Properties and quantitative characteristics of strange nonchaotic attractors
Lyapunov exponents
For quasiperiodically driven one-dimensional maps,
there is one non-trivial Lyapunov exponent L.
For SNA, as well as for a torus, it is negative. However, the attractor
consists of a set of orbits, hence, a presence of individual
unstable trajectories, which have
L>0, cannot be excluded. (A set of such orbits should be
thin, of zero measure, so that the Lyapunov exponent averaged
over the entire attractor should be negative.)
For example, in the model
xn+1=l th xncos 2pqn,
qn+1=qn+w (mod 1),
at l>1, a set of points on the axis x=0
with phases
qn=1/4+nw
forms just such an exclusive trajectory.
This is just the property, which is a distinctive feature of SNA
in comparison with the torus-attractor. In numerical computations,
presence of the unstable orbits on the attractor
may be detected from statistics of
local, or finite-time Lyapunov exponents.
For a one-dimensional quasiperiodically forced map
xn+1=f(xn, qn),
qn+1=qn+w (mod 1),
let us define a value
where a dash designates derivative over the
first argument, and an initial point
(x0, q0)
belongs to the attractor.
Performing the computation many times, we get a set
of values
LN, for which
we can introduce a distribution function F(L).
Let us consider as an example the driven quadratic map
xn+1=l - xn2+ecos 2pqn,
qn+1=qn+w (mod 1).
In the following figure to the left we show a portrait of SNA at
l=0.8, e=0.45
with the estimated value of the Lyapunov exponent L=-0.010. The right
diagram shows the distribution functions obtained
numerically LN at
N=250, 500 and 1000. They have a form of a one-hump curve,
and location of the maximum just corresponds to the
Lyapunov exponent L. With grows of N, a width of the
distribution decreases, but it keeps a 'tail' in positive
LN.
It indicates presence of locally unstable orbits on the attractor.
A fraction of events of generation of positive local exponents
LN decreases, but never vanishes although
tends to zero limit for infinitely large N.
Phase sensitivity exponent
Let us consider a possibility to reveal quantitatively a non-smooth dependence of
the coordinate variable on the phase intrinsic to SNA.
For this, it is natural to trace evolution in a course of iterations for the
value
In the following figure, the left diagram shows a plot of
DN versus a number of iterations in the model with
f(x, q)=l th xcos 2pq
in the case of SNA, at l=1.5,
To obtain a characteristic relating to the entire attractor, Pikovsky and Feudel
(Chaos, 5, 1995, 253) suggested determining the phase sensitivity exponent as a maximal value
of the power index over all orbits on the attractor:
For the present example, m=0,97. On the other hand,
for a torus-attractor, obviously, m=0. Thus, in principle,
the phase sensitivity exponent is a tool to distinguish SNA and torus.
Fractal structure and dimension of SNA
Although qualification of SNA as a 'geometrically strange object'
is one of the main points in its definition, the question on fractal properties
of SNA is studied not so well. In accordance to M.Ding et al. (Phys. Lett., 1989, A137, 167),
SNAs in the pitchfork bifurcation model and in the driven circle map
has fractal dimension (capacity)
and information dimension
Here N is a number of elements for covering by boxes of size
e, pi is a probability
of presense in the i-th box. As an example, the following figure
shows in the left diagram a portrait of SNA in the driven circle map at
K=0.95, e=1.2,
r=0.2841, and the right diagram is the double logarithmic plot used for the estimate
of the dimensions.
Crosses relate to the estimate of the capacity D0,
and stars to the information dimension D1, which are determined by
a slope of the approximating straight lines.
Spectral properties of SNA
Fourier analysis is a commonly used method of signal processing in studies
of dynamics, in particular, in experiments.
Discussing possible types of spectra, let us take the following way of reasoning.
As we have some sequence xn to be analyzed, let us construct
an 'accumulating sum'
S(W,N)=SxneiWn,
where W is a parameter,
and examine a dependence of the complex value
S(W,N) on the number of terms in the sum N.
For periodic and quasiperiodic sequences on some discrete set of values
W
we get
Rational approximations and nature of SNA
Any irrational number from interval (0,1) can be represented as a continued fraction
where ai are positive integers. The rational approximant of order k
corresponds to the sequence of elements cut at the k-th position:
Then,
If we take
a rational approximant of order k instead of the irrational frequency in a model
of forced map
then, the external driving will be periodic, of period qk.
In contrast to the the quasiperiodic case, when the phase variable in ergodic manner visits
a dense set on the unit interval, now the set is
finite:
The following figure shows charts of dynamical regimes on the parameter plane of the
forced quadratic map at rational approximations
(a-c) and at irrational frequency
Pikovsky and Feudel (Chaos, 5, 1995, 253) suggested a hypothesis for
a necessary condition of existence of SNA. Namely, at a rational approximation of the
frequency parameter, the system demonstrates bifurcations in dependence
on the initial phase, and this property persists under increase of the order of the
approximation. Qualitatively, in terms of the systems obtained from the rational approximations,
one can imagine somewhat like a slow drift of the phase parameter
accompanied with permanent bifurcations in a course of state evolution in the system.
This may be regarded as a mechanism of appearance of SNA.
.
Scheme of the computation consists in simultaneous iterations of
the original map together with the relation obtained as a derivative in respect to the
phase: