This homework is due Friday of week 10. You must solve all problems. Take the sy

Question

This homework is due Friday of week 10. You must solve all problems. Take the system x. = x - y - x(x2 + y2) y. = x + y - y(x2 + y2) Find all fixed points of the system and the linearization of the equation at (0, 0), Find one periodic solution (consider anything starting on the circle x2 + y2 = 1). Describe the stability of the origin. *Are solutions starting near the circle x2 + y2 = 1 going away from it or towards it? Consider the svstem x. = x(x2 - 1)(1 + y) y. = - 2 / 1+x2y Find all fixed points of the system and describe the stability of the linearization of the equation at each of them. Are they stable? unstable?... Describe all solutions of the system that either move only vertically or horizontally. Do all solutions (x(t), y(t)) approach the horizontal axis y = 0 as t rightarrow +infinity? Why? Consider the linear equation y. = Ay where yis a two-dimensional vector and A a 2x2 constant matrix. Suppose that thus equation has one periodic solution (i.e. a solution y(t) which is not constant and such that y(t + P) = y(t) t for some number P > 0). Then explain why all other non-zero solutions are periodic with the same period P and that the matrix A must have purely imaginary eigenvalues. The physical pendulum is the non-linear equation x. = y y. = -sin(x) Describe the linearization of the system at x = 0, y = 0. What type of fixed point is the origin? Do solutions to the linearize equation correctly predict the behavior of solutions to the non-linear equation for (x, y) near (0, 0)? (i.e. are solutions in case periodic or doing the same thing as t rightarrow +infinity?)

Explanation / Answer

You cannot ask multiple questions in a single post according to cramster/chegg's rules..

so split up your question to get positive responses//.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.