| Title:
|
Bifurcations of invariant measures in discrete-time parameter dependent cocycles (English) |
| Author:
|
Maltseva, Anastasia |
| Author:
|
Reitmann, Volker |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
140 |
| Issue:
|
2 |
| Year:
|
2015 |
| Pages:
|
205-213 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
discrete-time parameter-dependent cocycles |
| Keyword:
|
Hausdorff dimension estimate |
| Keyword:
|
invariant measure |
| MSC:
|
35B15 |
| MSC:
|
35B32 |
| MSC:
|
35B41 |
| MSC:
|
35K20 |
| MSC:
|
37H05 |
| idZBL:
|
Zbl 06486934 |
| idMR:
|
MR3368494 |
| DOI:
|
10.21136/MB.2015.144326 |
| . |
| Date available:
|
2015-06-30T12:19:30Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144326 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
[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. 10.1111/j.1749-6632.1987.tb48731.x |
| Reference:
|
[7] Kloeden, P. E., Schmalfu{ß}, B.: Nonautonomous systems, cocycle attractors and variable time-step discretization.Numer. Algorithms 14 (1997), 141-152. Zbl 0886.65077, MR 1456499, 10.1023/A:1019156812251 |
| Reference:
|
[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. Zbl 1317.39008, MR 3372683, 10.1134/S0012266114130035 |
| Reference:
|
[9] Reitmann, V.: Dynamical Systems, Attractors and Estimates of Their Dimension.Saint Petersburg State University Press Saint Petersburg (2013), Russian. |
| Reference:
|
[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. Zbl 1303.37009, MR 2918529, 10.3103/S1063454111040091 |
| Reference:
|
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| . |
