Posted: January 10th, 2017

Consider the second order nonlinear differential equation x ′′(t) + x ′ (t) − µx(t) + x 3 (t) = 0, where µ ∈ R. (a) Rewrite the above equation as a system of first order equations (in matrix form) and find all the equilibrium (critical) points of the resulting system

Consider the second order nonlinear differential equation x ′′(t) + x ′ (t) − µx(t) + x 3 (t) = 0, where µ ∈ R. (a) Rewrite the above equation as a system of first order equations (in matrix form) and find all the equilibrium (critical) points of the resulting system.

[2] (b) Obtain the eigenvalues of the corresponding linear system, in terms of µ, at each equilibrium point. [3] (c) In each case classify the systems equilibrium (critical) points in terms of its type and stability (or instability) and identify the bifurcation values for the system. [5] (ii) Determine the type of bifurcation that occurs for the following systems: (a) dx dt = rx − x 2 , dy dt = x − y, [4] (b) dx dt = rx − x 3 , dy dt = −y. [4] Confirm your analytical results with qualitatively different phase portraits before and after the bifurcation point. If instructed, draw the bifurcation diagram as well as the phase portraits by using Matlab.