seminal work by Dumortier and Roussarie in 1996, whose aim was to give a geometric approach and interpretation of canards in the van der Pol oscillator. Following Dumortier and Roussarie, many efforts have been performed to expand the capabilities of the method and to use it in a wide range of scenarios. Our goal is to present in a concise and compact form those results that, based on the blow-up method, are now the foundation of the geometric theory of fast-slow systems with non-hyperbolic singularities. Due to their great importance in the theory of fast-slow systems, we cover fold points as one of the main topics. Furthermore, we also present several other singularities such as Hopf, pitchfork, transcritical, cusp, and Bogdanov-Takens, in which the blow-up method has been proved to be extremely useful. Finally, we survey further directions as well as examples of specific applied models, where the blow-up method has been used successfully.

Modeling the transport of radioactive pollution in the Ussuri Gulf during the first days after the nuclear accident in the Chazhma Bay in August 1985

Представлены результаты лагранжевого моделирования распространения радиоактивного загрязнения в Уссурийском заливе на различных горизонтах по глубине на основе численной региональной модели циркуляции ROMS с использованием эмпирических данных выпадения радиоактивных осадков из атмосферы на поверхность акватории в день аварии на атомной подводной лодке в бухте Чажма 10 августа 1985 г. Показано, что радиоактивное пятно могло оставаться в Уссурийском заливе в течение первых четырех суток после аварии. Установлено, что эволюция и деформация начального пятна загрязнения на разных горизонтах обусловлены влиянием вихрей разных полярностей и размеров (мезомасштабный циклон в центре залива, субмезомасштабный антициклон на севере и мезомасштабный антициклон на юге) и лагранжевых когерентных структур, связанных с гиперболическими точками в заливе. The results of Lagrangian modeling of the transport of radioactive pollution in the Ussuri Gulf at various depths based on a regional ROMS numerical model of circulation using the empirical data on the radioactive fallout from the atmosphere at the sea surface on the day of the accident at a nuclear submarine in the Chazhma Bay in August 10, 1985. It was shown that the radioactive particles remain in the Ussuri Gulf for the first 4 days after the accident. It has been shown that the evolution and deformation of the initial pollution patch on various horizons was influenced by vortices of different polarity and size in the Ussuri Gulf (a mesoscale cyclone in the center of the Gulf, a sub-mesoscale anticyclone in the north and a mesoscale anticyclone in the south) and by Lagrangian coherent structures connected with hyperbolic points in the Gulf.

Bistable curvature potential at hyperbolic points of nematic shells

At hyperbolic points, where the Gaussian curvature is negative, nematic shells with unequal elastic constants can exhibit two preferred alignments.

SRB measures for almost Axiom A diffeomorphisms

We consider a diffeomorphism $f$ of a compact manifold $M$ which is almost Axiom A, i.e. $f$ is hyperbolic in a neighborhood of some compact $f$-invariant set, except in some singular set of neutral points. We prove that if there exists some $f$-invariant set of hyperbolic points with positive unstable Lebesgue measure such that for every point in this set the stable and unstable leaves are ‘long enough’, then $f$ admits an SRB (probability) measure.

Set of wires to simulate tokamaks with poloidal divertor

Simple wire models have been proposed to simulate magnetic configurations in tokamaks. Here we consider electric currents in five parallel infinite wires to obtain double-null magnetic surfaces with specific choices of magnetic axis positions, triangularity, and elongation. As an example, we choose the position and the electric current of each wire to obtain magnetic surfaces similar to those expected in the tokamak international thermonuclear experimental reactor. Moreover, we also integrate the perturbed field line differential equation to simulate chaotic layers near the hyperbolic points and deposition patterns at the divertor plate observed in tokamaks. To simulate that, we add to the model a perturbing error field, due to asymmetries in the tokamak coils, and introduce a random collisional term to the field line mapping to reproduce escape pattern alterations due to particle collisions.