dy dt = x + ky where k is a (real) parameter. (a) Write the system in matrix form d = AY. (b) Write the determinant and trace of A in terms of k and use these to find all values of k where the system has a bifurcation. in (c) Determine the type of equilibrium at the origin for all values of k which are not the hifungo

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
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This question concerns the linear system of differential equations
dx
kx + (k² + k)y
dt
dy
dt
= x + ky
where k is a (real) parameter.
(a) Write the system in matrix form d
(b) Write the determinant and trace of A in terms of k and use these to find all values of
k where the system has a bifurcation.
= AY.
(c) Determine the type of equilibrium at the origin for all values of k which are not the
bifurcation values given in (b).
Transcribed Image Text:This question concerns the linear system of differential equations dx kx + (k² + k)y dt dy dt = x + ky where k is a (real) parameter. (a) Write the system in matrix form d (b) Write the determinant and trace of A in terms of k and use these to find all values of k where the system has a bifurcation. = AY. (c) Determine the type of equilibrium at the origin for all values of k which are not the bifurcation values given in (b).
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