Interpretation:
To calculate
Concept Introduction:
The linearized system for
The matrix
Answer to Problem 11E
Solution:
The system in Cartesian coordinate is
The linearized system at the origin is
Explanation of Solution
Using the first equation
Integrating both sides,
When
Solve the stability of the origin by solving
Thus the origin is a stable point.
Solve the nature of the origin by solving
Thus the origin is spiral.
The system in
Comparing the above polar coordinates with the given coordinates,
Multiply equation 1) by x and equation 2) by y and subtract,
Multiply equation 1) by y and equation 2) by x and add,
The Jacobian matrix at any fixed point
Substituting values of
It is proved and origin is a stable star for the linearized system.
Integrating given polar equations we can find
Want to see more full solutions like this?
Chapter 6 Solutions
Nonlinear Dynamics and Chaos
- Suppose parametric equations for the line segment between (7, 4) and (0, 6) have the form: Sx(t) = a + bt ly(t) = c+ dt If the parametric curve starts at (7, 4) when t O and ends at (0, 6) at t = 1, then find a, b, c, and d. a = b = с — d =arrow_forwardI I I I I I I I I I I I - Your answer is partially correct. Find parametric equations of the line through (-3,1 parallel to 4i-8j+ k. www ***** x(t) = 4t-3 y(t) = -8t+ z(t) = t + 6 k 18 X Xarrow_forward1.) Given the parametric equations: x=t²-1 and y=r+t, find the 1* and 2nd derivatives.arrow_forward
- 3. Find the parametric equation of the line passing through the point (-1,0,3) and in the direction N = (7,0, –2).arrow_forwardA stunt cyclist needs to make a calculation for an upcoming cycle jump. The cyclist is traveling 100 ft/sec toward an inclined ramp which ends 10 feet above a level landing zone. Assume the cyclist maintains a constant speed up the ramp and the ramp is inclined A° (degrees) above horizontal. With the pictured imposed coordina system, the parametric equations of the cyclist will be: x(t) = 100t cos(A) y(t)=-16t² + 100t sin(A) + 10. (These are the parametric equations for the motion of the stunt cyclist.) y-axis 10 feet (a) Calculate the horizontal velocity of the cyclist at time t; this is the function x'(t)= (b) What is the horizontal velocity if A=20 degrees? (c) What is the horizontal velocity if A=45 degrees? (Four decimal places.) (d) Calculate the vertical velocity of the cyclist at time t; this is the function y'(t)= (e) What is the vertical velocity if A=20 degrees? (f) What is the vertical velocity if A=45 degrees? (Four decimal places.) (Four decimal places.) (g) The vertical…arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage