Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
discrete-time parameter-dependent cocycles; Hausdorff dimension estimate; invariant measure
discrete-time parameter-dependent cocycles; Hausdorff dimension estimate; invariant measure
Summary:
We consider parameter-dependent cocycles generated by nonautonomous difference equations. One of them is a discrete-time cardiac conduction model. For this system with a control variable a cocycle formulation is presented. We state a theorem about upper Hausdorff dimension estimates for cocycle attractors which includes some regulating function. We also consider the existence of invariant measures for cocycle systems using some elements of Perron-Frobenius theory and discuss the bifurcation of parameter-dependent measures.
References:
[1] Arnold, L.: Random Dynamical Systems. Springer Monographs in Mathematics Springer, Berlin (1998). MR 1374107 | Zbl 0938.37031
[2] Baladi, V., Viana, M.: Strong stochastic stability and rate of mixing for unimodal maps. Ann. Sci. Éc. Norm. Supér. (4) 29 (1996), 483-517. DOI 10.24033/asens.1745 | MR 1386223 | Zbl 0868.58051
[3] Bandtlow, O. F., Antoniou, I., Suchanecki, Z.: Resonances of dynamical systems and Fredholm-Riesz operators on rigged Hilbert spaces, Computational Tools of Complex Systems I. Comput. Math. Appl. 34 (1997), 95-102. MR 1478754
[4] Boichenko, V. A., Leonov, G. A., Reitmann, V.: Dimension Theory for Ordinary Differential Equations. Teubner Texts in Mathematics 141 Teubner, Wiesbaden (2005). MR 2381409 | Zbl 1094.34002
[5] Crauel, H., Flandoli, F.: Hausdorff dimension of invariant sets for random dynamical systems. J. Dyn. Differ. Equations 10 (1998), 449-474. DOI 10.1023/A:1022605313961 | MR 1646622 | Zbl 0927.37031
[6] Glass, L., Guevera, M. R., Shrier, A.: Universal bifurcations and the classification of cardiac arrhythmias. Ann. N. Y. Acad. Sci. 504 (1987), 168-178. DOI 10.1111/j.1749-6632.1987.tb48731.x
[7] Kloeden, P. E., Schmalfu{ß}, B.: Nonautonomous systems, cocycle attractors and variable time-step discretization. Numer. Algorithms 14 (1997), 141-152. DOI 10.1023/A:1019156812251 | MR 1456499 | Zbl 0886.65077
[8] Maltseva, A., Reitmann, V.: Global stability and bifurcations of invariant measures for the discrete cocycles of the cardiac conduction system's equations. Differ. Equ. 50 (2014), 1718-1732. DOI 10.1134/S0012266114130035 | MR 3372683 | Zbl 1317.39008
[9] Reitmann, V.: Dynamical Systems, Attractors and Estimates of Their Dimension. Saint Petersburg State University Press Saint Petersburg (2013), Russian.
[10] Reitmann, V., Slepukhin, A. S.: On upper estimates for the Hausdorff dimension of negatively invariant sets of local cocycles. Vestn. St. Petersbg. Univ., Math. 44 (2011), 292-300 translation from Vestn. St.-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2011 (2011), 61-70. DOI 10.3103/S1063454111040091 | MR 2918529 | Zbl 1303.37009
[11] Sun, J., Amellal, F., Glass, L., Billete, J.: Alternans and period-doubling bifurcations in atrioventricular nodal conduction. J. Theor. Biol. 173 (1995), 79-91. DOI 10.1006/jtbi.1995.0045
Search
Partner of
