36. Show that the fixed point at the origin of the system 4 2,2,.2 is unstable by using the function for a suitable choice of the constants α and β. 36. Show that the fixed point at the origin o...
27.(a) State and prove Liapunov's theorem for a continuous-time dynamical system. (b) By finding a suitable Liapunov function show that the origin is a stable fixed point for the dynamical system What is the domain of stability? 27.(a) State and prove Liapunov's theorem for a continuous-time dynamical system. (b) By finding a suitable Liapunov function show that the origin is a stable fixed point for the dynamical system What is the domain of stability?
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...
7. Consider the system 1 2 y (a) Show that the origin is a fixed point, and determine its stability (b) Show that the origin is the only fixed point. Hint: Argue using a theorem or result based on properties of the matrix. 7. Consider the system 1 2 y (a) Show that the origin is a fixed point, and determine its stability (b) Show that the origin is the only fixed point. Hint: Argue using a theorem or result...
2. Consider a linear time-invariant system with transfer function H(s)Find the (s + α)(s + β) impulse response, h(t), of the system 2. Consider a linear time-invariant system with transfer function H(s)Find the (s + α)(s + β) impulse response, h(t), of the system
4. Consider the system ry(1 +) -(42 2) (a) Show that the origin is an equilibrium point and it is a source (b) (For practice only, not to be handed in). Show that the system has no other equilibrium points (c) Find a value of C such that the set is positively invariant respect to the flow of (2) (d) Show that (2) has a periodic orbit in S. 4. Consider the system ry(1 +) -(42 2) (a) Show that...
Given the system x'=y+ux and y'= -x + uy - x2y, with u representing the greek letter mu: a) Show that the origin is a fixed point b) Linearize the system near the fixed point and determine the eigenvalues c) Show that a bifurcation exists at u=0 and determine the behavior of the fixed point for u<0, u=0, and u>0
2 This system has an equilibrium point at the origin (you do not have to show this) For parts (d)-(e), consider the system This system also has an equilibrium point at the origin (you do not have to show this). (d) (4 pts) Compute the linearization of this system, and conclude that the Jacobian yields no relevant infor- mation regarding the equilibrium at (0,0). (e) (3 pts) Sketch the nullcline diagram for this system. Conclude from the diagram that the...
For the system x1' = -2 xz' = 4 x1 O A. the origin is a saddle point. O B. the origin is a center and the trajectories' flows are clockwise. O c. the origin is an improper nodal sink. OD. the origin is an improper nodal source. O E. the origin is a center and the trajectories' flows are counterclockwise.
1. Suppose an object is subjected to a force F that varies with position: where β 20N/m is a constant. a) The object begins at the origin (x,y) = (0,0) and travels to point P located at (z,y)-(1m,3m): 3m 4m Calculate the work done by F along each of the three possible paths shown in the figure above by the dashed lines. Note that there could be other forces acting on the object in order to ensure the object travels...
A point charge q2 = -4.5 μC is fixed at the origin of a co-ordinate system as shown. Another point charge q1 = 3.5 μC is is initially located at point P, a distance d1 = 7.2 cm from the origin along the x-axis 1) What is ΔPE, the change in potenial energy of charge q1 when it is moved from point P to point R, located a distance d2 = 2.9 cm from the origin along the x-axis as...