1,412 research outputs found
Fractal analysis of hyperbolic saddles with applications
No full textIn this paper we express the Minkowski dimension of spiral trajectories near hyperbolic saddles and semi-hyperbolic singularities in terms of the dimension of intersections of such spirals with transversals near these singularities. We apply these results to hyperbolic saddle-loops and hyperbolic 2-cycles to obtain, using Minkowski dimension of a single spiral trajectory, some known upper bounds on the cyclicity of such limit periodic sets.This research was supported by Croatian Science Foundation (HRZZ) Grant PZS-2019-02-3055 from Research Cooperability program funded by the European Social Fund. The first two authors are supported by the Special Research Fund (BOF number:
BOF21BL01) of Hasselt University. The third author is supported by Croatian Science Foundation (HRZZ) Grant UIP-2017-05- 1020. The first and the third author are also supported by the bilateral Hubert-Curien Cogito grant 2023-24
Realization of analytic moduli for parabolic Dulac germs
No full textInternational audienceIn a previous paper [P. Mardešić and M. Resman. Analytic moduli for parabolic Dulac germs. Russian Math. Surveys , to appear, 2021, arXiv:1910.06129v2.] we determined analytic invariants, that is, moduli of analytic classification, for parabolic generalized Dulac germs. This class contains parabolic Dulac (almost regular) germs, which appear as first-return maps of hyperbolic polycycles. Here we solve the problem of realization of these moduli
Reading multiplicity in unfoldings from epsilon-neighborhoods of orbits
No full textAbstract.We consider generic analytic 1-parameter unfoldings of saddle-node germs of analytic vector fields on the real line, their time-one maps and the Lebesgue measure ofε-neighborhoods of the orbits of these time-one maps. The box dimension of an orbit gives the asymptotics of the principal term of this Lebesgue measure and it is known that it is dis-continuous at bifurcation parameters. To recover continuous dependence of the asymptotics on the parameter, here we expand asymptotically the Lebesgue measure ofε-neighborhoodsof orbits of time-one maps in a Chebyshev system, uniformly with respect to the bifurcation parameter. We use ́Ecalle-Roussarie-type compensators. We show how the number of fixed points of the time-one map born in the universal analytic unfolding of the parabolic point cor-responds to the number of terms vanishing in this uniform expansion of the Lebesgue measure of ε-neighborhoods of orbits
Maja Pelević : Nevidljiva
No full textPredstava Nevidljiva Maje Pelević Kazališta Virovitica i Kazališta lutaka Zadar u režiji Nikole Zavišića diplomski je ispit te tema diplomskog rada Gorana Vučka pod mentorstvom doc. dr. art. Maje Lučić Vuković na Umjetničkoj akademiji u Osijeku na Odsjeku za kazališnu umjetnost, smjer gluma i lutkarstvo. Početak rada zauzela je biografija autorice teksta Maje Pelević, nagrađivane srpske dramske autorice te analiza teksta po poglavljima, koja su bila prethodno zadana tekstom. Goran Vučko rad na predstavi započinje analizom režije te režijskim postupcima koji su bili netipični za njegovo dosadašnje iskustvo, proširivali su njegovo poimanje lutkarstva te su bili vrlo inspirativni za njega kao mladog umjetnika. Vučko ističe kako su vizualni elementi predstave također mnogo utjecali na sami razvoj te poetiku predstave, pogotovo silikonske LED lutke, reflektiranje svjetla na bijelo platno, kostimi roditelja. Vlastiti rad objasnio je kroz sve lutkarske tehnike i metode koje su u predstavi korištene. Opisao je rad sa silikonskim LED lutkama, rad sa svjetlom i sjenama, rad sa UV tehnikom, visećim slikama, te metalnim konstrukcijama. Vučko smatra bitnim dotaknuti se i svakog kolege s kojim je na sceni zajedno stvarao jer je njihovo stvaralaštvo direktno i indirektno utjecalo i na njegovo stvaralaštvo. Na kraju rada zaključuje da je Nevidljiva predstava koju je teško žanrovski kategorizirati, ali i da je lutkarstvo njezin veliki i važni elementThe play Invisible by Maja Pelević performed by Theater of Virovitica and Puppet Theater Zadar directed by Nikola Zavišić is graduate exam and the topic of graduate work by Goran Vučko under the mentor doc. dr. art. Maja Lučić Vuković at the Academy of Arts in Osijek at the Department of Theater Art, Acting and Puppetry. Graduate work begins with the biography of the text author Maja Pelević, the award-winning Serbian drama author, followed by the analysis of the text by chapters, which had been prefixed with the text. Goran Vučko's work on the performance began with the analysis of the rehearsals and rehearsal methods that were unusual for his experience so far, expanded his understanding of puppetry and were very inspirational for him as the young artist. Vučko points out that the visual elements of the play had also greatly influence the development itself and the poetics of the play, especially the silicone LED puppets, the reflection of the light on the white cloth, the costumes of the parents.He explained his own work through all the puppetry techniques and methods used in the play. He described the work with silicone LED puppets, light and shadows, UV technique, hanging pictures, and metal constructions. Vučko thinks it is important to refer on work of each of the colleagues he had worked with on the performance, because their creativity directly and indirectly influenced his own creativity. At the end of his work, he concludes that the play Invisible is difficult to categorize genres, but that puppetry is a major and important element
