On the Integrability of Liénard systems with a strong saddle

We study the local analytic integrability for real Li\'{e}nard systems, $\dot x=y-F(x),$ $\dot y= x$, with $F(0)=0$ but $F'(0)\ne0,$ which implies that it has a strong saddle at the origin. First we prove that this problem is equivalent to study the local analytic integrability of the $[p:-q]$ resonant saddles. This result implies that the local analytic integrability of a strong saddle is a hard problem and only partial results can be obtained. Nevertheless this equivalence gives a new method to compute the so-called resonant saddle quantities transforming the $[p:-q]$ resonant saddle into a strong saddle.The first author is partially supported by a MINECO/FEDER grant number MTM2014- 53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR-1204. The second author is partially supported by a FEDER-MINECO grant MTM2016-77278-P, a MINEC0 grant MTM2013-40998-P, and an AGAUR grant number 2014SGR-568

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao

The double copy and classical solutions

The Bern-Carrasco-Johansson (BCJ) double copy, which relates the scattering amplitudes of gauge and gravity theories has been an active area of research for a few years now. In this thesis, we extend the formalism of BCJ to consider classical solutions to the field equations of motion, rather than scattering amplitudes. One first approach relies on a family of solutions to the Einstein equations, namely Kerr-Schild metrics, which linearise the Ricci tensor. Using them we propose a simple ansatz to construct a gauge theory vector field which, in a stationary limit, satisfies linearised Yang-Mills equations. Using such ansatz, that we call the Kerr-Schild double copy, we are able to relate, for example, colour charges in Yang-Mills with the Schwarzschild and Kerr black holes. We extend this formalism to describe the Taub-NUT solution (which is dual to an electromagnetic dyon), perturbations over curved backgrounds and accelerating particles, both in gauge and gravity theories. A second, more utilitarian approach consists on using the relative simplicity of gauge theory to efficiently compute relevant quantities in a theory of perturbative gravity. Working along this lines, we review an exercise by Duff to obtain a spacetime metric using tree-level graphs of a quantum theory of perturbative gravity, and repeat it using a BCJ inspired gravity Lagrangian. We find that the computation is notably simplified, but a new formalism must be developed to remove the unwanted dilaton information, that naturally appears in the double copy

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Réduire la dimension des systèmes complexes : un regard sur l'émergence de la synchronisation

Les systèmes complexes se caractérisent par l’émergence de phénomènes macroscopiques qui ne s’expliquent pas uniquement par les propriétés de leurs composantes de base. La synchronisation est l’un de ces phénomènes par lequel les interactions entre des oscillateurs engendrent des mouvements collectifs coordonnés. Une représentation sous forme de graphe permet de modéliser précisément les interactions, alors que les oscillations se décrivent par des dynamiques non linéaires. Étant donné le lien subtil entre le graphe et la dynamique des oscillateurs, il est difficile de prédire l’émergence de la synchronisation. L’objectif de ce mémoire est de développer de nouvelles méthodes pour simplifier les systèmes complexes d’oscillateurs afin de révéler les mécanismes engendrant la synchronisation. À cette fin, nous introduisons des notions de la théorie des graphes et des systèmes dynamiques pour définir la synchronisation sur des bases mathématiques. Nous présentons ensuite des approches existantes sophistiquées pour réduire la dimension de dynamiques d’oscillateurs. Ces techniques sont toutefois limitées lorsque les dynamiques évoluent sur des graphes plus complexes. Nous développons alors une technique originale—spécialement adaptée pour les graphes aux propriétés plus hétérogènes—pour réduire la dimension de dynamiques non linéaires. En plus de généraliser des approches récentes, notre méthode dévoile plusieurs défis théoriques reliés à la simplification d’un système complexe. Entre autres, la réduction de la dimension du système se bute à une trichotomie : il faut favoriser la conservation des paramètres dynamiques, des propriétés locales du graphe ou des propriétés globales du graphe. Finalement, notre méthode permet de caractériser mathématiquement et numériquement l’émergence d’états exotiques de synchronisation.Complex systems are characterized by the emergence of macroscopic phenomena that cannot be explained by the properties of their basic constituents. Synchronization is one of these phenomena in which the interactions between oscillators generate coordinate collective behaviors. Graphs allow a precise representation of the interactions, while nonlinear dynamics describe the oscillations. Because of the subtle relationship between graphs and dynamics of oscillators, it is challenging to predict the emergence of synchronization. The goal of this master’s thesis is to develop new methods for simplifying complex systems of oscillators in order to reveal the mechanism causing synchronization. To this end, we introduce notions of graph theory and dynamical systems to define synchronization on sound mathematical basis. We then present existing sophisticated approaches for reducing the dimension of oscillator dynamics. Yet, the efficiency of these techniques is limited for dynamics on complex graphs. We thus develop an original method—specially adapted for graphs with heterogeneous properties—for reducing the dimensions of nonlinear dynamics. Our method generalizes previous approaches and uncovers multiple challenges related to the simplification of complex systems. In particular, the dimension reduction of a system comes up against a trichotomy: one must choose to conserve either the dynamical parameters, the local properties of the graph, or the global properties of the graph. We finally use our method to characterize mathematically and numerically the emergence of exotic synchronization states

