. La-notion-d-'attractivité-signifie-que, après un certains temps et de possibles larges déviations, toute trajectoire du système convergera vers son état d'équilibre. Cependant, un système attractif n'est pas nécessairement stable, au sens de la définition A.1.2, c'est pourquoi on introduit alors la notion de stabilité asymptotique qui garantit que l'état du système converge vers son état d'équilibre sans trop s'en éloigner

. La-philosophie-de-cette-méthode-s, appuie sur une observation fondamentale de la physique : si l'énergie totale d'un système est dissipée de manière continue alors le système devra rejoindre finalement un point d'équilibre. On pourra donc conclure à la stabilité d'un système par l'examen de la variation d'une fonction scalaire V

V. (. Soit and . Une-fonction-continue, V est dite propre définie positive si : (i) ?t ? R +

B. Sommaire, 1 Caractérisation du chaos, p.150

]. G. Alvarez-06a, &. S. Alvarez, and . Li, SOME BASIC CRYPTOGRAPHIC REQUIREMENTS FOR CHAOS-BASED CRYPTOSYSTEMS, International Journal of Bifurcation and Chaos, vol.7, issue.08, pp.2129-2151, 2006.
DOI : 10.1109/82.877154

G. Álvarez and &. Li, Some basic cryptographic requirements for chaosbased cryptosystems. accepted by International Journal of Bifurcation and Chaos, 2005.

M. Arcak and &. P. Kokotovic, Nonlinear observers: a circle criterion design and robustness analysis, Automatica, vol.37, issue.12, pp.1923-1930, 2001.
DOI : 10.1016/S0005-1098(01)00160-1

A. and ]. D. Aubry, Contribution à la synthèse d'observateurs pour les systèmes non linéaires, Thèse, 1999.

W. T. Baumann and &. W. Rugh, Feedback control of nonlinear systems by extended linearization, IEEE Transactions on Automatic Control, vol.31, issue.1, pp.40-46, 1986.
DOI : 10.1109/TAC.1986.1104100

G. Besançon, Q. Zhang, and &. H. Hammouri, High-gain observer based state and parameter estimation in nonlinear systems, Proceedings of NOLCOS, IFAC Symposium on Nonlinear Control Systems, 2004.

D. Bestle and &. M. Zeitz, Canonical form observer design for non-linear time-variable systems, International Journal of Control, vol.27, issue.2, pp.419-431, 1983.
DOI : 10.1080/00207177308932395

S. P. Bhattacharyya, Observer design for linear systems with unknown inputs, IEEE Transactions on Automatic Control, vol.23, issue.3, pp.483-484, 1978.
DOI : 10.1109/TAC.1978.1101758

]. S. Biagiola and &. J. Figueroa, A high gain nonlinear observer: application to the control of an unstable nonlinear process, Computers & Chemical Engineering, vol.28, issue.9, pp.1881-1898, 2004.
DOI : 10.1016/j.compchemeng.2004.03.004

S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and &. C. Zhou, The synchronization of chaotic systems, Physics Reports, vol.366, issue.1-2, pp.1-101, 2002.
DOI : 10.1016/S0370-1573(02)00137-0

M. Darouach, Observers design for non linear descriptor systems, Proceedings of the 34th IEEE Conference on Decision and Control, 1995.

&. H. Darouach and . Rafaralahy, Generalized state-space observers for chaotic synchronization and secure communication, IEEE Transactions on Circuits and Systems I, vol.49, issue.3, pp.345-349, 2002.

S. Boyd, L. Elghaoui, E. Feron, and &. V. Balakrishnan, Linear matrix inequalities in systems and control theory, 1994.
DOI : 10.1137/1.9781611970777

H. Hammouri, M. Kinaert, and &. E. Yaagoubi, Observer-based approach to fault detection and isolation for nonlinear systems, IEEE Transactions on Automatic Control, vol.44, issue.10, pp.1879-1884, 1999.
DOI : 10.1109/9.793728

G. Hostetter and &. J. Meditch, Observing systems with unmmeasurable inputs, IEEE Transactions on Automatic Control, vol.18, issue.3, pp.307-308, 1973.
DOI : 10.1109/TAC.1973.1100296

M. Hou and &. P. Muller, Design of observers for linear systems with unknown inputs, IEEE Transactions on Automatic Control, vol.37, issue.6, pp.871-875, 1992.
DOI : 10.1109/9.256351

]. E. Inoue and &. T. Ushio, Chaos communication using unknown input observers, Electronics and Communications in Japan (Part III: Fundamental Electronic Science), vol.79, issue.215, pp.1801-1807, 2001.
DOI : 10.1109/9.76372

A. H. Jazwinski, Stochastic processes and filtering theory, 1970.

]. Jiang, A note on chaotic secure communication systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.49, issue.1, pp.92-96, 2002.
DOI : 10.1109/81.974882

