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Unformatted text preview: 6. Discuss symmetry and reversibility properties (if any) of the equations in prob-lems 3 and 4 7. Consider the following dynamical system in the plane that depends on the real parameter μ . ˙ x =-x ( x 2 + y 2-μ )-y ( x 2 + y 2 ) ˙ y =-y ( x 2 + y 2-μ ) + x ( x 2 + y 2 ) (a) Show that the system has a periodic orbit for μ > 0. 2 (b) Is it stable? 8. Discuss the evolution of the phase portraits in the preceding question as μ varies from negative to positive. 9. Consider the system ¨ x = αx-x 3 . Study the evolution of the phase portraits of this system as α varies from negative to positive. 10. Add a ﬁxed amount of dissipation to the preceding system and repeat the question. Include a study of symmetries as well....
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- Fall '09
- phase portrait, Dynamical systems