Matter manipulation with extreme terahertz light: Progress in the enabling THz technology

Terahertz (THz) light has proven to be a fine tool to probe and control quasi-particles and collective excitations in solids, to drive phase transitions and associated changes in material properties, and to study rotations and vibrations in molecular systems. In contrast to visible light, which usually carries excessive photon energy for collective excitations in condensed matter systems, THz light allows for direct coupling to low-energy (meV scale) excitations of interest, The development of light sources of strong-field few-cycle THz pulses in the 2000s opened the door to controlled manipulation of reactions and processes. Such THz pulses can drive new dynamic states of matter, in which materials exhibit properties entirely different from that of the equilibrium. In this review, we first systematically analyze known studies on matter manipulation with strong-field few-cycle THz light and outline some anticipated new results. We focus on how properties of materials can be manipulated by driving the dynamics of different excitations and how molecules and particles can be controlled in useful ways by extreme THz light. Around 200 studies are examined, most of which were done during the last five years. Secondly, we discuss available and proposed sources of strong-field few-cycle THz pulses and their state-of-the-art operation parameters. Finally, we review current approaches to guiding, focusing, reshaping and diagnostics of THz pulses. (C) 2019 The Author(s). Published by Elsevier B.V

Fraktalna analiza neomeđenih skupova u euklidskim prostorima i Lapidusove zeta funkcije

