{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

homework1

# homework1 - 6 Discuss symmetry and reversibility...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}