a) Show that the nonlinear dynamical system i = 2y – x² + 1, ý = 1 – x² – y? has two equilibrium points and find their locations. b) Use the linearisation theorem to classify the nature of the equilibrium points and give two separate sketches showing phase portraits of the linearised systems. Your sketches should include isoclines (of the linearised systems) and all the straight line phase paths of the linearised systems (if any exist). c) Sketch the phase portrait of the nonlinear dynamical system indicating directions of the trajectories and the horizontal and vertical isoclines.
a) Show that the nonlinear dynamical system i = 2y – x² + 1, ý = 1 – x² – y? has two equilibrium points and find their locations. b) Use the linearisation theorem to classify the nature of the equilibrium points and give two separate sketches showing phase portraits of the linearised systems. Your sketches should include isoclines (of the linearised systems) and all the straight line phase paths of the linearised systems (if any exist). c) Sketch the phase portrait of the nonlinear dynamical system indicating directions of the trajectories and the horizontal and vertical isoclines.
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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Transcribed Image Text:a) Show that the nonlinear dynamical system
ý = 1 – x² – y?
,2
i = 2y – x* + 1,
-
has two equilibrium points and find their locations.
b) Use the linearisation theorem to classify the nature of the equilibrium points and
give two separate sketches showing phase portraits of the linearised systems. Your
sketches should include isoclines (of the linearised systems) and all the straight line
phase paths of the linearised systems (if any exist).
c) Sketch the phase portrait of the nonlinear dynamical system indicating directions
of the trajectories and the horizontal and vertical isoclines.
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