Fraktalne dimenzije
No full textU ovom radu bavimo se fraktalnim dimenzijama. U uvodu objašnjavamo pojam fraktala te navodimo neke poznate primjere fraktalnih skupova. U drugom poglavlju definiramo topološku dimenziju te određujemo topološke dimenzije raznih objekata kako bismo ih kasnije mogli usporediti s fraktalnim dimenzijama. U trećem poglavlju uvodimo pojam Hausdorffove mjere i pomoću njega definiramo Hausdorffovu dimenziju. Potom proučavamo i dokazujemo važna svojstva Hausdorffove dimenzije. Nakon toga, definiramo box dimenziju, navodimo razne ekvivalentne definicije box dimenzije te potom dokazujemo njezina svojstva. Na kraju poglavlja, kratko uspoređujemo Hausdorffovu i box dimenziju te navodimo primjer za koji se one razlikuju. Konačno, u četvrtom poglavlju određujemo fraktalne dimenzije raznih objekata, počevši od klasičnih geometrijskih objekata. Zatim određujemo Hausdorffovu i box dimenziju nekih fraktalnih skupova te na kraju spirale koja nije fraktalni skup u pravom smislu riječi, ali box dimenzija dobro opisuje njezinu gustoću oko ishodišta.In this thesis we study fractal dimensions. In the introduction we explain the concept of fractals and we introduce some well-known fractal sets. In the second chapter we define topological dimension and we calculate topological dimensions of some sets, in order to compare them later with their fractal dimensions. In the third chapter we introduce the concept of the Hausdorff measure and we define the Hausdorff dimension. We further study and prove some important properties of Hausdorff dimension. After that, we define the box dimension, we list equivalent definitions of box dimension and then we prove its properties. At the end of the chapter, we shortly compare Hausdorff and box dimension and we give an example of a set for which they are different. Finally, in the fourth chapter, we calculate fractal dimensions of different sets, starting with classical geometric objects. After that, we calculate Hausdorff and box dimensions of some fractal sets. Finally, we compute the box dimension of a spiral which is not really a fractal set, but its box dimension nevertheless describes its density of accumulation at the origin
Predator-prey dynamics of population models
No full textU ovom radu promatramo modele ponašanja populacija u vremenu. Prvo promatramo ponašanje jedne populacije kroz Malthusov i logistički model rasta. Zatim promatramo interakcije dvije populacije pomoću grabežljivac-plijen modela. Radimo kvalitativnu analizu ponašanja trajektorija u vremenu bez eksplicitnog rješavanja jednadžbi. Kroz rad proučavamo ekvilibrije linearnih i nelinearnih sustava te pripadajuće fazne portrete. Analizu faznih portreta linearnih sustava radimo s obzirom na svojstvene vrijednosti matrice sustava. Koristimo Hartman-Grobmanov teorem koji povezuje izgled trajektorija nelinearnih sustava i njegove linearizacije u slučaju hiperboličkog ekvilibrija. Objasnili smo da za grabežljivac-plijen (Lotka-Volterra) model postoje dva ekvilibrija, jedan je sedlo, a drugi centar. Na drugi ekvilibrij grabežljivac-plijen modela ne možemo primijeniti Hartman-Grobmanov teorem, nego dokazujemo da su trajektorije rješenja oko ekvilibrija zatvorene. To objašnjava periodično ponašanje populacije grabežljivaca i populacije plijena u vremenu.In this thesis, we consider population behavior models over time. First, we examine the behavior of a single population through the Malthusian and logistic growth models. Next, we consider the interactions of two populations using the predator-prey model. We conduct a qualitative analysis of the system over time without explicitly solving the equations. Throughout the work, we study the equilibria of linear and nonlinear systems, along with their corresponding phase portraits. The analysis of phase portraits of linear systems is based on the eigenvalues of the matrix of the system. We use the Hartman-Grobman theorem, which connects the trajectories of nonlinear systems and their linearizations in the case of hyperbolic equilibrium. We explain that for the predator-prey model, two equilibria exist: one is a saddle, and the other is a center. For the second equilibrium of the predator-prey model, the Hartman-Grobman theorem cannot be applied. We show that around second equlibrium solution trajectories are closed, which explains the periodic behavior of the predator and prey populations over time
