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

• Autorzy: Oprocha P. , Wilczyński P.
• Języki publikacji: EN

Abstrakty

EN

In the present paper we prove distributional chaos for the Poincaré map in the perturbed equation [formula]. Heteroclinic and homoclinic connections between two periodic solutions bifurcating from the stationary solution 0 present in the system when N = 0 are also discussed.

Słowa kluczowe

EN

distributional chaos; isolating segments; fixed point index; bifurcation

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2010
• Tom: Vol. 30, no. 1
• Strony: 5--36
• Opis fizyczny: Bibliogr. 17 poz., rys., wykr.

Twórcy

autor

Oprocha P.

autor

Wilczyński P.

• Universidad de Murcia Departamento de Matematicas Campus de Espinardo, 30100 Murcia, Spain AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30, 30-059 Krakow, Poland,

Bibliografia

• [1] F. Balibrea, J. Smítal, M. Štefánková, The three versions of distributional chaos, Chaos Solitons Fractals 23 (2005) 5, 1581–1583.
• [2] W. Bauer, K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math. 79 (1975), 81–92.
• [3] R. Bowen, Topological entropy and axiom A, [in:] Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), pages 23–41. Amer. Math. Soc., Providence, R.I., 1970.
• [4] M. Denker, C. Grillenberger, K. Sigmund, Ergodic theory on compact spaces, Springer-Verlag, Berlin, 1976, Lecture Notes in Mathematics, vol. 527.
• [5] A. Dold, Lectures on algebraic topology, Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition.
• [6] N. Jonoska, B. Marcus, Minimal presentations for irreducible sofic shifts, IEEE Trans. Inform. Theory 40 (1994) 6, 1818–1825.
• [7] P. Ku rka, Topological and symbolic dynamics, Cours Spécialisés [Specialized Courses], vol. 11, Société Mathématique de France, Paris, 2003.
• [8] T.Y. Li, J.A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975) 10, 985–992.
• [9] D. Lind, B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995.
• [10] P. Oprocha, P. Wilczyński, A study of chaos for processes under small perturbations, Publ. Math. Debrecen 76 (2010) 1–2, 101–116.
• [11] B. Schweizer, J. Smital, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994) 2, 737–754.
• [12] K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc. 190 (1974), 285–299.
• [13] R. Srzednicki, A geometric method for the periodic problem in ordinary differential equations, Séminaire d’Analyse Moderne [Seminar on Modern Analysis], vol. 22, Uni-versité de Sherbrooke Département de Mathématiques et d’Informatique, Sherbrooke,QC, 1992.
• [14] R. Srzednicki, Ważewski method and Conley index, [in:] Handbook of differential equations, Elsevier/North-Holland, Amsterdam, 2004, 591–684.
• [15] R. Srzednicki, K. Wójcik, P. Zgliczyński, Fixed point results based on the Ważewski method, [in:] Handbook of topological fixed point theory, Springer, Dordrecht, 2005, 905–943.
• [16] K.Wójcik, P. Zgliczyński, Isolating segments, fixed point index, and symbolic dynamics,J. Differential Equations 161 (2000) 2, 245–288.
• [17] K.Wójcik, P. Zgliczyński, Isolating segments, fixed point index, and symbolic dynamics. III. Applications, J. Differential Equations 183 (2002) 1, 262–278.