非线性动力学（英文） (11).pdf

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.
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) = Fψ[ψx(t)], ψx Rn (23)
dt ≤
Assume that the flow ψx(t) is dissipative, with attractor A.
100
Properties of the attractor A:
A is invariant with flow (., 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
Γ
θ
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 dimensi