Applications of ordinary differential equations and solving methods
No full textZa rješavanje običnih diferencijalnih jednadžbi možemo koristiti različite analitičke, kvalitativne i numeričke metode. Analitičke metode uključuju pronalaženje eksplicitnog rješenja običnih diferencijalnih jednadžbi pomoću algebarskih i računskih tehnika, ali možda neće biti izvedive za složenije jednadžbe. S druge strane, kod kvalitativnih metoda na temelju oblika jednadžbe procjenjujemo ponašanje rješenja u vremenu, bez eksplicitnog rješavanja jednadžbe. Jedna od kvalitativnih metoda je grafički prikaz polja smjerova i to će nam predstavljati glavni alat u kvalitativnoj analizi. Numeričke metode uključuju diskretizaciju i aproksimaciju rješenja običnih diferencijalnih jednadžbi pomoću numeričkih algoritama. Prilikom odabira odgovarajuće metode, važno je poznavati prednosti i nedostatke svake pojedine metode.To solve ordinary differential equations, we use different analytical, qualitative, and numerical methods. Analytical methods involve finding an explicit solution of ordinary differential equations using algebraic and computational techniques, but they may not be feasible for more complicated equations. On the other hand, using qualitative methods, we estimate the behavior of the solution over time based on the equation, without explicit solving it. One of the qualitative methods is drawing the direction field of slopes for the given equation, which is our main tool in qualitative analysis. Numerical methods involve discretization and approximation of solutions of ordinary differential equations using numerical algorithms. When selecting the appropriate method, it is important to be aware of the advantages and disadvantages of each method
Cantor set and applications
No full textOvaj diplomski rad bavi se raznim analitičkim konstrukcijama Cantorovog skupa. U prvom poglavlju upoznajemo se s Cantorovim skupom i konstruiramo ga na tzv. geometrijski način te objašnjavamo neka njegova svojstva (duljinu, kardinalitet i gustoću u skupu ). U drugom poglavlju bavimo se prvom konstrukcijom Cantorovog skupa preko tzv. verižnih razlomaka. Definiramo verižne razlomke, dokazujemo postojanje konačnog prikaza svakog racionalnog broja u obliku verižnog razlomka i postojanje razvoja u verižni razlomak za svaki realni broj, odnosno, dokazujemo konvergenciju beskonačnih verižnih razlomaka. Naposlijetku dokazujemo da je, za cijeli broj , , skup svih realnih brojeva , takvih da je i takvih da razvoj broja , u verižni razlomak ne sadrži parcijalni kvocijent manji od k, Cantorov skup. U trećem poglavlju konstruiramo Cantorov skup kao homeomorfnu sliku prstena cijelih -adskih brojeva , prost broj. Prvo definiramo -adske brojeve i cijele -adske brojeve. Definiramo -adsku normu i -adsku udaljenost na polju te dokazujemo da polje s -adskom normom nije potpuno. Upotpunjujemo polje poljem klasa ekvivalencije Cauchyjevih nizova, . Nakon toga, pokazujemo ekvivalenciju polja kao skupa klasa ekvivalencija nizova i beskonačnih -adskih ekspanzija koje konvergiraju u -adskoj normi na . Konačno, pokazujemo postojanje homeomorfizma izmedu prstena cijelih -adskih brojeva i Cantorovog skupa u naslijeđenoj euklidskoj topologiji.This thesis deals with various analytic constructions of the Cantor set. In the first chapter, we get acquainted with the Cantor set and construct it on the so-called geometric way and we explain some of its properties (length, cardinality and density in the set ). In the second chapter, we deal with the first construction of the Cantor set through the so-called continued fractions. We define continued fractions, we prove the existence of a finite representation of every rational number in the form of a continued fraction and the existence of a development into a continued fraction for every real number, that is, we prove the convergence of infinite continued fractions. Finally, we prove that, for an integer , , the set of all real numbers , such that and such that the development of the number into a continued fraction does not contain a partial quotient smaller than k, the Cantor set. In the third chapter, we construct the Cantor set as a homeomorphic image of the ring of -adic integers , a prime number. First we define -adic numbers and whole -adic numbers. We define the -adic norm and the -adic distance on the field and prove that the field with the -adic the norm is not complete. We complete the field with the field of equivalence classes of Cauchy sequences, . After that, we show the equivalence of the field as a set of equivalence classes of sequences and infinite -adic expansions that converge in the -adic norm on . Finally, we show the existence of a homeomorphism between the ring of -adic integers and the Cantor set in the inherited Euclidean topology
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