Author | Murdock, James. author |
---|---|

Title | Normal Forms and Unfoldings for Local Dynamical Systems [electronic resource] / by James Murdock |

Imprint | New York, NY : Springer New York : Imprint: Springer, 2003 |

Connect to | http://dx.doi.org/10.1007/b97515 |

Descript | XX, 500 p. online resource |

SUMMARY

The subject of local dynamical systems is concerned with the following two questions: 1. Given an nร{151}n matrix A, describe the behavior, in a neighborhood of the origin, of the solutions of all systems of di?erential equations having a rest point at the origin with linear part Ax, that is, all systems of the form x ? = Ax+ยทยทยท , n where x? R and the dots denote terms of quadratic and higher order. 2. Describethebehavior(neartheorigin)ofallsystemsclosetoasystem of the type just described. To answer these questions, the following steps are employed: 1. A normal form is obtained for the general system with linear part Ax. The normal form is intended to be the simplest form into which any system of the intended type can be transformed by changing the coordinates in a prescribed manner. 2. An unfolding of the normal form is obtained. This is intended to be the simplest form into which all systems close to the original s- tem can be transformed. It will contain parameters, called unfolding parameters, that are not present in the normal form found in step 1. vi Preface 3. The normal form, or its unfolding, is truncated at some degree k, and the behavior of the truncated system is studied

CONTENT

Preface -- 1. Two Examples -- 2. The splitting problem for linear operators -- 3. Linear Normal Forms -- 4. Nonlinear Normal Forms -- 5. Geometrical Structures in Normal Forms -- 6. Selected Topics in Local Bifurcation Theory -- Appendix A: Rings -- Appendix B: Modules -- Appendix C: Format 2b: Generated Recursive (Hori) -- Appendix D: Format 2c: Generated Recursive (Deprit) -- Appendix E: On Some Algorithms in Linear Algebra -- Bibliography -- Index

Mathematics
Differential equations
Applied mathematics
Engineering mathematics
Physics
Mathematics
Ordinary Differential Equations
Applications of Mathematics
Theoretical Mathematical and Computational Physics