0000000000102133

AUTHOR

Maja Resman

showing 5 related works from this author

Multiplicity of fixed points and growth of ε-neighborhoods of orbits

2012

We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on epsilon of the length of epsilon-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered before (Elezovic, Zubrinic, Zupanovic) in the differentiable case, and related to the box dimension of the orbit. Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity o…

Critical Minkowski orderDynamical Systems (math.DS)Fixed pointsymbols.namesakeMinkowski spaceFOS: MathematicsCyclicityDifferentiable functionHomoclinic orbitlimit cycles; multiplicity; cyclicity; Chebyshev scale; Critical Minkowski order; box dimension; homoclinic loopMathematics - Dynamical SystemsAbelian groupPoincaré mapMathematicsBox dimensionApplied MathematicsMathematical analysisMultiplicity (mathematics)Limit cyclesMultiplicityPoincaré conjecturesymbols37G15 34C05 28A75 34C10Homoclinic loopAnalysisChebyshev scaleJournal of Differential Equations
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The Fatou coordinate for parabolic Dulac germs

2017

We study the class of parabolic Dulac germs of hyperbolic polycycles. For such germs we give a constructive proof of the existence of a unique Fatou coordinate, admitting an asymptotic expansion in the power-iterated log scale.

Pure mathematicsMonomialClass (set theory)Mathematics::Dynamical SystemsConstructive proofLogarithmTransseries[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]orbitsDulac germAsymptotic expansionDynamical Systems (math.DS)01 natural sciencesMSC: 37C05 34C07 30B10 30B12 39A06 34E05 37C10 37C1537C05 34C07 30B10 30B12 39A06 34E05 37C10 37C15Mathematics::Algebraic GeometryFOS: Mathematics0101 mathematicsMathematics - Dynamical SystemsMathematicsDulac germ ; Fatou coordinate ; Embedding in a flow ; Asymptotic expansion ; TransseriesdiffeomorphismsMathematics::Complex VariablesApplied Mathematics010102 general mathematicsFatou coordinate010101 applied mathematicsclassificationnormal formsepsilon-neighborhoodsEmbedding in a flowAsymptotic expansionAnalysis
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Linearization of complex hyperbolic Dulac germs

2021

We prove that a hyperbolic Dulac germ with complex coefficients in its expansion is linearizable on a standard quadratic domain and that the linearizing coordinate is again a complex Dulac germ. The proof uses results about normal forms of hyperbolic transseries from another work of the authors.

Pure mathematicsMathematics::Dynamical SystemsMathematics::Complex VariablesApplied Mathematics010102 general mathematicsMathematics::Classical Analysis and ODEsDynamical Systems (math.DS)01 natural sciencesDomain (mathematical analysis)Dulac germs and series ; Hyperbolic fixed point ; Linearization ; Koenigs' sequenceQuadratic equationLinearization0103 physical sciencesFOS: MathematicsGerm010307 mathematical physics0101 mathematicsMathematics - Dynamical SystemsAnalysisMathematics
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Normal forms and embeddings for power-log transseries

2016

First return maps in the neighborhood of hyperbolic polycycles have their asymptotic expansion as Dulac series, which are series with power-logarithm monomials. We extend the class of Dulac series to an algebra of power-logarithm transseries. Inside this new algebra, we provide formal normal forms of power-log transseries and a formal embedding theorem. The questions of classifications and of embeddings of germs into flows of vector fields are common problems in dynamical systems. Aside from that, our motivation for this work comes from fractal analysis of orbits of first return maps around hyperbolic polycycles. This is a joint work with Pavao Mardešić, Jean-Philippe Rolin and Vesna Župano…

Mathematics::Dynamical Systems[ MATH.MATH-CA ] Mathematics [math]/Classical Analysis and ODEs [math.CA]TransseriesGeneral Mathematics[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS][ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS]MSC: 34C20 37C10 39B12 46A19 28A75 58K50 26A12[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA]Normal forms01 natural sciencesIteration theory ; Dulac map ; normal forms ; embedding in a flow ; transseries.0101 mathematicsAlgebra over a fieldMathematicsSeries (mathematics)Dulac mapIteration theoryformal normal forms parabolic transseriesMathematics::History and Overview010102 general mathematicsPower (physics)010101 applied mathematicsAlgebraEmbeddingEmbedding in a flowIteration theoryAdvances in Mathematics
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Fixed points of diffeomorphisms, singularities of vector fields and epsilon-neighborhoods of their orbits, the thesis

2013

The thesis deals with recognizing diffeomorphisms from fractal properties of discrete orbits, generated by iterations of such diffeomorphisms. The notion of fractal properties of a set refers to the box dimension, the Minkowski content and their appropriate generalizations, or, in wider sense, to the epsilon-neighborhoods of sets, for small, positive values of parameter epsilon. In the first part of the thesis, we consider the relation between the multiplicity of the fixed point of a real-line diffeomorphism, and the asymptotic behavior of the length of the epsilon-neighborhoods of its orbits. We establish the bijective correspondence. At the fixed point, the diffeomorphisms may be differen…

FOS: MathematicsDynamical Systems (math.DS)Mathematics - Dynamical Systems
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