Group of  Nonlinear Dynamics & Complex Systems

| Introduction | Research fields | Faculties | Research staff | Visitors | Publications | Seminar Info.| Links |

Research Fields —Chaos Control, Synchronization  & Communications

Controlling chaos is of great interest and importance from both fundamental point of view and application point of view. Recently, we have introduced wavelet controlled dynamics as a new paradigm of dynamical control.

Mathematically, wavelets are sets of L2 functions generated from a single function by translation and dilation. Some of the most important features of wavelets include time-frequency localization and multiresolution analysis. Physically, wavelet transform can split a function into different frequency bands or components so that each component can be studied with a resolution matched to its scale, thus providing excellent frequency and spatial resolution, and achieving high efficiency. Moreover, we can devise a wavelet system for representing physical information at various levels of details, leading to the so-called mathematical microscopy.

We considered the wavelet decompositon on the coupling matrix of several chaotic dynamical systems. We found that by modifying a tiny fraction of the wavelet subspaces of the coupling matrix, we could dramatically enhance the transverse stability of the synchronous manifold of a chaotic system. Wavelet controlled Hopf bifurcation from chaos is observed.

The impact of modifying a wavelet subspace to the coupling matrix A of chaotic oscillators. The original coupling matrix; the three-scale wavelet decomposition of the original coupling matrix; the modified wavelet decomposition the physical space image of the wavelet enhanced coupling matrix.	Wavelet controlled dynamics showing the transition from chaos to periodicity. (a) The original chaotic states; (b) Wavplet induced Hopf bifurcation from

Recently, this paper has been highlighted at Physical Review Letters and featured in Nature (Vol 422, P384) by Peter Ashwin. As he stated, "It isn't easy to create a semblance of order in interconnected dynamical systems. But this mathematical tool could be the means to synchronize systems more effectively - and keep chaos at bay."

Since the pioneer work of Ott, Grebogi, and Yorke in the 1990, controlling chaos has been extensively investigated. A variety of approaches, such as the OGY scheme, open-loop strategy, feedback technique and adaptive method. Have been developed for the purpose of chaos control. Most early work deals with low-dimensional chaotic systems such as the logistic map and the Lorenz system. Recently chaos control has been gradually carried out in spatially extended dynamical systems, such as coupled map lattices and partial differential equations. These works are motivated by potential applications in laser and plasma physics, chemical reactions, electric circuits, neuronal networks as well as secure communication. One of the most complicated spatio-temporal systems is the real-world fluid turbulence

However, the possibility of controlling real-world flow turbulence by using the control strategies developed in the low-dimensional chaotic system has not been studied. In this study, we show that the flow turbulence governed by the Navier-Stokes equations can be effectively controlled by global and local pinning methods.

Figure Contour plots of the vorticity field of the flow system before and after the control. (a) t=100 and (b) t=250 before the control; (c) t=120 and (d) t=200 after the control showing the gradual convergence to the spatially periodic target.

S. G. Guan, Y. C. Zhou, G. W. Wei, and C.- H Lai, Chaos 13, 64 (2003).

Cryptography and communication by using chaos synchronization have been one of the most important applications for chaos study. And how to design a more secure and effective communication scheme is always an open question and still a challenge forscientists in this area. In our recent work, we compare and analyze systemically the former works in this domain by study their global performance which includes the security, the encryption speed, and the bit error propagation, and so on. Our conclusion is that, no scheme is workable in case of in comparison with the classical methods. Our work is mainly concentrate on designing an effective and reliable encryption and communication scheme which comparable or better in some aspects than the classical methods. 

New encryption scheme is proposed and designed based on the synchronization of spatiotemporal chaos. The scheme has excellent performance in each aspect. High and adjustable security under the most severe attacks like linear attack and differential attack and even also with the cryptanalysis special for chaotic systems, the error function attack. This encryption scheme performs incomparable advantage than former chaos encryptions and also has comparable ability as the most advanced encryption method, the advanced encryption standard (AES). Moreover, this new scheme owns more flexibility than AES by choosing more or less parameters as the secret key, and we regard this as a resolution for the crisis of "key short" which troubling the safety in today’s encryptions.

Figure: the original picture;  encrypted picture; the picture received; security of encryption,  with parameter with order 1, the key basin only with order 10-12.

Fast encryption speed which even fast than AES, in our numerical simulations, the speed for our scheme is about 216.7Mb/sec, while under the same situation (SGI OCTANE workstation (two 195MHZ IP30 CPU 256M RAM, Fortran90 compiler), the encryption speed for AES is about 180Mb/sec. By choosing parameters within some suitable

ranges, the problem of error bitpropagation can be constrained within in a very low level and is also comparable with AES. The digital sequence generated by this scheme also performs high quality when used as digital spread spectrum communication. Except its high security and fast code generation, the correlation performances, both auto-correlation and cross-correlation, are excellent and the bit error rate (BER) is also comparable with classical codes like golden code both for single user situation and multi-user situation.