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Problem 3. Linearization of a nonlinear system at a non-hyperbolic fixed point] Consider the nonlinear system...
Consider the nonlinear system: - x + (x – 1) y y + 4x° (1 –...
Consider the nonlinear system: - x + (x – 1) y y + 4x° (1 – x). (a) Show that the system has a unique fixed point at the origin (0, 0). (b) Use a linear approximation to determine the stability of the fixed point. (c) Apply the Liapunov direct method to determine the stability of the fixed point. Is your conclusion different form that of Part (a)? Why? (d) Can the system have closed orbits (trajectories)? Explain.
Problem 5 (40 pts). Given the system of nonlinear differential equations Se=y+ 2(x2 + y2 -...
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).
Question 410 marks Consider the nonlinear system ェ=(1-y)2(4-12), ỳ=(1-z)y(y2-4) (0<x<2, o<y<2), which has a single fixed...
Question 410 marks Consider the nonlinear system ェ=(1-y)2(4-12), ỳ=(1-z)y(y2-4) (0<x<2, o<y<2), which has a single fixed point at (1,1) (a) Show that the following expression for K(x, y) is a constant of motion for this system: K(x, y)- 2 ln(ry) + Inl( 2)(y- 2)]-3In(2)(y+2)]. (b) Use the constant of motion to show that the fixed point is a centre of the nonlinear system.
Consider the system of coupled ODES: x' = x - y, y = x + xy...
Consider the system of coupled ODES: x' = x - y, y = x + xy - 6y (+) (a) Find the critical points (C+, Y*) € R2 of this system. [3 marks] Hint: One critical point is (0,0) and there are two more critical points. (b) For each critical point, find the approximate linear ODE system that is valid in a small neighbourhood of it. [6 marks] (c) Find the eigenvalues of each of the linear systems found in...
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 linear...
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...
5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(...
5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(t)-2y(t)l, y'(t) = y(t)12-y(t)--2(t)] (a) Find all critical points of this system, and at each determine whether or not the system is locally stable or unstable. (b) We proved in class, using the Bendixson-Dulac theorem, that this system has no periodic solution with trajectory in the first quadrant of the plane. Assuming this, use the Poincare-Bendixson theorem to prove that all trajectories (z(t),y(t)) of the system (2) with initial...
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change...
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change to polar coordinates and find the solution r(t) and (t) of the system (b) As t goes to infinity, (x(t). y(t)) will follow the circle trajectory. Find the radius and period of the circle trajectory. (limit behavior of the solution (a))
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change to...
(1 point) Find the linearization of f(x) = V3x + 5 at the point x =...
(1 point) Find the linearization of f(x) = V3x + 5 at the point x = 1. L(x) = 10+8% Preview My Answers Subraut Answers Your score was recorded. You have attempted this problem 4 times. You received a score of 0% for this attempt. Your overall recorded score is 0%. You have 2 attempts remaining. MacBook 금이 F3 888 F4 she 76 Du F2 F5 86 99 VB A ) # 3 $ 4 % 5 & 7 2...
5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(t)-2y(t)l, y'(t) = y(t)12-y(t)--2(t)] (a)...
5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(t)-2y(t)l, y'(t) = y(t)12-y(t)--2(t)] (a) Find all critical points of this system, and at each determine whether or not the system is locally stable or unstable. (b) We proved in class, using the Bendixson-Dulac theorem, that this system has no periodic solution with trajectory in the first quadrant of the plane. Assuming this, use the Poincare-Bendixson theorem to prove that all trajectories (z(t),y(t)) of the system (2) with initial...
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...
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...
Consider the nonlinear system: - x + (x – 1) y y + 4x° (1 – x). (a) Show that the system has a unique fixed point at the origin (0, 0). (b) Use a linear approximation to determine the stability of the fixed point. (c) Apply the Liapunov direct method to determine the stability of the fixed point. Is your conclusion different form that of Part (a)? Why? (d) Can the system have closed orbits (trajectories)? Explain.
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).
Question 410 marks Consider the nonlinear system ェ=(1-y)2(4-12), ỳ=(1-z)y(y2-4) (0<x<2, o<y<2), which has a single fixed point at (1,1) (a) Show that the following expression for K(x, y) is a constant of motion for this system: K(x, y)- 2 ln(ry) + Inl( 2)(y- 2)]-3In(2)(y+2)]. (b) Use the constant of motion to show that the fixed point is a centre of the nonlinear system.
Consider the system of coupled ODES: x' = x - y, y = x + xy - 6y (+) (a) Find the critical points (C+, Y*) € R2 of this system. [3 marks] Hint: One critical point is (0,0) and there are two more critical points. (b) For each critical point, find the approximate linear ODE system that is valid in a small neighbourhood of it. [6 marks] (c) Find the eigenvalues of each of the linear systems found in...
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...
5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(t)-2y(t)l, y'(t) = y(t)12-y(t)--2(t)] (a) Find all critical points of this system, and at each determine whether or not the system is locally stable or unstable. (b) We proved in class, using the Bendixson-Dulac theorem, that this system has no periodic solution with trajectory in the first quadrant of the plane. Assuming this, use the Poincare-Bendixson theorem to prove that all trajectories (z(t),y(t)) of the system (2) with initial...
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change to polar coordinates and find the solution r(t) and (t) of the system (b) As t goes to infinity, (x(t). y(t)) will follow the circle trajectory. Find the radius and period of the circle trajectory. (limit behavior of the solution (a))
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change to...
(1 point) Find the linearization of f(x) = V3x + 5 at the point x = 1. L(x) = 10+8% Preview My Answers Subraut Answers Your score was recorded. You have attempted this problem 4 times. You received a score of 0% for this attempt. Your overall recorded score is 0%. You have 2 attempts remaining. MacBook 금이 F3 888 F4 she 76 Du F2 F5 86 99 VB A ) # 3 $ 4 % 5 & 7 2...
5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(t)-2y(t)l, y'(t) = y(t)12-y(t)--2(t)] (a) Find all critical points of this system, and at each determine whether or not the system is locally stable or unstable. (b) We proved in class, using the Bendixson-Dulac theorem, that this system has no periodic solution with trajectory in the first quadrant of the plane. Assuming this, use the Poincare-Bendixson theorem to prove that all trajectories (z(t),y(t)) of the system (2) with initial...
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...
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