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Interpretation:
Sketch the arrangement of the given cycles
Concept Introduction:
Index of a closed curve C is an integer that measures the windings of the
To find the index of the curve, draw a close contour C on the phase portrait. Then on each point “x” on the contour C, vector field
The index of closed curve C with respect to the vector field is given as
Here
Properties of Indices
If contour C is continuously deformed into C’ without passing through a fixed point, then
If contour C doesn’t enclose any fixed points then
If the directions of all arrows in the vector field are reversed by changing
If the closed curved C is a trajectory for the system, then
Theorem: Any closed orbit in the phase plane mustenclose fixed points whose indices sum is
Theorem: If a close curve C surrounds n isolated fixed point
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Chapter 6 Solutions
EBK NONLINEAR DYNAMICS AND CHAOS WITH S
- Show that the three points (x1,y1)(x2,y2) and (x3,y3) in the a plane are collinear if and only if the matrix [x1y11x2y21x3y31] has rank less than 3.arrow_forwardFind the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forwardConsider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the closest to u that is, show that d(u,projvu) is a minimum.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
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