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Consider the following system of equations dx dt = x(3- 3x - y), dy dt y(3x-3y), for 0≤ x ≤2, 0 ≤ y ≤2. 4 (a) Find all the fixed points of this system. (b) Compute the Jacobian matrix of (4) and determine the linearisation about each fixed point. Calculate the eigenvalues and eigenvectors and hence classify the type and stability of each fixed point, sketching the phase portrait of each linearised system. = = = x are invariant, with respect to 4 (c) Show that the lines x 0, y O and y (d) Sketch a phase portrait of the nonlinear system 4 indicating fixed points, invari- ant lines and the qualitative behaviour.

Consider the following system of equations dx dt = x(3- 3x - y), dy dt y(3x-3y), for 0≤ x ≤2, 0 ≤ y ≤2. 4 (a) Find all the fixed points of this system. (b) Compute the Jacobian matrix of (4) and determine the linearisation about each fixed point. Calculate the eigenvalues and eigenvectors and hence classify the type and stability of each fixed point, sketching the phase portrait of each linearised system. = = = x are invariant, with respect to 4 (c) Show that the lines x 0, y O and y (d) Sketch a phase portrait of the nonlinear system 4 indicating fixed points, invari- ant lines and the qualitative behaviour.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.1: Matrix Operations
Problem 20EQ: Referring to Exercise 19, suppose that the unit cost of distributing the products to stores is the...
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can you please do c and d

Consider the following system of equations dx dt = x(3- 3x - y), dy dt y(3x-3y), for 0≤ x ≤2, 0 ≤ y ≤2. 4 (a) Find all the fixed points of this system. (b) Compute the Jacobian matrix of (4) and determine the linearisation about each fixed point. Calculate the eigenvalues and eigenvectors and hence classify the type and stability of each fixed point, sketching the phase portrait of each linearised system. = = = x are invariant, with respect to 4 (c) Show that the lines x 0, y O and y (d) Sketch a phase portrait of the nonlinear system 4 indicating fixed points, invari- ant lines and the qualitative behaviour.
Transcribed Image Text:Consider the following system of equations dx dt = x(3- 3x - y), dy dt y(3x-3y), for 0≤ x ≤2, 0 ≤ y ≤2. 4 (a) Find all the fixed points of this system. (b) Compute the Jacobian matrix of (4) and determine the linearisation about each fixed point. Calculate the eigenvalues and eigenvectors and hence classify the type and stability of each fixed point, sketching the phase portrait of each linearised system. = = = x are invariant, with respect to 4 (c) Show that the lines x 0, y O and y (d) Sketch a phase portrait of the nonlinear system 4 indicating fixed points, invari- ant lines and the qualitative behaviour.
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