An Algorithm for Higher Order Hopf Normal Forms
Normal form theory is important for studying the qualitative behavior of nonlinear oscillators. In some cases, higher order normal forms are required to understand the dynamic behavior near an equilibrium or a periodic orbit. However, the computation of high-order normal forms is usually quite compl...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
1995-01-01
|
| Series: | Shock and Vibration |
| Online Access: | http://dx.doi.org/10.3233/SAV-1995-2405 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832567861811347456 |
|---|---|
| author | A.Y.T. Leung T. Ge |
| author_facet | A.Y.T. Leung T. Ge |
| author_sort | A.Y.T. Leung |
| collection | DOAJ |
| description | Normal form theory is important for studying the qualitative behavior of nonlinear oscillators. In some cases, higher order normal forms are required to understand the dynamic behavior near an equilibrium or a periodic orbit. However, the computation of high-order normal forms is usually quite complicated. This article provides an explicit formula for the normalization of nonlinear differential equations. The higher order normal form is given explicitly. Illustrative examples include a cubic system, a quadratic system and a Duffing–Van der Pol system. We use exact arithmetic and find that the undamped Duffing equation can be represented by an exact polynomial differential amplitude equation in a finite number of terms. |
| format | Article |
| id | doaj-art-52438593089945349ca5190f9a680f39 |
| institution | Kabale University |
| issn | 1070-9622 1875-9203 |
| language | English |
| publishDate | 1995-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Shock and Vibration |
| spelling | doaj-art-52438593089945349ca5190f9a680f392025-02-03T01:00:22ZengWileyShock and Vibration1070-96221875-92031995-01-012430731910.3233/SAV-1995-2405An Algorithm for Higher Order Hopf Normal FormsA.Y.T. Leung0T. Ge1Department of Civil and Structural Engineering, University of Hong Kong, Hong KongDepartment of Civil and Structural Engineering, University of Hong Kong, Hong KongNormal form theory is important for studying the qualitative behavior of nonlinear oscillators. In some cases, higher order normal forms are required to understand the dynamic behavior near an equilibrium or a periodic orbit. However, the computation of high-order normal forms is usually quite complicated. This article provides an explicit formula for the normalization of nonlinear differential equations. The higher order normal form is given explicitly. Illustrative examples include a cubic system, a quadratic system and a Duffing–Van der Pol system. We use exact arithmetic and find that the undamped Duffing equation can be represented by an exact polynomial differential amplitude equation in a finite number of terms.http://dx.doi.org/10.3233/SAV-1995-2405 |
| spellingShingle | A.Y.T. Leung T. Ge An Algorithm for Higher Order Hopf Normal Forms Shock and Vibration |
| title | An Algorithm for Higher Order Hopf Normal Forms |
| title_full | An Algorithm for Higher Order Hopf Normal Forms |
| title_fullStr | An Algorithm for Higher Order Hopf Normal Forms |
| title_full_unstemmed | An Algorithm for Higher Order Hopf Normal Forms |
| title_short | An Algorithm for Higher Order Hopf Normal Forms |
| title_sort | algorithm for higher order hopf normal forms |
| url | http://dx.doi.org/10.3233/SAV-1995-2405 |
| work_keys_str_mv | AT aytleung analgorithmforhigherorderhopfnormalforms AT tge analgorithmforhigherorderhopfnormalforms AT aytleung algorithmforhigherorderhopfnormalforms AT tge algorithmforhigherorderhopfnormalforms |