THE CONSTRUCTION OF SOLUTIONS OF PIECEWISE-LINEAR PHASE SYSTEMS
The creation of methods for the study of nonlinear phase systems has a long history, since the 60s of the last century (V.I. Tikhonov, V. Lindsay, M.V. Kapranov, B.I. Shakhtarin, etc.). By now, rigorous and approximate analysis methods of such systems have been developed. However, most methods are l...
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Moscow State Technical University of Civil Aviation
2019-02-01
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| Series: | Научный вестник МГТУ ГА |
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| Online Access: | https://avia.mstuca.ru/jour/article/view/1450 |
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| author | A. F. Gribov B. I. Shakhtarin |
| author_facet | A. F. Gribov B. I. Shakhtarin |
| author_sort | A. F. Gribov |
| collection | DOAJ |
| description | The creation of methods for the study of nonlinear phase systems has a long history, since the 60s of the last century (V.I. Tikhonov, V. Lindsay, M.V. Kapranov, B.I. Shakhtarin, etc.). By now, rigorous and approximate analysis methods of such systems have been developed. However, most methods are limited to the analysis of low order systems. Only in recent years attempts have been made to create methods, which allow to carry out the analysis of high-order phase systems. The material of this article deals with these methods. The article considers the construction of solutions of phase systems on the example of phase-locked frequency of arbitrary dimension with piecewise linear approximation of the nonlinear function. This approximation allowed to use an explicit form of solutions in the linearity and to obtain analytical conditions for the existence of various types of system behavior. The analytical conditions for the existence of solutions leading to the emergence of complex limit sets of the trajectories of phase systems and their bifurcations are obtained. These are homoclinic trajectories in the case of the saddle-focus equilibrium state, which play a decisive role in the occurrence of chaos. It is also shown that it is possible to obtain analytical conditions for the bifurcation of the birth and the existence of multi-pass rotational cycles in a piecewise linear phase system, on the basis of which a criterion for the transition to chaos through bifurcations cascade of doubling the stable cycle period can be obtained; in accordance with the Sharkovsky theorem it ends with the bifurcation of the cycle birth of the period three and the occurrence of developed chaos. It should be noted that the research methods of piecewise linear systems described in the paper were applied by the authors not only to phase systems, but, for example, to the Chua system, allowing various chaotic behavior. |
| format | Article |
| id | doaj-art-c84c6d4af09249b5a3a6c4aa3d30ec2a |
| institution | Matheson Library |
| issn | 2079-0619 2542-0119 |
| language | Russian |
| publishDate | 2019-02-01 |
| publisher | Moscow State Technical University of Civil Aviation |
| record_format | Article |
| series | Научный вестник МГТУ ГА |
| spelling | doaj-art-c84c6d4af09249b5a3a6c4aa3d30ec2a2025-08-04T10:35:16ZrusMoscow State Technical University of Civil AviationНаучный вестник МГТУ ГА2079-06192542-01192019-02-0122110612310.26467/2079-0619-2019-22-1-106-1231286THE CONSTRUCTION OF SOLUTIONS OF PIECEWISE-LINEAR PHASE SYSTEMSA. F. Gribov0B. I. Shakhtarin1Bauman Moscow State Technical University (BMSTU)Bauman Moscow State Technical University (BMSTU)The creation of methods for the study of nonlinear phase systems has a long history, since the 60s of the last century (V.I. Tikhonov, V. Lindsay, M.V. Kapranov, B.I. Shakhtarin, etc.). By now, rigorous and approximate analysis methods of such systems have been developed. However, most methods are limited to the analysis of low order systems. Only in recent years attempts have been made to create methods, which allow to carry out the analysis of high-order phase systems. The material of this article deals with these methods. The article considers the construction of solutions of phase systems on the example of phase-locked frequency of arbitrary dimension with piecewise linear approximation of the nonlinear function. This approximation allowed to use an explicit form of solutions in the linearity and to obtain analytical conditions for the existence of various types of system behavior. The analytical conditions for the existence of solutions leading to the emergence of complex limit sets of the trajectories of phase systems and their bifurcations are obtained. These are homoclinic trajectories in the case of the saddle-focus equilibrium state, which play a decisive role in the occurrence of chaos. It is also shown that it is possible to obtain analytical conditions for the bifurcation of the birth and the existence of multi-pass rotational cycles in a piecewise linear phase system, on the basis of which a criterion for the transition to chaos through bifurcations cascade of doubling the stable cycle period can be obtained; in accordance with the Sharkovsky theorem it ends with the bifurcation of the cycle birth of the period three and the occurrence of developed chaos. It should be noted that the research methods of piecewise linear systems described in the paper were applied by the authors not only to phase systems, but, for example, to the Chua system, allowing various chaotic behavior.https://avia.mstuca.ru/jour/article/view/1450piecewise-linear phase systemhomoclinic trajectoryrotational cycleschaosbifurcations |
| spellingShingle | A. F. Gribov B. I. Shakhtarin THE CONSTRUCTION OF SOLUTIONS OF PIECEWISE-LINEAR PHASE SYSTEMS Научный вестник МГТУ ГА piecewise-linear phase system homoclinic trajectory rotational cycles chaos bifurcations |
| title | THE CONSTRUCTION OF SOLUTIONS OF PIECEWISE-LINEAR PHASE SYSTEMS |
| title_full | THE CONSTRUCTION OF SOLUTIONS OF PIECEWISE-LINEAR PHASE SYSTEMS |
| title_fullStr | THE CONSTRUCTION OF SOLUTIONS OF PIECEWISE-LINEAR PHASE SYSTEMS |
| title_full_unstemmed | THE CONSTRUCTION OF SOLUTIONS OF PIECEWISE-LINEAR PHASE SYSTEMS |
| title_short | THE CONSTRUCTION OF SOLUTIONS OF PIECEWISE-LINEAR PHASE SYSTEMS |
| title_sort | construction of solutions of piecewise linear phase systems |
| topic | piecewise-linear phase system homoclinic trajectory rotational cycles chaos bifurcations |
| url | https://avia.mstuca.ru/jour/article/view/1450 |
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