Opuscula Math. 30, no. 1 (2010), 5-36

Opuscula Mathematica

A study of chaos for processes under small perturbations II: rigorous proof of chaos

Piotr Oprocha
Paweł Wilczyński

Abstract. In the present paper we prove distributional chaos for the Poincaré map in the perturbed equation \[\dot{z}=\left(1 + e^{i\kappa t} |z|^2\right)\bar{z}^2 - N e^{-i\frac{\pi}{3}}.\] Heteroclinic and homoclinic connections between two periodic solutions bifurcating from the stationary solution \(0\) present in the system when \(N = 0\) are also discussed.

Keywords: distributional chaos, isolating segments, fixed point index, bifurcation.

Mathematics Subject Classification: 34C28, 37B30.

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  • Piotr Oprocha
  • Universidad de Murcia, Departamento de Matemáticas, Campus de Espinardo, 30100 Murcia, Spain
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
  • Paweł Wilczyński
  • Jagiellonian University, Institute of Mathematics, ul. Łojasiewicza 6, 30-348 Kraków, Poland
  • Received: 2009-04-11.
  • Revised: 2009-10-05.
  • Accepted: 2009-10-14.
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Cite this article as:
Piotr Oprocha, Paweł Wilczyński, A study of chaos for processes under small perturbations II: rigorous proof of chaos, Opuscula Math. 30, no. 1 (2010), 5-36, http://dx.doi.org/10.7494/OpMath.2010.30.1.5

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