On transverse stability of synchronized chaotic attractors
Tomasz Kapitaniak
Division of Dynamics, Technical University of Lodz,
Stefanowskiego 1/15, 90-924 Lodz, Poland
The problem of synchronization of chaotic systems can be understood as a
problem of stability of n-dimensional chaotic attractor in m-dimensional phase
space (m;SPMgt;n). Let A be a chaotic attractor. The basin of attraction
is the set of points whose 
-limit set is contained in A.
In Milnor's definition [1] of an attractor the basin of attraction need not
include the neighbourhood of the attractor. For example, a riddle basin [4,5]
has positive Lebesque measure but does not contain any neighbourhood of the
attractor. Attractor A is an asymptotically stable attractor if it is Lyapunov
stable 
has positive Lebesque measure) and contains neighbourhood
of A.
In this letter we define monotonic stability of attractor as a special case of asymptotic stability and show that it is characteristic for coupled systems.
Consider a system
consisting of two coupled identical subsystems governed by
where x,y 
R 
, 
, 
R 
. Assume that
x=f(x) and y=f(y) have asymptotically stable chaotic attractor A in
invariant subspace R 
N.
As it was shown in [3] chaotic attractor A is asymptotically stable in
R 
if 
where 
is the largest Lyapunov
exponent of chaotic state. In this case the synchronized state
x(t)=y(t) is achieved for all initial conditions in the
neighbourhood of A. The detailed description of the dynamics of the system
(1) for 
can be found in [2] and will not be discussed here.
Dynamical phenomena characteristic for this range of D values have been also
described in [4-6].
With further increase of 
we can observe a special case of
asymptotic stability of attractor A. Let z 
[x 
,
y ] 
be the initial perturbation of the trajectory of
the system (1) and let us define the distance of the perturbed trajectory
z(t) from the attractor A as
If d(z(t), A) is a monotonically decreasing function of time t
then attractor A is monotonically stable. It should be noted here that stability
depends on a metric d(z(t), A). In this sense monotonicity is a
quantitative property of the attractor and may depend on the observables. For example
the linear system (x,y)=(-ax+by, -ay-bx) with 
exhibits monotonic
decay of 
, but if we choose 
with 
this decay will
no longer be monotonic. In this example eigenvalues are complex (with
negative real part) so the transition from monotonic to asymptotic stability is no
connected with well-known in linear analysis transition from spiral to node.
For the simplified analysis of the stability of chaotic attractor A let us introduce new variable
With this transformation one replaces system (1) with the equivalent system
The first equation (5) describes evolution in the neighbourhood of n-dimensional
invariant suvspace N while the second equation (6) describes the evolution
transverse to subspace N. The spectrum of Lyapunov exponents of eq. (5,6) can be
easily divided into two subsets 
associated with the evolution
of x(t) describing dynamics in subspace N while the other set 
describes propagation of perturbation normal to N.
Let us linearize eq. (6) in the neighbourhood of the attractor A, i.e. in the
neighbourhood of the fixed point e=0 with the condition 
.
In this case one obtains equation,
where
and x = [x,e] 
.
The introduced concept of linearization in the neighbourhood of the attractor allows us to reduce the problem of analysis of the stability of the attractor to the problem of investigation of the stability of the fixed point e=0 of the eq. (6).
The main result of this paper is as follows
The chaotic attractor A is asymptotically and monotonically stable in metric
(3) in R if for all x A, e=0 is the asymptotically stable
fixed point of eq. (3).
Examples of application of this results will be given.