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Unformatted text preview: yz ˙ z = x + y + z + z 2 + cos xz exist for all time for any set of initial conditions. 8. Consider the Hopf example ˙ x =-y + x ( μ-x 2-y 2 ) ˙ y = x + y ( μ-x 2-y 2 ) where μ is a real parameter. Calculate the linearization at the origin and the associated eigenvalues and show that the result is consistent with the phase portrait. 2 9. Let ( X ( t,μ ) ,Y ( t,μ )) be the solution of the Hopf equation in the preceding exercise with initial condition ( X (0 ,μ ) ,Y (0 ,μ )) = (1 , 1) Find a diﬀerential equation that determines ∂X ∂μ and ∂Y ∂μ . 10. Using the results of Proposition 1.3.10 in the class notes or otherwise, show the following for solutions of a smooth vector ﬁeld X . If γ ( t ) is a periodic orbit of X with period say τ , then for any T > 0, there is an ± > 0 such that a trajectory with initial conditions a distance no more than ± from the periodic orbit exists for time at least T ....
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- Fall '09
- phase portrait, Dynamical systems