Normal form for the Poincar\303\251-Andronov-Hopf Bifurcation and the Neimark-Sacker Torus Bifurcation Veronika Hajnov\303\241 and Lenka P\305\231ibylov\303\241, Masaryk University This sheet implements a method to find a normal form of a dynamical system. The essential background can be found in chapter 19 in [1]. Sources: [1] Stephen Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, ISBN 0-387-00177-8 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
<Text-field style="Heading 1" layout="Heading 1">Tranformations that enables to eliminate k-order therms</Text-field>
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">the Poincar\303\251-Andronov-Hopf Bifurcation</Font></Text-field> 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
<Text-field style="Heading 1" layout="Heading 1">the Neimark-Sacker Torus Bifurcation</Text-field> 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
<Text-field style="Heading 1" layout="Heading 1">Matrix representation of tranformation map in given basis</Text-field> 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
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">the Poincar\303\251-Andronov-Hopf Bifurcation</Font></Text-field> 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEwb3BlcmF0b3JfbWF0cml4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRjY2Ji1GIzYuLUklbXN1cEdGJDYlLUYsNiVRInpGJ0YvRjItRiM2JC1JI21uR0YkNiRRIjNGJy9GM1Enbm9ybWFsRidGSi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGSi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZVLyUqc3ltbWV0cmljR0ZVLyUobGFyZ2VvcEdGVS8lLm1vdmFibGVsaW1pdHNHRlUvJSdhY2NlbnRHRlUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GPzYlRkEtRiM2JC1GRzYkUSIyRidGSkZKRkwtRlA2LVEnJnNkb3Q7RidGSkZTL0ZXRlVGWEZaRmZuRmhuRmpuRlxvL0Zgb0Zeby1GLDYlUSJ3RidGL0YyRk9GQUZpby1GPzYlRl5wRmRvRkxGTy1GPzYlRl5wRkRGTEZKRkovJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGTy1GLDYlUSp0cmFuc2Zvcm1GJ0YvRjIvJStleGVjdXRhYmxlR0ZVRkpGSkZecUZK 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
<Text-field style="Heading 2" layout="Heading 2">the Neimark-Sacker Torus Bifurcation</Text-field> 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