6533b7d9fe1ef96bd126cc5b

RESEARCH PRODUCT

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

Vesna ŽUpanovićPavao MardešićMaja Resman

subject

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 scale

description

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 of fixed points and the dependence on epsilon of the length of epsilon-neighborhoods of orbits in non-differentiable cases. Applications include in particular Poincare maps near homoclinic loops and hyperbolic 2-cycles, and Abelian integrals. This is a new approach to estimate the cyclicity, by computing the length of the epsilon-neighborhood of one orbit of the Poincare map (for example numerically), and by comparing it to the appropriate scale.

yearjournalcountryeditionlanguage
2012-10-01Journal of Differential Equations
https://doi.org/10.1016/j.jde.2012.06.020