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hw6 - Homework 6 Cover Sheet Name Space below is for the...

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Homework 6 Cover Sheet Name: Space below is for the instructor. Problem Score
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CDS 140a: Homework Set 6 Due: Wednesday, November 21, 2007. Nonlinear Differential Equations and Dynamical Systems , Ferdinand Ver- hulst, Second Edition. Consider two systems ˙ x = A x and ˙ x = B x with x R n and A,B constant matrices. Assume that all the eigenvalues of A,B have nonzero real parts. Find conditions on A,B so that there exists a diffeomorphism H : U mapsto→ V from a neighborhood of origin U to V which takes orbits of one system to those of other and preserves parametrization by time. Consider ˙ x = ax and ˙ x = bx with x R and a,b nonzero real numbers. Find a homeomorphism H : R mapsto→ R which takes the orbits of one system to those of the other and preserves parametrization by time. Show that H is not a diffeomorphism in general. Consider the vector field ˙ x = 2 x + y 2 (1) ˙ y = y Denote the linearization of (1) at the origin by J and the flow generated by (1) by ϕ t .
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