In this thesis , we consider relative fractal drums and their corresponding Lapidus fractal zeta functions, as well as a generalization of this notions to the case of unbounded sets at infinity. Relative fractal drums themselves are a generalization of the notion of a bounded subset in an Euclidean space. Here, we continue the ongoing research into their properties and the higher-dimensional theory of their fractal zeta functions and complex dimensions which started as a collaboration between M. L. Lapidus and D. Žubrinić in 2009 with the later addition of the author of this thesis. The theory of complex dimensions is already well developed for fractal strings; that is, for fractal subsets of the real line. The complex dimensions of a relative fractal drum are defined as poles of a meromorphic continuation of its corresponding distance or tube zeta function. Complex dimensions of a relative fractal drum generalize, in a way, the notion its box (or Minkowski) dimension. More precisely, under some mild conditions, the value of the box dimension of a relative fractal drum is a pole of its corresponding fractal zeta function with maximal real part. Moreover, the residue computed at this pole is closely related to its Minkowski content. Here we derive important results which further justify the notion of ‘complex dimensions’ and connect it to fractal properties of a given relative fractal drum. More precisely, we establish fractal tube formulas for a class of relative fractal drums which express their relative tube function; that is, the Lebesgue measure of their relative δ-neighborhood for small values of δ, as a sum over the residues of their fractal zeta function. These formulas are given with or without an error term and hold pointwise or distributionally depending on the growth properties of the corresponding fractal zeta function. The importance of these formulas is that they show how the complex dimensions are related to the asymptotic development of the relative tube function of a given relative fractal drum. As an application we derive a Minkowski measurability criterion for a large class of relative fractal drums. Furthermore, we also show that the complex dimensions of a relative fractal drum are, as expected, invariant to the dimension of the ambient space. We introduce a further generalization of the theory of complex dimensions to the context of unbounded sets at infinity which can be used as a new approach of applying fractal analysis to unbounded subsets in Euclidean spaces. This is done for unbounded sets of finite Lebesgue measure by introducing the notions of Minkowski content at infinity and Minkowski (or box) dimension at infinity which describe their fractal properties. Furthermore, we proceed by introducing an appropriate Lapidus (or distance) zeta function at infinity and show that it is well connected with the fractal properties of unbounded sets. We proceed by constructing interesting examples of quasiperiodic sets at infinity with arbitrary number (even infinite) of quasiperiods that exhibit complex fractal behavior. We also address the natural question which arises when dealing with unbounded sets and their fractal properties; that is, establish results about the fractal properties of their images under the one-point compactification and under the geometric inversion. Furthermore, we also investigate fractal properties of unbounded sets of infinite Lebesgue measure by introducing notions of the parametric φ-shell Minkowski content at infinity and the corresponding parametric φ-shell Minkowski (or box) dimension at infinity and we establish results connecting these notions with the distance zeta function at infinity. Finally we demonstrate how fractal analysis of unbounded sets via the geometric inversion may be applied to investigate bifurcations of dynamical systems occurring at infinity.U ovoj disertaciji bavimo se relativnim fraktalnim bubnjevima i njihovim fraktalnim zeta funkcijama Lapidusovog tipa, kao i generalizacijama ovih pojmova za slučaj neomeđenih skupova u beskonačnosti. Relativni fraktalni bubnjevi su sami po sebi generalizacija pojma omeđenog skupa u Euklidskom prostoru. Ovdje nastavljamo istraživanje njihovih svojstava i višedimenzionalne teorije njihovih fraktalnih zeta funkcija te pripadajućih kompleksnih dimenzija koje je započeto suradnjom M. L. Lapidusa i D. Žubrinića 2009. godine a kojoj se autor disertacije pridružio nešto kasnije. Teorija kompleksnih dimenzija već je vrlo dobro razvijena za slučaj fraktalnih struna, odnosno, fraktalnih podskupova realnog pravca. Kompleksne dimenzije relativnog fraktalnog bubnja definirane su kao polovi meromorfnog proširenja pripadajuće razdaljinske ili cijevne zeta funkcije. Na određeni način kompleksne dimenzije relativnog fraktalnog bubnja generaliziraju pojam njegove box dimenzije (ili dimenzije Minkowskog). Preciznije, uz neke blage uvjete, vrijednost box dimenzije relativnog fraktalnog bubnja jest pol njegove pripadajuće fraktalne zeta funkcije s maksimalnom vrijednošću realnog dijela. Štoviše, reziduum u tom polu usko je povezan sa sadržajem Minkowskog danog relativnog fraktalnog bubnja. U ovoj radnji izvodimo važne rezultate koji donose daljnje opravdanje pojma ‘kompleksnih dimenzija’ i povezuju ga s fraktalnim svojstvima danog relativnog fraktalnog bubnja. Preciznije, kao rezultat dobivamo fraktalne cijevne formule za klasu relativnih fraktalnih bubnjeva koje izražavaju njihovu relativnu cijevnu funkciju, odnosno, Lebesgueovu mjeru njihove relativne δ-okoline za male vrijednosti δ, kao sumu po reziduumima njihove fraktalne zeta funkcije. Te formule su dane s greškom ili bez greške i vrijede po točkama ili distribucijski ovisno svojstvima rasta pripadajuće fraktalne zeta funkcije. Važnost ovih formula je u tome što pokazuju kako su kompleksne dimenzije povezane s asimptotikom relativne cijevne funkcije danog relativnog fraktalnog bubnja. Kao primjenu izvodimo kriterij za Minkowskivljevu izmjerivost velike klase relativnih fraktalnih bubnjeva. Nadalje, očekivano, pokazujemo da su kompleksne dimenzije danog relativnog fraktalnog bubnja invarijantne u odnosu na dimenziju ambijentnog prostora. U nastavku radnje uvodimo generalizaciju teorije kompleksnih dimenzija u kontekstu neomeđenih skupova u beskonačnosti koja može poslužiti kao novi pristup primjeni fraktalne analize na neomeđene skupove u Euklidskim prostorima. U slučaju neomeđenih skup ova konačne Lebesgueove mjere, generalizaciju provodimo uvođenjem pojmova sadržaja Minkowskog u beskonačnosti i box dimenzije u beskonačnosti (ili dimenzije Minkowskog u beskonačnosti) koji opisuju njihova fraktalna svojstva. Nadalje, uvodimo i pripadajuću Lapidusovu (ili razdaljinsku) zeta funkciju u beskonačnosti te pokazujemo da je dobro povezana s fraktalnim svojstvima neomeđenih skupova. Nastavljamo s konstrukcijom zanimljivih primjera kvaziperiodičkih skupova u beskonačnosti s proizvoljnim brojem (moguće i beskonačnim) kvaziperioda koji posjeduju složena fraktalna svojstva. Također se bavimo i prirodnim pitanjem koje se postavlja prilikom istraživanja neomeđenih skupova i njihovih fraktalnih svojstava, u vidu pronalaženja rezultata koji ih povezuju s fraktalnim svojstvima njihovih slika po jednotočkovnoj kompaktifikaciji i po geometrijskoj inverziji. Nadalje, također istražujemo i fraktalna svojstva neomeđenih skupova beskonačne Lebesgueove mjere uvođenjem pojmova parametarskog φ-omotačkog sadržaja Minkowskog u beskonačnosti i pripadajuće parametarske φ-omotačke dimenzije Minkowskog u beskonačnosti (ili φ-omotačke box dimenzije u beskonačnosti) te izvodimo rezultate koji povezuju ove pojmove s razdaljinskom zeta funkcijom u beskonačnosti. Naposljetku, demonstriramo kako se fraktalna analiza neomeđenih skupova preko geometrijske inverzije može primijeniti u istraživanju bifurkacija dinamičkih sustava koje se događaju u beskonačnosti