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Math119BProject - Bang Dinh Nhan Math 119B 11 September...

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Bang Dinh Nhan Math 119B 11 September 2008 Prof. Björn Birnir University of California at Santa Barbara Final Project: An Account on Ruelle-Takens-Newhouse Scenario I. Ruelle-Takens-Newhouse Scenario Theorem (Newhouse, Ruelle, Takens, 1978). Let v = (v 1 , …, v n ) be a constant vector field on the torus T n = R n / Z n . If n ≥ 3, every C 2 neighborhood of v contains a vector field v' with a strange Axiom A 1 attractor. If n ≥ 4, we may take C instead of C 2 . Assumptions for the Scenario i) Assume a system x′(t) = F μ (x) has a steady-state solution x μ for μ < μ c , μ c is critical value of the parameter. ii) Assume this steady-state solution loses its stability through Hopf bifurcation (Ruelle and Takens, 1971). Namely, the complex eigenvalues of the Jacobian matrix of the system 1 Axiom A (Smale, 1967). (a) the nonwandering set Ω is hyperbolic. (b) the periodic points of f are dense in Ω. 1
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will cross the pure imaginary axis, or the flow of the system will cross the unit circle as the parameter μ is increasing. iii) Assume that this happens three times in succession, and the three newly created modes are essentially independent. II. Analysis For μ < μ c , solution x μ is stable solution and the Poincaré map of this orbit is a fixed point. Figure 1 According to the assumption ii), the stability of the solution is lost through the Hopf bifurcation. The flow will be the periodic orbits whose Poincaré section is an unstable curve. Figure 2 If the map of the flow is iterated once more, the flow will become a torus. The Poincaré section of the torus is rings. Figure 3 When the number of iteration reaches three, the rings on the previous Poincaré section will break and become resonant curves (Figure 4). In fact, for small parameter, there exist both resonant bands and normal rings. As the parameter is increased, more resonant bands will take
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