Show that the zero solution of y' =-y+y is asymptotically stable, but not globally, i.e. not...
13. Use a Lyapunov function to show that the origin is globally asymptotically stable: x' = -y - xemy y' = x - y Hint: Try V = x2 + y2. x' = 2y – 2.3 y = –23 – 45 Hint: Try V = ax+ + by2 for an appropriate choice of a, b > 0.
5. Show that the zero solution of is asymptotically stable if b > 0 and unstable if b < 0. Does this depend on the sign of the constant k?
5. Show that the zero solution of is asymptotically stable if b > 0 and unstable if b
4 Consider the autonomous differential equation y f(v) a) (3 points) Find all the equilibrium solutions (critical points). b) (3 points) Use the sign of y f(z) to determine where solutions are increasing / decreasing. Sketch several solution curves in each region determined by the critical points in c) (3 points) the ty-plane. d) (3 points) Classify each equilibrium point as asymptotically stable, unstable, or semi-stable and draw the corresponding phase line.
4 Consider the autonomous differential equation y f(v)...
7. Answer the questions below for the following initial value problem: y (t) = sin y, 0 <y(0) < 27. (a) [1 pt) Determine the equilibrium (i.e., critical or steady-state) solutions. (b) (2 pts) Construct a sign chart for y' = sin y. Hy' = sin y 21 (c) (3 pts] Now construct a sign chart for y", and find the inflection points (if any). Hy" = f(y) 271 (d) [5 pts] Draw the phase line, and sketch a graph...
Using Differential
Equations.
6. For y, = y3 _ y, y(0) = 30, -00 <30 < 00, draw the graph of (y) = y3-y versus y, determine the equilibrium solutions (critical points) and classify each one as unstable or asymptotically stable. Draw the phase line, and sketch several representative integral curves (graphs of solutions) in the (t, y) plane. Hint: None of this requires explicit formulas for solutions y = φ(t) of the initial value problem.]
3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using the Lyapunov function V(x, y, z) = ρ「2 + ơy2 + ơz?, show that the origin is globally asymptotically stable. (Hint. You may need to use the Invariance Principle as well.) στ
3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using...
Determine the values of a, if any, for which all solutions of the differential equation y'' – (2a – 9)y' + (a? – 9a + 14)y = 0 tend to zero as t → 00; also determine the values of a, if any, for which all (nonzero) solutions become unbounded as t → . Solutions tend to zero as t + op as long as a (Click for List) A QE Solutions become unbounded as t → as long as...
Need answer to part (c).
QUESTION 3. Consider the following system (a) Convert to polar coordinates and find a periodic orbit. Write the corresponding periodic solution as a function of time t. (b) Show that the periodic orbit found in (a) is locally asymptotically stable by calculating a Floquet multiplier of the variational equation (or eigenvalue of derivative of Poincaré map) corresponding to (1) at the periodic solution. (c) Can you prove further that all non-zero solutions converge asymptotically to...
3. Show tha the system of equations where r r2 + y2 has a limit cycle at r - To for each ro such that F(ro) = 0 and F,(m)メ0. The orbit is called asymptotically orbitally stable if F(ro) >0 and unstable if F(ro)0. For the case when F(r)r2)(2- 4r 3) find all limit cycles, determine the orbital stability, and sketch the orbits in the phase plane.
3. Show tha the system of equations where r r2 + y2 has...
= 3x +0.75y, = 1.66667x + y. For this system, the smaller eigenvalue is 1/2 and the larger eigenvalue is 7/2 [Note-- you may want to view a phase plane plot (right click to open in a new window).] If y' = Ay is a differential equation, how would the solution curves behave? All of the solutions curves would converge towards 0. (Stable node) All of the solution curves would run away from 0. (Unstable node) The solution curves would...