Evaluation of the largest Lyapunov exponent in dynamical systems with time delay

Chaos, Solitons and Fractals

Volumn 23, Issue 5, 2005, Pages 1651-1659

  Stefanski, Andrzej ^a   Dabrowski, Artur ^a   Kapitaniak, Tomasz ^a  

^a LODZ UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; CHAOS THEORY; DIFFERENTIAL EQUATIONS; EQUATIONS OF MOTION; FEEDBACK; MATHEMATICAL MODELS; NONLINEAR CONTROL SYSTEMS; PARAMETER ESTIMATION; PERTURBATION TECHNIQUES; SYNCHRONIZATION; THEOREM PROVING;

DYNAMICAL SYSTEMS; IDENTICAL SYSTEMS; PERTURBATION COEFFICIENT;

DELAY CIRCUITS;

EID: 9544230703     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0960-0779(04)00428-X     Document Type: Article

References (25)

1. Awrejcewicz J, Wojewoda J. Observation of chaos in a nonlinear oscillator with delay: numerical study. KSME J 1989;3(1):15-24.
2. Benettin G, Galgani L, Strelcyn JM. Kolmogorov entropy and numerical experiment. Phys Rev A 1976;14:2338-45.
3. Benettin G, Galgani L, Giorgilli A, Strelcyn JM. Lyapunov exponents for smooth dynamical systems and Hamiltonian systems; a method for computing all of them, part I: theory. Meccanica 1980; 15:9-20.
4. Benettin G, Galgani L, Giorgilli A, Strelcyn JM. Lyapunov exponents for smooth dynamical systems and Hamiltonian systems; a method for computing all of them, part II: numerical application. Meccanica 1980;15:21-30.
5. Eckmann JP, Kamphorst SO, Ruelle D, Ciliberto S. Lyapunov exponents from a time series. Phys Rev Lett 1986;34(9):4971-9.
6. Fujisaka H, Yamada T. Stability theory of synchronized motion in coupled-oscillator systems. Prog Theor Phys 1983;69(1):32-47.
7. Henon M. A two dimensional map with a strange attractor. Commun Math Phys 1976;50:69.
8. Henon M, Heiles C. The applicability of the third integral of the motion: some numerical results. Astron J 1964;69:77.
9. Hinrichs N, Oestreich M, Popp K. Dynamics of oscillators with impact and friction. Chaos, Solitons & Fractals 1997;4(8):535-58.
10. Kapitaniak T, Sekieta M, Ogorzalek. Monotone synchronization of chaos. Bifurcat Chaos 1996;6:211-5.
11. Müller P. Calculation of Lyapunov exponents for dynamical systems with discontinuities. Chaos, Solitons & Fractals 1995;5(9): 1671-81.
12. Nusse HE, Yorke JA. Dynamics: numerical explorations. New York: Springer-Verlag; 1997.
13. Oseledec VI. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans Moscow Math Soc 1968;19:197-231.
14. Parlitz U. Identification of true and spurious Lyapunov exponents from time series. J Bifurcat Chaos 1992;2(1):155-65.
15. Pecora LM, Carroll TL. Synchronization of chaos. Phys Rev Lett 1990;64:221-4.
16. Plauth RH, Hsieh J-C. Chaos in a mechanism with time delays under parametric and external excitation. J Sound Vibrat 1987;114(1):73-90.
17. Rosenstein MT, Collins JJ, De Luca CJ. A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 1993;65(1,2):117-34.
18. Sano M, Sawada Y. Measurement of the Lyapunov spectrum from a chaotic time series. Phys Rev Lett 1985;55:1082-5.
19. Shimada I, Nagashima T. A numerical approach to ergodic problem of dissipative dynamical systems. Prog Theor Phys 1979;61(6):1605-16.
20. Stefanski A, Kapitaniak T. Using chaos synchronization to estimate the largest Lyapunov exponent of non-smooth systems. Discrete Dyn Nat Soc 2000;4:207-15.
21. Stefanski A. Estimation of the largest Lyapunov exponent in systems with impacts. Chaos, Solitons & Fractals 2000;11(15):2443-51.
22. Stefanski A, Kapitaniak T. Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization. Chaos, Solitons & Fractals 2003;15:233-44.
23. Takens F. Detecting strange attractors in turbulence. Lect Notes Math 1981;898:366.
24. Wolf A, Swift JB, Swinney HL, Vastano JA. Determining Lyapunov exponents from a time series. Physica D 1985;16:285-317.
25. Wolf A. Quantifying chaos with Lyapunov exponents. In: Holden V, editor. Chaos. Manchester: Manchester University Press; 1986. p. 273-90.