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homework1 - 6 Discuss symmetry and reversibility...

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1 CDS 140a: Homework Set 1 Due: Friday, October 9th, 2009. 1. Consider the following planar system for ( x, v ) R 2 : ˙ x = v ˙ v = - x 3 (a) Find the equilibrium points for the system (b) Find a conserved energy for the system (c) Draw the phase portrait (d) Argue informally that all the trajectories outside the origin are periodic 2. Draw the phase portrait for the system ¨ x = - x 3 - ˙ x and comment on its structure. 3. Consider the following second order equation for x R : ¨ x = 2 x + x 2 - x 3 (a) Find the equilibrium points for the system (b) Find a conserved energy for the system (c) Draw the phase portrait and comment on the periodic orbit structure 4. Draw the phase portrait for the system ¨ x = 2 x + x 2 - x 3 - 2 ˙ x and comment on its structure. 5. Discuss symmetry and reversibility properties (if any) of the equations in prob- lems 1 and 2
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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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