Bang Dinh Nhan
Math 119B
11 September 2008
Prof. Björn Birnir
University of California at Santa Barbara
Final Project:
An Account on RuelleTakensNewhouse Scenario
I.
RuelleTakensNewhouse 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 steadystate solution x
μ
for μ < μ
c
, μ
c
is critical value
of the parameter.
ii)
Assume this steadystate 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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 Spring '11
 Math, Chaos Theory, Manifold, Attractor, periodic orbits, Kifer

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