
Concept explainers
Interpretation:
To show that each of the following fixed points has an index equal to +1.
a) stable spiral b) unstable spiral c) center d) star e) degenerate node
Concept Introduction:
Index theory provides the global information about the phase portrait.
The index of the closed curve can be defined as the net number of counter-clockwise revolutions made by
Where,

Answer to Problem 1E
Solution:
For each of the fixed points, the index is equal to +1 is proved.
Explanation of Solution
Following fixed points have an index equal to
Sketch for the stable spiral,
The above sketch shows that every vector is always pointing inwards. It is seen from the direction of arrow that the vector field is rotating in counter-clockwise direction. Hence, the index for the stable spiral is +1.
Sketch for an unstable spiral
The above sketch shows that every vector is always pointing outwards. It is seen from the direction of arrow that the vector field is rotating in counter-clockwise direction. Hence, the index for the unstable spiral is +1.
Sketch for a Centre
The above sketch shows that every vector is making tangent to the closed orbit. It is seen from the direction of arrow that the vector field is rotating in counter-clockwise direction by
Sketch for a star
The above sketch shows that every vector is always pointing outwards. It is seen from the direction of arrow that the vector field is rotating in counter-clockwise direction by
Sketch for a degenerate node:
The vector field at the point is pretty fastidious where the trajectory turns around and starts moving in opposite direction. Due to this, the graph of the only vector head is plotted instead of the individual vectors. These vectors bunch around the angle
Above sketches showthat each of the fixed point have an index equal to +1.
Want to see more full solutions like this?
Chapter 6 Solutions
Nonlinear Dynamics and Chaos
- Explore this statement by constructing at least three examples, one of which must be a negative integer. Indicate if the statement is true or false for each example.arrow_forward2. Consider the following statement: For each natural number n, (3.2n+2.3n+1) is a prime number. (a) Explore this statement by completing the table below for n = 2,3 and two additional values of n of your choosing (notice n = 1 has been completed for you). One of your rows should contain a counterexample. n 1 3.2 2.3 +1 3.212.31 + 1 = 13 prime or composite? prime 2 3 (b) Write a formal counterexample argument for the statement using the template fromarrow_forwardPlease ensure that all parts of the question are answered thoroughly and clearly. Include a diagram to help explain answers. Make sure the explanation is easy to follow. Would appreciate work done written on paper. Thank you.arrow_forward
- Q4 4 Points 3 Let A = 5 -1 Let S : R³ → R² be the linear transformation whose standard matrix is A. Let U : R² → R³ be the linear transformation whose standard matrix is AT (the transpose of A). Let P: R³ → R³ be the linear transformation which first applies S and then applies U. Let Q: R² → R² be the linear transformation which first applies U and then applies S. Find the standard matrix of P and the standard matrix of Q. Clearly indicate which is which in your work. Please select file(s) Select file(s) Save Answerarrow_forwardQ3 4 Points Let T: R4 → R³ be the linear transformation defined by the formula 11 x1+x3+2x4 T x2 + 3 + 24 Is −1 +222 +23 I i. (2 points) Find the standard matrix of T. ii (2 points) Determine if I is one-to-one and determine if I' is onto. Please select file(s) Select file(s)arrow_forwardx 1.1 1.2 1.3 f 3.1 3.9 य find numerical f'(1) by using approximation.arrow_forward
- Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
- Elementary Geometry For College Students, 7eGeometryISBN:9781337614085Author:Alexander, Daniel C.; Koeberlein, Geralyn M.Publisher:Cengage,Algebra for College StudentsAlgebraISBN:9781285195780Author:Jerome E. Kaufmann, Karen L. SchwittersPublisher:Cengage LearningElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning




