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## MATHEMATICAL METHODS FOR ENGINEERS AND SCIENTISTS

by: Mrs. Dedric Little

18

0

1

# MATHEMATICAL METHODS FOR ENGINEERS AND SCIENTISTS MTH 481

Marketplace > Oregon State University > Mathematics (M) > MTH 481 > MATHEMATICAL METHODS FOR ENGINEERS AND SCIENTISTS
Mrs. Dedric Little
OSU
GPA 3.79

Staff

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COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
1
WORDS
KARMA
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## Popular in Mathematics (M)

This 1 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 481 at Oregon State University taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/224454/mth-481-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15
MTH 481581 SUPPLEMENTARY NOTE 1 Consider the plane autonomous system a d d7 9I7y where f and g are twice continuously differentiable Let 3 3 I y 7 I7y 75I7y 7 3 3 7 55 y 55 y be the Jacobi matrixi Suppose 10 yo is an isolated critical point of the system that is fzo yo 0 910 yo 0 Suppose in addition this critical point is not degenerate that is det Jzoyo 0 In this case we may write the solutions of our system in the form zt 10 u1 t W yo u2t where d u 7 7 7 ul EiAuRu A7Jzoyo ui 12 l Here Ru consists of the remainder terms from the Taylor expansions of f and g For small we have lAul 2 clul and S clulQi Hence we expect that we can neglect Thus we are led to the linean39zatz39on of our system at 1034 dill Aui The following table gives the relation between the critical point of the linear system at the origin and the critical point of the original nonlinear system at 1030 same unstable Jacobi matrix not unstable proper proper Jacobi matrix unstable nonzero real unstable pure unstable If there is a neighborhood U of 10 yo and a function h continuously diffrentiable in U such that ah ah i i 0 39 U f 81 l 9 8y In and 8h E then zoy0 is a center or a saddle point for the nonlinear systemi The function h is called a rst integral Physically it corresponds to a conserved quantity 4 8h 67 f 0 in U except possibly at zoyo y Bent Petersen File ref 481u93n1tex July 28 1993 SUMMER

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