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6) For the nonlinear autonomous system dx/dt = f(x), where X = (X1,x2)" and f.(X) =...
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.
Construct a Liapunov function on the form V(x,y) = ax2 + cy2 for the nonlinear system dx dt dy dt 3 山 一一 and deduce that the critical point at the origin is asymptotically stable. Construct a Liapunov function on the form V(x,y) = ax2 + cy2 for the nonlinear system dx dt dy dt 3 山 一一 and deduce that the critical point at the origin is asymptotically stable.
Theoretical Part 1. Consider the problem of computing f(x)dx, where f(x) could be any function. Letting X1, X2 IID ~U[0, 2, define three very simple estimators: ff(0)f(2), i2= f(X1)f(X2), fi3 = f(X1/2) + f((X2+2)/2) (a) (5 points) Is ft an unbiased estimator of u? (b) (5 points) Is i2 an unbiased estimator of ? (c) (5 points) Is 3 an unbiased estimator of ? (d) (10 points) Compute the variance of each of the three estimator when f(x) x Theoretical...
Problem 3. Consider the following continuous differential equation dx dt = αx − 2xy dy dt = 3xy − y 3a (5 pts): Find the steady states of the system. 3b (15 pts): Linearize the model about each of the fixed points and determine the type of stability. 3b (15 pts): Draw the phase portrait for this system, including nullclines, flow trajectories, and all fixed points. Problem 2 (25 pts): Two-dimensional linear ODEs For the following linear systems, identify the...
dx/dt= -2 -5 5 -2 x(0)= 2 4 Solve the system dx/dt = [-2 -5; 5 -2]x with x(0) = [2 4]. NOTE: *Give Solutions in real form x1=_____ and x2=________
In Problems 3-6, find the critical point set for the given system. dx 4. dx = x-y, 3. dt y1 dt dy dy = x2 y2 - 1 dt = x + y + 5 dt dx dx x2- 2xy y2- 3y 2 6. 5. dt dt dy dy 3xy - y2 (x- 1)(y 2) dt dt
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).
#10 all parts In each of Problems 5 through 18: (a) Determine all critical points of the given system of equations. (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system? (d) Draw a phase portrait of the nonlinear system to confirm your conclusions or to extend them in those cases where the linear system does not provide definite information about the...
Let S f(w)dt = 6, f(x)dx = -4, log(x)dt = 12, 9(x) dx = 9 Use these values to evaluate the given definite integral: -3 (f(x) f(x) + g(x)) dx
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?