Consider the nonlinear differential equation 30 x(x – y)² + y?) 2 -2 (- - y)² + y²) 1 =:A whose right hand side is written as the sum of the linear part with coefficient matrix A and a nonlinearity. (i) Show that (x*, y*) := (0,0) is the only equilibrium. (ii) Calculate the real Jordan normal form of the coefficient matrix A using an invertible transformation matrix T E R²×2. (iii) Explain why the equilibrium (x*, y*) = (0,0) is repulsive.

Consider the nonlinear differential equation 30 x(x – y)² + y?) 2 -2 (- - y)² + y²) 1 =:A whose right hand side is written as the sum of the linear part with coefficient matrix A and a nonlinearity. (i) Show that (x, y) := (0,0) is the only equilibrium. (ii) Calculate the real Jordan normal form of the coefficient matrix A using an invertible transformation matrix T E R²×2. (iii) Explain why the equilibrium (x, y) = (0,0) is repulsive.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Consider the nonlinear differential equation
(-2(x – y)² + v°)
2 -2
+
1
+
=:A
whose right hand side is written as the sum of the linear part with coefficient matrix A and a
nonlinearity.
(i)
Show that (x*, y*) := (0, 0) is the only equilibrium.
(ii) Calculate the real Jordan normal form of the coefficient matrix A using an invertible
transformation matrix T E R²×2.
(iii) Explain why the equilibrium (x*, y*) = (0,0) is repulsive.

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