Generalized Lorenz System: History, Classification and Synchronization

Abstract

Generalized Lorenz system (GLS) was first mentioned by Celikovsky and Vanecek in 1994 in order to provide a class of system which would be broader one than the well-known classical Lorenz system, but at the same time it would keep its basic equations structure. In 1999, Chen and Ueta introduced a new chaotic system with equations resembling the classical Lorenz one, yet with topologically very different chaotic attractors, called by others, later on, as the Chen system. In 2002, Celikovsky and Chen derived common canonical form for both Chen and classical Lorenz system, thereby showing that all previous systems are covered by the GLS class and its canonical form may be characterized by a single scalar parameter. Moreover, this scalar parameter could be used to tune subtle nonlinear phenomena like chaos and various kinds of bifurcations leading to it, while remaining parameters were eigenvalues of linear approximation at the origin. Furthermore, Celikovsky and Chen discovered and analyzed in 2002 the so-called hyberbolic type generalized Lorenz system (HGLS), being in a certain sense a complementary to the GLS. Finally, the same authors provided in 2005 yet another canonical form, enabling global exponential synchronization of two copies of GLS's or HGL's via a scalar synchronizing signal. This result was, later on, used for a novel digital encryption scheme based on continuous time chaotic system. Technique for the complete GLS and HGLS classification is, in particular, based on the special classification tools using the ideas stemming from the well-known LaSalle principle. In this keynote talk, this classification tool will be presented in a more detail, together with the resulting complete classification of GLS and HGLS. The technique for the synchronization based on nonlinear transformation and output injection leading to the observer canonical form of GLS will be explained and the synchronization of two copies of GLS demonstrated. Brief discussion of possible applications in encryption will be presented as well.

About the Speaker

Sergej Celikovsky
Institute of Information Theory and Automation
Academy of Sciences of the Czech Republic Faculty of Electrical Engineering
Czech Technical University in Prague, the Czech Republic
celikovs@utia.cas.cz

Expertise: Nonlinear systems, chaos control and synchronization, nonlinear stability and stabilization, nonlinear observers, modelling, analysis and control of underactuated systems with applications to walking robots.

Degrees: MSc. from Faculty of Numerical Mathematics and Cybernetics of the Moscow State University, Department of Optimal Control 1984; RNDr. degree (Rerum Naturalium Doctoris) from the Mathematical and Physical Faculty of Charles University in Prague 1985; Ph.D degree from the Institute of Information Theory and Automation of the Czechoslovak Academy of Sciences 1989.

Visiting positions: Research associate at the Faculty of Mathematics, University of Twente, Enschede, NL, 1994, and at the Department of Mechanical and Automation Engineering of the Chinese University of Hong Kong, 1996; Visiting professor at CINVESTAV-IPN, Unidad Guadalajara, Mexico, 1998-2000.

Currently: Associate member of the Centre for Chaos and Complex Networks at the City University of Hong Kong; Chief research fellow and the Head of Department of Control Theory in the Institute of Information Theory and Automation of the Academy of Sciences of the Czech Republic; Associate Professor at the Czech Technical University in Prague. Senior Member IEEE, member of the IFAC TC on Nonlinear Systems, Secretary of the Czech IFAC NMO.

Associate Editors: Dynamics of Continuous, Discrete an Impulsive Systems (2004-2006); IEEE Transaction on Automatic Control (2006-2009); from 2004 Member of Editorial Board of Kybernetika; from 2010 Guest Associate Editor of International Journal of Bifurcation and Chaos. Subarea Chair of the IPC of the: IFAC Nonlinear Control Symposium NOLCOS 2007, Pretoria, South Africa and IFAC Nonlinear Control Symposium NOLCOS 2010, Bologna, Italy.

IPC members: numerous conferences, in particular, IFAC Conference on Chaos Control 2006 in Reims, FR, and2009, London, GB; 3rd International Conference on Dynamics, Vibration and Control (ICDVC-2010), Hangzhou, China; 5th Asia-Pacific Workshop on Chaos Control and Synchronization, Kunming, China.

Publications: co-author of one book and two book chapters, co-editor of the book, over 40 papers in international journals, over 80 papers in international conference proceeding, over 800 SCI citations (auto-citations excluded).