1. Effect of Parameter Mismatch and Noise on The Loss of
Chaos Synchronization
W. Lim and S.-Y. Kim
Department of Physics
Kangwon National University
 Unidirectionally-coupled 1D maps
: Mismatching parameter
: Noise strength
: A uniform random variable with
and a unit variance
Invariant synchronization line: y = x
A strongly stable synchronous
chaotic attractor (SCA)

2. (Transcritical contact bifurcation)
Riddling and Bubbling Transitions
Weakly stable SCA
Strongly stable SCA
Weakly stable SCA c cr
Riddling Transition
(Transcritical contact bifurcation) cb
Bubbling Transition
(Supercritical PDB)
Riddling transition when c decreases through cr (= …)
Basin of the SCA becomes riddled with a dense set of “holes” leading to the divergent orbits.
Bubbling transition when c increases through cb (= …)
A typical trajectory starting near the diagonal first exhibits a transient intermittent bursting.
But, it is eventually attracted to the SCA.

3. Chaotic Transients for The Riddling Case
Parameter mismatch
SCA with the riddled basin
Chaotic transient

4. (long lived chaotic transient)
Characterization of The Chaotic Transients
: Average life-time of the chaotic transient
c ~ -2.84 cr c
Algebraic scaling
Exponential scaling
(long lived chaotic transient)
Crossover

5. Effect of Noise for The Riddling Case
Chaotic transient
Effect of noise is the same as that of
parameter mismatch.
SCA with the riddled basin
Noise
Chaotic transient
 Characterization of the chaotic transient

6. Effect of Parameter Mismatch and Noise for The Bubbling Case
Parameter mismatch or noise  The SCA is broken up, and then it exhibits a persistent
intermittent bursting.  Attractor bubbling
The maximum bursting amplitude increases when passing c=cb.
|y-x|max
|y-x|max

7. Abrupt Change of The Maximum Bursting Amplitude
The maximum bursting amplitude increases abruptly
through the interior crisis of the absorbing area for c ~
Small absorbing area before the crisis
Large absorbing area after the crisis
Abrupt increase of the maximum bursting amplitude is in contrast to the case of symmetric coupling.
Unidirectional coupling
Symmetric coupling
|y-x|max
|y-x|max

8. Summary Effect of parameter mismatch and noise
 Riddling case: Chaotic transient
Average life-time of the chaotic transient 
Crossover
Exponential scaling Algebraic scaling
(long lived chaotic transient)
 Bubbling case: Attractor bubbling (Persistent intermittent bursting)
Abrupt increase of the maximum bursting amplitude
through the interior crisis of an absorbing area