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Unformatted text preview: 10 Introduction to Strange Attractors Thus far, we have studied only classical attractors such as fixed points and limit cycles. In this lecture we begin our study of strange attractors . We emphasize their generic features. 10.1 Dissipation and attraction Our studies of oscillators have revealed explicitly how forced systems can reach a stationary (yet dynamic) state characterized by an energy balance: average energy supplied = average energy dissipated An example is a limit cycle: θ θ Initital conditions inside or outside the limit cycle always evolve to the limit cycle. Limit cycles are a specific way in which dissipation ∞ attraction . More generally, we have an n-dimensional ﬂow d ψx ( t ) = F ψ [ ψx ( t )] , ψx ≤ R n (23) d t Assume that the ﬂow ψx ( t ) is dissipative, with attractor A . 100 Properties of the attractor A : • A is invariant with ﬂow (i.e., it does not change with time). • A is contained within B , the basin of attraction . B is that part of phase space from which all initial conditions lead to A as t ∗ → : A B A has dimension d < n . • Consider, for example, the case of a limit cycle: θ θ Γ The surface is reduced by the ﬂow to a line segment on the limit cycle (the attractor). Here d = attractor dimension = 1 n = phase-space dimension = 2 . This phenomenon is called reduction of dimensionality . Consequence: loss of information on initial conditions....
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This note was uploaded on 02/24/2012 for the course MECHANICAL 12.006J taught by Professor Danielrothman during the Fall '06 term at MIT.
- Fall '06