G. Jiang, &. Chen, and . Tang, A New Criterion for Chaos Synchronization Using Linear State Feedback Control, International Journal of Bifurcation and Chaos, vol.13, issue.08, pp.2343-2351, 2003.
DOI : 10.1142/S0218127494000691

&. W. Jiang and . Zheng, An LMI criterion for linear-state-feedback based chaos synchronization of a class of chaotic systems, Chaos, Solitons & Fractals, vol.26, issue.2, pp.437-443, 2005.
DOI : 10.1016/j.chaos.2005.01.012

]. N. Jo and &. J. Seo, Input output linearization approach to state observer design for nonlinear system, IEEE Transactions on Automatic Control, vol.45, issue.12, pp.2388-2393, 2000.
DOI : 10.1109/9.895580

]. C. Johnson, Accomodation of external disturbances in linear regulator and servomechanism problems, IEEE Transactions on Automatic Control, vol.16, issue.6, pp.635-644, 1971.
DOI : 10.1109/TAC.1971.1099830

]. R. Kalman, A New Approach to Linear Filtering and Prediction Problems, Journal of Basic Engineering, vol.82, issue.1, pp.35-45, 1960.
DOI : 10.1115/1.3662552

N. Kazantzis and &. C. Kravaris, Nonlinear observer design using Lyapunov???s auxiliary theorem, Systems & Control Letters, vol.34, issue.5, pp.241-247, 1998.
DOI : 10.1016/S0167-6911(98)00017-6

&. W. Kelber and . Schwarz, General design rules for chaos-based encryption systems, Proceedings of NOLTA2005, International Symposium on Nonlinear Theory and its Applications, pp.465-468, 2005.

J. Levine and &. R. Marino, Nonlinear system immersion, observers and finite-dimensional filters, Systems & Control Letters, vol.7, issue.2, pp.133-142, 1986.
DOI : 10.1016/0167-6911(86)90019-8

C. Li, X. Liao, and &. Wong, Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication, Physica D: Nonlinear Phenomena, vol.194, issue.3-4, pp.133-142, 1986.
DOI : 10.1016/j.physd.2004.02.005

&. Liao and . Huang, An observer-based approach for chaotic synchronization with applications to secure communications, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.46, issue.9, pp.1144-1150, 1999.
DOI : 10.1109/81.788817

&. H. Liu and . Peng, Inverse-Dynamics Based State and Disturbance Observers for Linear Time-Invariant Systems, Journal of Dynamic Systems, Measurement, and Control, vol.21, issue.3, pp.375-381, 2002.
DOI : 10.1109/TAC.1976.1101230

D. G. Luenberger, An introduction to observers, IEEE Transactions on Automatic Control, vol.16, issue.6, pp.596-602, 1971.
DOI : 10.1109/TAC.1971.1099826

R. Mainieri and &. J. Rehacek, Projective Synchronization In Three-Dimensional Chaotic Systems, Physical Review Letters, vol.54, issue.15, pp.3042-3045, 1999.
DOI : 10.1017/S0305004100033223

G. Millérioux and &. C. Mira, Coding Scheme Based on Chaos Synchronization from Noninvertible Maps, International Journal of Bifurcation and Chaos, vol.12, issue.7, pp.2019-2029, 1998.
DOI : 10.1142/S0218127496000187

]. G. Millérioux and &. J. Daafouz, An observer-based approach for input-independent global chaos synchronization of discrete-time switched systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.50, issue.10, pp.1270-1279, 2003.
DOI : 10.1109/TCSI.2003.816301

]. G. Millérioux and &. J. Daafouz, UNKNOWN INPUT OBSERVERS FOR MESSAGE-EMBEDDED CHAOS SYNCHRONIZATION OF DISCRETE-TIME SYSTEMS, International Journal of Bifurcation and Chaos, vol.10, issue.04, pp.1357-1368, 2004.
DOI : 10.1109/9.1278

]. G. Millérioux and &. J. Daafouz, CHAOS SYNCHRONIZATION: FROM THE GENESIS TO POLYTOPIC OBSERVERS, Proceedings Chaos'06, IFAC Conference on Analysis and Control of Chaotic Systems, pp.345-350, 2006.
DOI : 10.3182/20060628-3-FR-3903.00060

]. E. Misawa-89, &. J. Misawa, and . Heydrick, Nonlinear Observers???A State-of-the-Art Survey, Journal of Dynamic Systems, Measurement, and Control, vol.111, issue.3, pp.344-352, 1989.
DOI : 10.1115/1.3153059

S. Mondie and &. V. Kharitonov, Exponential estimates for retarded time-delay systems: an LMI approach, IEEE Transactions on Automatic Control, vol.50, issue.2, pp.268-273, 2005.
DOI : 10.1109/TAC.2004.841916

