4. Consider the system ry(1 +) -(42 2) (a) Show that the origin is an equilibrium point and it is...
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...
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...
Consider the system: x' = y(1 + 2x) y' = x + x2 - y2 a. Find all the equilibrium points, and linearize the system about each equilibrium point to find the type of the equilibrium point. b. Show that the system is a gradient system, and conclude that it has no periodic solutions. c. Sketch the phase portrait. Explain how you determined what the phase portrait looks like.
5 Stripes Consider the following dynamical system. State space: 2 Dynamical map: Each 0 that's followed by a 0 turns into a 1, and each 1 that's followed by a 1 turns into a o Let's call this map E. As a demonstration, here's what E does to one point in 2N 001110011011110000101110 E(u) 100010001000011110100010 a. Find two fixed points of E, and convince the grader they're the only two. (Corrected The previous version claimed, incorrectly, that there was only...
3) Given the systemxx2-x,y'-2y, find all fixed points. For each fixed point linearize the system near the fixed point, shift the fixed point to the origin, determine the eigenvalues of the linearized system, and determine whether the fixed point is a source, sink, saddle, stable orbit, or spiral. Attach a phase plane diagram to verify the behavior you found. 3) Given the systemxx2-x,y'-2y, find all fixed points. For each fixed point linearize the system near the fixed point, shift the...
Consider the system x'=xy+y2 and y'=x2 -3y-4. Find all four equilibrium points and linearize the system around each equilibrium point identifying it as a source, sink, saddle, spiral source or sink, center, or other. Find and sketch all nullclines and sketch the phase portrait. Show that the solution (x(t),y(t)) with initial conditions (x(0),y(0))=(-2.1,0) converges to an equilibrium point below the x axis and sketch the graphs of x(t) and y(t) on separate axes. Please write the answer on white paper...
4. Consider a general planar system (might not be linear): Show that (o, yo) is an equilibrium point of this system if and only if 4. Consider a general planar system (might not be linear): Show that (o, yo) is an equilibrium point of this system if and only if
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
1. Consider the system 2(t)--3i(t) +z2(t) +3() (a) (i) Find the linearised system at the equilibrium point (0, 0). (ii) What type of equilibrium point is (0,0)? (State your reasons fully.) (ii) Sketch the phase portrait for the linearised system near (0,0). (b) Repeat part (a) for the equilibrium point at (1,0). (c) (i) Are there any other equilibria? (ii) Read the Grobman-Hartman theorem and confirm that it applies to the above equilibria. 1. Consider the system 2(t)--3i(t) +z2(t) +3()...
Problem 5. Consider the system = I-? – ry, = 3y - xy - 2y? Please answer the following questions. (a) Determine all critical points of the system of equations. (b) Find the corresponding linear system near each critical point. (c) Discuss the stability of each critical point for the nonlinear system.