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

Autori
Mardešić, Pavao ; Resman, Maja ; Županović, Vesna

Izvornik
Journal of differential equations (0022-0396) 253 (2012), 8; 2493-2514

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
limit cycles; multiplicity; cyclicity; Chebyshev scale; Critical Minkowski order; box dimension; homoclinic loop

Sažetak
We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on ε of the length of ε-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered in Elezović, Žubrinić and Županović (2007) [5] 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 ε of the length of ε-neighborhoods of orbits in non-differentiable cases. Applications include in particular Poincaré 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 ε-neighborhood of one orbit of the Poincaré map (for example numerically), and by comparing it to the appropriate scale.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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