O. Morgül and &. E. Solak, Observer based synchronization of chaotic systems, Physical Review E, vol.28, issue.5, pp.4803-4811, 1996.
DOI : 10.1016/0005-1098(92)90177-H

O. Morgül and &. M. Feki, A chaotic masking scheme by using synchronized chaotic systems, Physics Letters A, vol.251, issue.3, pp.169-176, 1999.
DOI : 10.1016/S0375-9601(98)00868-8

A. Nemirovskii and &. P. Gahinet, The projective method for solving linear matrix inequalities, Proceedings of 1994 American Control Conference, ACC '94, pp.840-844, 1994.
DOI : 10.1109/ACC.1994.751861

]. S. Nicosia-89, P. Nicosia, &. A. Tomei, and . Tornambé, An approximate observer for a class of nonlinear systems, Systems & Control Letters, vol.13, issue.1, pp.43-51, 1989.
DOI : 10.1016/0167-6911(89)90019-4

H. Nijmeijer and &. I. Mareels, An observer looks at synchronization, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.44, issue.10, pp.882-890, 1997.
DOI : 10.1109/81.633877

R. Nikoukhah, A new methodology for observer design and implementation, IEEE Transactions on Automatic Control, vol.43, issue.2, pp.229-234, 1998.
DOI : 10.1109/9.661071

URL : https://hal.archives-ouvertes.fr/inria-00074014

M. Nørgaard, N. K. Poulsen, and &. O. Ravn, New developments in state estimation for nonlinear systems, Automatica, vol.36, issue.11, pp.1627-1638, 2000.
DOI : 10.1016/S0005-1098(00)00089-3

M. J. Ogorzalek, Taming chaos. I. Synchronization, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.40, issue.10, pp.693-699, 1993.
DOI : 10.1109/81.246145

A. V. Oppenheim, G. W. Wornell, S. H. Isabelle, and &. K. Cuomo, Signal processing in the context of chaotic signals, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, pp.117-120, 1992.
DOI : 10.1109/ICASSP.1992.226472

E. Ott, C. Grebogi, and &. J. Yorke, Controlling chaos, Physical Review Letters, vol.34, issue.11, pp.1196-1199, 1990.
DOI : 10.1109/9.21099

P. C. Parks and &. V. Hahn, Stability theory, 1993.

U. Parlitz, L. O. Chua, L. Kocarev, K. S. Halle, and &. A. Shang, TRANSMISSION OF DIGITAL SIGNALS BY CHAOTIC SYNCHRONIZATION, International Journal of Bifurcation and Chaos, vol.02, issue.04, pp.973-977, 1993.
DOI : 10.1142/S0218127492000562

]. U. Parlitz-99a, L. Parlitz, &. L. Junge, and . Kocarev, Chaos synchronization
DOI : 10.1007/BFb0109942

É. Fossen, New directions in nonlinear observer design, pp.511-525

]. U. Parlitz-99b, &. L. Parlitz, and . Kocarev, Synchronization of Chaotic Systems, Handbook of Chaos Control, pp.271-303, 1999.
DOI : 10.1002/3527607455.ch11

L. M. Pecora and &. T. Carroll, Synchronization in chaotic systems, Physical Review Letters, vol.10, issue.8, pp.821-824, 1990.
DOI : 10.1017/S0140525X00047336

L. M. Pecora, T. L. Carroll, G. Johnson, and &. D. Mar, Fundamentals of synchronization in chaotic systems, concepts, and applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, vol.54, issue.5, pp.520-543, 1997.
DOI : 10.1103/PhysRevE.54.6708

A. M. Pertew, H. J. Marquez, and &. Q. Zhao, synthesis of unknown input observers for non-linear Lipschitz systems, International Journal of Control, vol.13, issue.15, pp.1155-1165, 2005.
DOI : 10.1109/TAC.1975.1100968

J. Primbs, Survey of nonlinear observer design techniques, 1996.

S. Raghavan and &. J. Hedrick, Observer design for a class of nonlinear systems, International Journal of Control, vol.59, issue.2, pp.515-528, 1994.
DOI : 10.1109/9.53535

R. Rajamani, Observers for Lipschitz nonlinear systems, IEEE Transactions on Automatic Control, vol.43, issue.3, pp.397-401, 1998.
DOI : 10.1109/9.661604

A. Rapaport and &. A. Maloum, Embedding for exponential observers of nonlinear systems, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), 2000.
DOI : 10.1109/CDC.2000.912867

K. Reif, F. Sonnemann, and &. R. Unbehauen, An EKF-Based Nonlinear Observer with a Prescribed Degree of Stability, Automatica, vol.34, issue.9, pp.1119-1123, 1998.
DOI : 10.1016/S0005-1098(98)00053-3

J. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica, vol.39, issue.10, pp.1667-1694, 2003.
DOI : 10.1016/S0005-1098(03)00167-5

A. S. Rosenblum, &. J. Pikovsky, and . Kurths, Phase Synchronization of Chaotic Oscillators, Physical Review Letters, vol.75, issue.11, pp.1804-1807, 1996.
DOI : 10.1103/PhysRevLett.75.3190

N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and &. H. , Generalized synchronization of chaos in directionally coupled chaotic systems, Physical Review E, vol.50, issue.2, pp.980-994, 1995.
DOI : 10.1103/PhysRevE.50.2580

R. Seliger and &. P. Frank, Fault-diagnosis by disturbance decoupled nonlinear observers, [1991] Proceedings of the 30th IEEE Conference on Decision and Control, 1991.
DOI : 10.1109/CDC.1991.261547

C. E. Shannon, Communication Theory of Secrecy Systems*, Bell System Technical Journal, vol.28, issue.4, pp.656-715, 1949.
DOI : 10.1002/j.1538-7305.1949.tb00928.x

R. Sharma and &. M. Aldeen, Estimation of unknown disturbances in nonlinear systems, Control, 2004.

K. M. Short, STEPS TOWARD UNMASKING SECURE COMMUNICATIONS, International Journal of Bifurcation and Chaos, vol.04, issue.04, pp.959-977, 1994.
DOI : 10.1142/S021812749400068X

K. M. Short, UNMASKING A MODULATED CHAOTIC COMMUNICATIONS SCHEME, International Journal of Bifurcation and Chaos, vol.06, issue.02, pp.367-375, 1996.
DOI : 10.1142/S0218127496000114

]. C. Silva-93 and . Silva, Shil'nikov's theorem-a tutorial, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.40, issue.10, pp.675-682, 1993.
DOI : 10.1109/81.246142

J. E. Slotine, J. K. Heydrick, and &. E. Misawa, On Sliding Observers for Nonlinear Systems, Journal of Dynamic Systems, Measurement, and Control, vol.109, issue.3, pp.245-252, 1987.
DOI : 10.1115/1.3143852

M. I. Sobhy and &. A. Shehata, Methods of attacking chaotic encryption and countermeasures, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221), 2001.
DOI : 10.1109/ICASSP.2001.941086

D. Stinson, Cryptography : theory and practice, 1995.

S. Tang, H. F. Chen, S. K. Hwang, and &. J. Liu, Message encoding and decoding through chaos modulation in chaotic optical communications, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.49, issue.2, pp.163-169, 2002.
DOI : 10.1109/81.983864

F. E. Thau, Observing the state of non-linear dynamic systems???, International Journal of Control, vol.82, issue.3, pp.471-479, 1973.
DOI : 10.1109/TAC.1966.1098323

]. B. Walcott-87a, M. J. Walcott, &. S. Corless, and . Zak, Comparative study of non-linear state-observation techniques, International Journal of Control, vol.45, issue.6, pp.2109-2132, 1987.
DOI : 10.1080/00207178708933870

]. B. Walcott-87b, &. S. Walcott, and . Zak, State observation of nonlinear uncertain dynamical systems, IEEE Transactions on Automatic Control, vol.32, issue.2, pp.166-170, 1987.
DOI : 10.1109/TAC.1987.1104530

]. Wang, E. J. Davison, and &. P. Dorato, Observing the states of systems with unmeasurable disturbances, IEEE Transactions on Automatic Control, vol.20, issue.5, pp.716-717, 1975.
DOI : 10.1109/TAC.1975.1101076

G. Wang, &. X. Chen, and . Yu, Anticontrol of chaos in continuous-time systems via time-delay feedback, Chaos: An Interdisciplinary Journal of Nonlinear Science, vol.40, issue.4, pp.771-779, 2000.
DOI : 10.1109/81.246149

X. F. Wang, G. Zhong, K. Tang, K. F. Man, and &. Liu, Generating chaos in Chua's circuit via time-delay feedback, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.48, issue.9, pp.1151-1156, 2001.
DOI : 10.1109/81.948446

S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos

A. Wolf, J. Swift, H. Swinney, and &. J. Vastano, Determining Lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, vol.16, issue.3, pp.285-317, 1985.
DOI : 10.1016/0167-2789(85)90011-9

URL : https://hal.archives-ouvertes.fr/hal-01654059

. W. Wu-93-]-c, . O. Wu-&-l, and . Chua, A SIMPLE WAY TO SYNCHRONIZE CHAOTIC SYSTEMS WITH APPLICATIONS TO SECURE COMMUNICATION SYSTEMS, International Journal of Bifurcation and Chaos, vol.03, issue.06, pp.1619-1627, 1993.
DOI : 10.1142/S0218127493001288

J. Wünnenberg, Observer-based fault detection in dynamic systems, Thèse, 1990.