
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

Want to see the full answer?
Check out a sample textbook solution
Chapter 6 Solutions
Nonlinear Dynamics and Chaos
- Q/ show that the system: x = Y + x(x² + y²) y° = =x+y (x² + y²) 9 X=-x(x²+ y²) 9 X Y° = x - y (x² + y²) have the same lin car part at (0,0) but they are topologically different. Give the reason.arrow_forwardQ/ Find the region where ODES has no limit cycle: -X = X + X3 y=x+y+y'arrow_forwardB:Show that the function 4H(x,y)= (x² + y2)2-2((x² + y²) is a first integral of ODES: x=y + y(x² + y²) y=x+x (x² + y²) and sketch the stability of critical points and draw the phase portrait of system.arrow_forward
- A: Show that the ODES has no limit cycle in a region D and find this region: x=y-2x³ y=x+y-2y3 Carrow_forwardoptımızatıon theoryarrow_forwardQ3)A: Given H(x,y)= x²-x4 + y² as a first integral of an ODEs, find this ODES corresponding to H(x,y) and show the phase portrait by using Hartman theorem and by drawing graph of H(x,y)=c. Discuss the stability of critical points of the corresponding ODEs.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage


