

4. Consider a general planar system (might not be linear): Show that (o, yo) is an equilibrium po...
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...
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...
1. Classify the stability type of the equilibrium point (0,0) of the following linear planar homogeneous systems (same notation as in the previous homework). -(G3) 1 -3 2 -1 A-(25 4 -5
5. Consider the following first order system: (a) Find the equilibrium point (s). (b) Obtain linear functions that approximate the system for each equilibrium point. (c) Use Matlab to plot the original function(+22 -2) and the linear approx- imations all on the same plot. For the x axis limits on the plot use -2.5 to 2.5
Consider the linear system x′=Px where P is a 2×2 constant
matrix. assume P has only one eigenpair
Consider the linear system x' = Px where P is a 2 x 2 constant matrix. Assume P has only one eigenpair Then the general solution of this system might have the form: (-20) 2 Ite (a) C 21 le-24 C[***]+O (b) c(-7*]+c=[e (6) c [**] +c[-e* (a) C[*]+c [64 le 21 (1+)e 24
4 -9 a. Find the most general real-valued solution to the linear system of differential equations a' = 31(t) . C1 + c2 22(t) b. In the phase plane, this system is best described as a source / unstable node sink / stable node O saddle O center point / ellipses spiral source O spiral sink O none of these
numbers 8 and 9
8) Assume the gaseous mercury(II) hydroxide molecular has a planar structure, with a linear O-Hg-O angle, but bent at both ends. Assume the O-Hg-O bonds are on the z-axis. This molecule belongs to the C2h point group. Using the character table for C2h, determine which irreducible representation the following Hg atomic orbitals belong to: a) an s orbital b) a p: orbital c) a px orbital d) a d-y? Orbital 9) Draw the three p orbitals...
Exercise 5.5. Consider the linear system 2 as in (5.44) with A-4 0 C [1 0 -1 4 1 a. Show that the system is not (internally) asymptotically stable b. Show that the system is both controllable and observable. c. Find matrices F e R1x2 and GE R2x1 such that o(A+ BF) C C_ and o (A GC) C C_ d. Find matrices (K, L, M, N) such that the feedback controller w(t) Kw(t) Ly(t) u(t) Mw(t)Ny(t) is internally stabilizing...
Part A {\rm CIO}_{4}^3} trigonal planar O bent trigonal pyramidal tetrahedral linear Submit Request Answer Part B.
a Find the most general real-valued solution to the linear system of differential equations a' -3 -4 -3 21(t) + 22(t) b. In the phase plane, this system is best described as a O source / unstable node O sink/stable node O saddle center point / ellipses spiral source spiral sink none of these