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Normal Forms, Holomorphic Linearization and Generic Bifurcations of Dynamic Equations on Discrete Time Scales

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Abstract

In this paper, we extend the classical theory of normal forms for continuous and difference dynamical systems to dynamic equations on discrete time scales. As consequences of the well known results from the theory of analytic differential equations, we obtain some versions of the Poincaré and Siegel theorems for dynamic equations on discrete time scales. Using these results and known results on the stability of dynamic equations on time scales, we obtain some stability results for the nonlinear dynamic equations. We also prove some results on generic properties of bifurcation curves and the saddle-node bifurcation for one-parameter families of dynamic equations on arbitrary time scales.

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Acknowledgements

The author would like to express their gratitude to the anonymous referee for their valuable comments which significantly improved the original manuscript. This work was supported by the Slovak Research and Development Agency under the Contract No. APVV-18-0308 and by the Slovak Grant Agency VEGA-SAV-MŠ No. 1/0358/20.

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  1. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius Unversity, Mlynská dolina, 842 48, Bratislava, Slovakia

    Milan Medveď

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  1. Milan Medveď
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Correspondence to Milan Medveď.

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Dedicated to the memory of professor Pavel Brunovský

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This work was supported by Grants VEGA 1/0358/20 and APVV-18-0308.

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Medveď, M. Normal Forms, Holomorphic Linearization and Generic Bifurcations of Dynamic Equations on Discrete Time Scales. J Dyn Diff Equat 36 (Suppl 1), 553–569 (2024). https://doi.org/10.1007/s10884-022-10177-8

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  • DOI: https://doi.org/10.1007/s10884-022-10177-8

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