7 7. (20 points) Consider the system of nonlinear equations: a) The system has 4 critical points. Find them. b) One of the critical points is (-1, -1). Linearize the system at that point. c) Based...
Consider the nonlinear system ?x′ = ln(y^2 − x) and y'=x-y-1 (a)Find all the critical points (b)Find the corresponding linearized system near the critical points. (c) Classify the (i) type (node, saddle point, · · · ), and (ii) stability of the critical points for the corresponding linearized system. (d) What conclusion can you obtain for the type and stability of the critical points for the original nonlinear system?
Consider the nonlinear System of differential equations di dt dt (a) Determine all critical points of the system (b) For each critical point with nonnegative x value (20) i. Determine the linearised system and discuss whether it can be used to approximate the ii. For each critical point where the approximation is valid, determine the general solution of iii. Sketch by hand the phase portrait of each linearised system where the approximation behaviour of the non-linear system the linearised system...
2. Consider the nonlinear autonomous system of DEs: dx dt dy dt (a) Find all critical points of this system. (Make sure that you have found all of them.) (b) Find the linearization (a linear system) at each critical point. Calculate the eigen- values of the contant coefficient matrix, classify the corresponding critical point, and state its stability.
(6 points) The point (-1,1) is a critical point of the nonlinear system of equations This critical point is a(n) (a) unstable saddle point. (b) asymptotically stable spiral point. (c) unstable node. (d) asymptotically stable proper node / star point. (6 points) The point (-1,1) is a critical point of the nonlinear system of equations This critical point is a(n) (a) unstable saddle point. (b) asymptotically stable spiral point. (c) unstable node. (d) asymptotically stable proper node / star point.
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...
Problem 5 (40 pts). Given the system of nonlinear differential equations Se=y+ 2(x2 + y2 - 1) y'= -r + y(x² + y2 - 1) (a) Find its critical point(s). (b) Linearize the system about each critical point. (c) Classify each critical point by discussing the zeros of the corresponding characteristic equa- tion. (d) Solve the linearized systems of differential equations about the critical point(s).
please give specific steps of all the questions, thanks Q1. The following nonlinear system of DE's can be interpreted as describing the inter- action of two species with population densities r and y, respectively. 1dy1 2dt 2 dt (a) Write the given system in the form where A is a matrix with constant enteries. Also, show that the system is locally linear. (b) This system has three equilibrium or critical points. Determine those critical points and give a physical interpretation...
consider the system of differential equations ; 1) Find the fixed points of the system , 2) Evaluate the Jacobian Matrix at each fixed point, 3) Classify stability of each fixed point, 4) Sketch the graph of the phase portrait,
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.
Consider the system given by dx/dt (1 -0.5y), dy/dx-y(2.5 1.5y +0.25 . Find the critical points . Find the Jacobian of this system and use it to find the linear approximation at each of the critical points. Determine the type and the stability. . Briefly describe the overall behavior of r and y Consider the system given by dx/dt (1 -0.5y), dy/dx-y(2.5 1.5y +0.25 . Find the critical points . Find the Jacobian of this system and use it to...