Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations |
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| Affiliation: | 1. School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, China;2. Key Laboratory of Micro-systems and Micro-structures Manufacturing, Ministry of Education, Harbin Institute of Technology, Harbin 150080, China;3. Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, Harbin 150080, China;4. School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China;1. FMI, Universität Stuttgart, Universitätsstr. 38, D-70569 Stuttgart, Germany;2. Institute for Software Engineering and Programming Languages, Universität zu Lübeck, Ratzeburger Allee 160, D-23562 Lübeck, Germany |
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| Abstract: | A new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations including ordinary and partial, autonomous and non-autonomous differential equations and differential equations with delay arguments is presented in this paper. Four corner-stones lie in the foundation of this theory: the Feigenbaum’s theory of period doubling bifurcations in one-dimensional mappings, the Sharkovskii’s theory of bifurcations of cycles of an arbitrary period up to the cycle of period three in one-dimensional mappings, the Magnitskii’s theory of rotor type singular points of two-dimensional non-autonomous systems of differential equations as a bridge between one-dimensional mappings and differential equations and the theory of homoclinic cascade of bifurcations of stable cycles in nonlinear differential equations. All propositions of the theory are strictly proved and illustrated by numerous analytical and computing examples. |
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