New Elements of Analysis of a Degenerate Chenciner Bifurcation
A new transformation of parameters for generic discrete-time dynamical systems with two independent parameters is defined, for when the degeneracy occurs. Here the classical transformation of parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-01-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/1/77 |
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| author | Sorin Lugojan Loredana Ciurdariu Eugenia Grecu |
| author_facet | Sorin Lugojan Loredana Ciurdariu Eugenia Grecu |
| author_sort | Sorin Lugojan |
| collection | DOAJ |
| description | A new transformation of parameters for generic discrete-time dynamical systems with two independent parameters is defined, for when the degeneracy occurs. Here the classical transformation of parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi>α</mi><mn>1</mn></msub><mo>,</mo><msub><mi>α</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>→</mo><mrow><mo>(</mo><msub><mi>β</mi><mn>1</mn></msub><mo>,</mo><msub><mi>β</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is not longer regular at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula>; therefore, implicit function theorem (IFT) cannot be applied around the origin, and a new transformation is necessary. The approach in this article to a case of Chenciner bifurcation is theoretical, but it can provide an answer for a number of applications of dynamical systems. We studied the bifurcation scenario and found out that, by this transformation, four different bifurcation diagrams are obtained, and the non-degenerate Chenciner bifurcation can be described by two bifurcation diagrams. |
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| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T00:26:24Z |
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| series | Symmetry |
| spelling | doaj.art-23b1fe2b4a7246b6a17858e7831750fc2023-11-23T15:33:03ZengMDPI AGSymmetry2073-89942022-01-011417710.3390/sym14010077New Elements of Analysis of a Degenerate Chenciner BifurcationSorin Lugojan0Loredana Ciurdariu1Eugenia Grecu2Department of Mathematics, Politehnica University of Timisoara, 300006 Timisoara, RomaniaDepartment of Mathematics, Politehnica University of Timisoara, 300006 Timisoara, RomaniaDepartment of Management, Politehnica University of Timisoara, 300006 Timisoara, RomaniaA new transformation of parameters for generic discrete-time dynamical systems with two independent parameters is defined, for when the degeneracy occurs. Here the classical transformation of parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi>α</mi><mn>1</mn></msub><mo>,</mo><msub><mi>α</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>→</mo><mrow><mo>(</mo><msub><mi>β</mi><mn>1</mn></msub><mo>,</mo><msub><mi>β</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is not longer regular at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula>; therefore, implicit function theorem (IFT) cannot be applied around the origin, and a new transformation is necessary. The approach in this article to a case of Chenciner bifurcation is theoretical, but it can provide an answer for a number of applications of dynamical systems. We studied the bifurcation scenario and found out that, by this transformation, four different bifurcation diagrams are obtained, and the non-degenerate Chenciner bifurcation can be described by two bifurcation diagrams.https://www.mdpi.com/2073-8994/14/1/77bifurcationchencinerdegeneracydiscrete dynamical systems |
| spellingShingle | Sorin Lugojan Loredana Ciurdariu Eugenia Grecu New Elements of Analysis of a Degenerate Chenciner Bifurcation Symmetry bifurcation chenciner degeneracy discrete dynamical systems |
| title | New Elements of Analysis of a Degenerate Chenciner Bifurcation |
| title_full | New Elements of Analysis of a Degenerate Chenciner Bifurcation |
| title_fullStr | New Elements of Analysis of a Degenerate Chenciner Bifurcation |
| title_full_unstemmed | New Elements of Analysis of a Degenerate Chenciner Bifurcation |
| title_short | New Elements of Analysis of a Degenerate Chenciner Bifurcation |
| title_sort | new elements of analysis of a degenerate chenciner bifurcation |
| topic | bifurcation chenciner degeneracy discrete dynamical systems |
| url | https://www.mdpi.com/2073-8994/14/1/77 |
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