Search results for "37G15"

showing 3 items of 3 documents

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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Planar maps whose second iterate has a unique fixed point

2007

Let a>0, F: R^2 -> R^2 be a differentiable (not necessarily C^1) map and Spec(F) be the set of (complex) eigenvalues of the derivative F'(p) when p varies in R^2. (a) If Spec(F) is disjoint of the interval [1,1+a[, then Fix(F) has at most one element, where Fix(F) denotes the set of fixed points of F. (b) If Spec(F) is disjoint of the real line R, then Fix(F^2) has at most one element. (c) If F is a C^1 map and, for all p belonging to R^2, the derivative F'(p) is neither a homothety nor has simple real eigenvalues, then Fix(F^2) has at most one element, provided that Spec(F) is disjoint of either (c1) the union of the number 0 with the intervals ]-\infty, -1] and [1,\infty[, or (c2) t…

Discrete mathematics37G10; 37G15; 34K18Algebra and Number TheoryApplied Mathematics37G15Dynamical Systems (math.DS)Fixed point37G10Homothetic transformationPlanar graphSet (abstract data type)symbols.namesakeMathematics - Classical Analysis and ODEsSimple (abstract algebra)Classical Analysis and ODEs (math.CA)FOS: MathematicssymbolsEmbeddingDifferentiable functionMathematics - Dynamical Systems34K18AnalysisEigenvalues and eigenvectorsMathematicsJournal of Difference Equations and Applications
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Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems

2015

In this paper we introduce new methods to prove the finite cyclicity of some graphics through a triple nilpotent point of saddle or elliptic type surrounding a center. After applying a blow-up of the family, yielding a singular 3-dimensional foliation, this amounts to proving the finite cyclicity of a family of limit periodic sets of the foliation. The boundary limit periodic sets of these families were the most challenging, but the new methods are quite general for treating such graphics. We apply these techniques to prove the finite cyclicity of the graphic $(I_{14}^1)$, which is part of the program started in 1994 by Dumortier, Roussarie and Rousseau (and called DRR program) to show that…

Pure mathematicsCenter (category theory)Boundary (topology)Dynamical Systems (math.DS)Type (model theory)FoliationNilpotentMathematics (miscellaneous)FOS: MathematicsLimit (mathematics)Point at infinityMathematics - Dynamical Systems34C07 37G15SaddleMathematicsTransactions of the Moscow Mathematical Society
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