5. Show that the zero solution of is asymptotically stable if b > 0 and unstable if b < 0. Does t...
Show that the zero solution of y' =-y+y is asymptotically stable, but not globally, i.e. not all solutions tend to zero as t + Sketch all solutions in the (ty) plane, taken to = 0. Also sketch all solutions in phase space. What can you conclude about the solution y = 1?
8. The van der Pol equation. The equation arises in the study of vacuum tubes. Show that if e < 0, the origin is asymptotically stable and that for0 it is unstable. 8. The van der Pol equation. The equation arises in the study of vacuum tubes. Show that if e
1 (c) (12 pts) Consider the logistic equation IP 3 Use phase portrait analysis to classify the equilibrium solutions as asymptotically stable, 10 unstable or semi-stable. (ii) Find the general solution to the ODE. (The solution may be expressed in implicit form.) 1 (c) (12 pts) Consider the logistic equation IP 3 Use phase portrait analysis to classify the equilibrium solutions as asymptotically stable, 10 unstable or semi-stable. (ii) Find the general solution to the ODE. (The solution may be...
4) Using the Routh-Hurwitz Criterion, determine whether the following Polynomials are Stable or Unstable. Please Show Supporting Work: 1) H(s) = s? + 10s + 5 = 0 Stable Unstable 11) H(s) = s4 +53 + 5s2 + 20s + 10 = 0 Stable Unstable 111) H(s) = 83 + 4Ks2 + (5 + K)s + 10 = 0 The Range of K for a Stable System is: a. b. K > 0.46 K< 0.46 0<K <0.46 Unstable for all...
Each of the following equations specifies an LTID system. Determine whether these systems are asymptotically stable, unstable, or marginally stable. 9.6-1 (a) yk 20.6y[k + 1] - 0.16y[k] = f k + 1 - 2flk] (b) (Е? (c) (E 1Ey{k] = (E + 2)fjk] (d) yk2y(k]0.96y(k - 2] 2flk - 1] +3f(k - 3] (e) (E2- 1)(E +E+1)уk] 3DEflk] +1)yk fk] Each of the following equations specifies an LTID system. Determine whether these systems are asymptotically stable, unstable, or marginally...
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.
Problem : Consider the systems A and B whose roots are shown below BI 1. Regarding stability, the systems are a) b) c) d) Both stable Both unstable A is unstable and B is stable A is stable and B is unstable 2. The responses of the systems to step input are characterized as follows: a) Both are underdamped b) Both are overdamped c) A is underdamped and B is overdamped d) A is overdamped and B is underdamped 3....
Do each of the following eight (8) problems. The problems have equal weight. For each problem, in order to receive maximum possible credit, show the steps of the solution clearly,and provide appropriate explanation. Return this exam with your answer sheets . Chapter continunous-time system, with time t in seconds () input fO, and output yo. is specified by the equation y(t) = 1.5cos(2x500 + 0.8ft). a. Is this system instantaneous (memoryless) or dynamic (with memory)? Justify your answer Show that...
-4 -6 -10 -10 -5 10 Stable node or Sink. (Eigenvalues are real and negative.) O Unstable node or Source. (Eigenvalues are real and positive.) Saddle Node. (Eigenvalues are real with opposite signs.) O Stable focus. (Eigenvalues are imaginary with negative real part.) O Unstable focus. (Eigenvalues are imaginary with positive real part.) O Center. (Eigenvalues are imaginary with zero real part.) None of these -4 -6 -10 -10 -5 10 Stable node or Sink. (Eigenvalues are real and negative.)...
Consider the nonlinear second-order differential equation x4 3(x')2 + k2x2 - 1 = 0, _ where k > 0 is a constant. Answer to the following questions. (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: stable / saddle unstable(not saddle)} (c) Show that there is no periodic solution in a simply connected region {(r, y) R2< 0} R =...