Generalized synchronization of chaos: The auxiliary system approach

Abstract

Synchronization of chaotic oscillators in a generalized sense leads to richer behavior than identical chaotic oscillations in coupled systems. It may imply a more complicated connection between the synchronized trajectories in the state spaces of coupled systems. We suggest a method here that can be used to detect and study generalized synchronization in drive-response systems. This technique, the auxiliary system method, utilizes a second, identical response system to monitor the synchronized motions. The method can be implemented both numerically and experimentally and in some cases it leads to analytical results for generalized synchronization. ͓S1063-651X͑96͒02505-6͔

Generalized and partial synchronization of coupled neural networks

“…Synchronization conditions (9) and (10) 3 o B 3 ) are shown, demonstrating the synchronous activity of neuron pairs (1 A 1 B ) a n d ( 2 A 2 B ). As this gure suggests, in addition to partial synchronization, according to de nition (1), this is at the same time an example for a generalized dynamics of two modules. In fact, the resulting dynamics of the coupled system is not 4-dimensional -as expected -but it is still constrained to a 3-dimensional manifold as can be seen from the (o A 3 o B 3 ) plot.…”

“…We choose modules and couplings as shown in gure 1. The corresponding dynamics of the coupled system is then given by a 1 then a completely synchronized dynamics exists for this con guration. This has been reported in 37].…”

“…We are considering neuromodules as discrete parameterized dynamical systems on an n-dimensional activity phase space R n given by the map a i (t + 1 ) = i + n X j=1 w ij (a j (t)) i = 1 : : : n (1) where a i 2 R n denotes the activity of the i-th neuron, and i = i + I i denotes the sum of its xed bias term i and its stationary external input I i , respectively. The output o i = (a i ) of a unit is given by the sigmoidal transfer function (a) : = ( 1 + e ;a ) ;1 , a 2 R, and w ij denotes the synaptic weight from unit j to unit i.…”

Chaos synchronization in fractional differential systems

“…, α n ), FPS becomes HPS. In the study by Deng [95], three methods for achieving GS of fractional systems were discussed from the so-called auxiliary system approach [96], where theorem 3.11 above was specifically used to realize GS.…”

Synchronization with positive conditional Lyapunov exponents

“…In the first case, a replica subsystem driven by chaotic signals of the chaotic system can synchronize identically with the drive system [1][2][3][4][5], if the largest conditional Lyapunov is negative. This is referred to as identical synchronization.Secondly, a driven system, which is not a replica of the drive system, however, may not achieve identical synchronization, but generalized synchronization [6][7][8], if the largest conditional Lyapunov exponent is negative. Two identical systems, driven by the same signal, thus may come to the same final state due to the negative largest conditional Lyapunov exponent.…”