Opuscula Math. 30, no. 1 (2010), 5-36
http://dx.doi.org/10.7494/OpMath.2010.30.1.5

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
• Revised: 2009-10-05.
• Accepted: 2009-10-14.