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## Determining Lyapunov Exponents from a Time Series (1985)

Venue: | Physica |

Citations: | 495 - 1 self |

### Citations

1558 |
Deterministic nonperiodic flow
- Lorenz
- 1963
(Show Context)
Citation Context ...-, -), a limit cycle; and (-, -, -), a fixed point. Fig. 1 illustrates the expanding, "slower than exponential," and contracting character of the flow for a three,dimensional system, the Lorenz model =-=[23]-=-. (All of the model systems that we will discuss are defined in table I.) Since Lyapunov exponents involve long-time averaged behavior, the short segments of the trajectories shown in the figure canno... |

828 |
Detecting strange attractors in turbulence.
- Takens
- 1981
(Show Context)
Citation Context ...tral estimation for experimental data Experimental data typically consist of discrete measurements of a single observable. The wellknown technique of phase space reconstruction with delay coordinates =-=[2, 33, 34]-=- makes it possible to obtain from such a time series an attractor whose Lyapunov spectrum is identical to that of the original attractor. We have designed a method, conceptually similar to the ODE app... |

428 | The art of computer programming. Vol. 3: Sorting and searching - Knuth - 1973 |

346 | Characterization of strange attractors. - Grassberger, Procaccia - 1983 |

301 |
Geometry from a time series,
- Packard, Crutchfield, et al.
- 1980
(Show Context)
Citation Context ...tral estimation for experimental data Experimental data typically consist of discrete measurements of a single observable. The wellknown technique of phase space reconstruction with delay coordinates =-=[2, 33, 34]-=- makes it possible to obtain from such a time series an attractor whose Lyapunov spectrum is identical to that of the original attractor. We have designed a method, conceptually similar to the ODE app... |

190 | A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems - Oseledec - 1968 |

126 |
An equation for continuous chaos,”
- Rössler
- 1976
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Citation Context ... motion Lyapunov Lyapunov System Parameter spectrum dimension* values (bits/s)t H~non: [25] ~1 = 0.603 X. +1 = 1 - aX;. + Yn { b = 1.4 h 2 = - 2.34 Y. + 1 = bX. = 0.3 (bits/iter.) 1.26 Rossler-chaos: =-=[26]-=- )( = - (Y + Z) [ a = 0.15 )k 1 = 0.13 )'= X+ aY I b = 0.20 ~2 =0.00 = b + Z(X- c) c = 10.0 h 3 = - 14.1 2.01 Lorenz: [23] )(= o(Y- X) [ o = 16.0 h 1 = 2.16 ~'= X( R- Z)- Y I R=45.92 X 2 =0.00 = XY - ... |

119 | Esitmation of Kolmogorov entropy from a chaotic signal", - Grassberger, Procaccia - 1983 |

91 |
Chaotic behavior of multidimensional difference equations. Functional Differential Equations and Approximations of Fixed Points.
- KAPLAN, YORKE
- 1979
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Citation Context ...imension-like quantities, including the fractal dimension, information dimension, and the correlation exponent; the difference between them is often small. It has been conjectured by Kaplan and Yorke =-=[28, 29]-=- that the information dimension d r is related to the Lyapunov spectrum by the equation Ei-- 1~i df=J+ I?~j+il ' (2) where j is defined by the condition that j j+l E)~i> 0 and EX,<O. (3) i--1 i--1 The... |

69 |
entropy and Lyapunov exponents’, Ergodic Theory Dynam
- Young, ‘Dimension
- 1982
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Citation Context ...imension-like quantities, including the fractal dimension, information dimension, and the correlation exponent; the difference between them is often small. It has been conjectured by Kaplan and Yorke =-=[28, 29]-=- that the information dimension d r is related to the Lyapunov spectrum by the equation Ei-- 1~i df=J+ I?~j+il ' (2) where j is defined by the condition that j j+l E)~i> 0 and EX,<O. (3) i--1 i--1 The... |

61 |
Strange attractors, chaotic behavior, and information flow Z.
- Shaw
- 1981
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Citation Context ...ost. Any system containing at least one positive Lyapunov exponent is defined to be chaotic, with the magnitude of the exponent reflecting the time scale on which system dynamics become unpredictable =-=[10]-=-. For systems whose equations of motion are explicitly known there is a straightforward technique [8, 9] for computing a complete Lyapunov spectrum. This method cannot be applied directly to experimen... |

57 | A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems. - Shimada, Nagashima, et al. - 1979 |

43 |
An equation for hyperchaos,”
- Rössler
- 1979
(Show Context)
Citation Context ...issipative system there are three possible types of strange attractors: their Lyapunov spectra are (+, +,0,-), (+,0,0,-), and (+,0,-,-). An example of the first type is Rossler's hyperchaos attractor =-=[24]-=- (see table I). For a given system a change in parameters will generally change the Lyapunov spectrum and may also change both the type of spectrum and type of attractor. The magnitudes of the Lyapuno... |

31 |
Oscillations and Chaos in
- Glass, Mackey
- 1977
(Show Context)
Citation Context ... ;k 3 = - 32.4 2.07 Rossler-hyperchaos: [24] Jr'= - (Y+ Z) ( a = 0.25 A t = 0.16 )'= X+ aY+ W [ b= 3.0 X 2 =0.03 = b + XZ | c = 0.05 h 3 = 0.00 if" = cW - dZ k d = 0.5 h4 = - 39.0 3.005 Mackey-Glass: =-=[27]-=- ( a = 0.2 h t = 6.30E-3 j( = aX(t + s ) - bX(t) / b = 0.1 )~2 = 2.62E-3 1 + [ X(t + s)] c ) c = 10.0 IX31 < 8.0E-6 s = 31.8 )'4 = - 1.39E-2 3.64 tA mean orbital period is well defined for Rossler cha... |

28 | Fluctuations and simple chaotic dynamics
- Crutchfield, Farmer
- 1982
(Show Context)
Citation Context ...rom a time series clear that the system possesses well defined exponents that are potentially recoverable. Strictly speaking, in the latter case Lyapunov exponents are not well defined, but some work =-=[37]-=- has suggested that a system may be characterized by numbers that are the Lyapunov exponents for the noise-free system averaged over the range of noise-induced states. Our first study of the effects o... |

22 |
Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors.
- GREENSIDE, WOLF
- 1982
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Citation Context ... 1. With experimental data that appear to define an approximately two-dimensional attractor, an independent calculation of df from its definition (feasible for attractors of dimension less than three =-=[35]-=-) may justify this approach to estimating hx. 6. Implementation details 6.1. Selection of embedding dimension and delay time In principle, when using delay coordinates to reconstruct an attractor, an ... |

15 | Chaotic mode competition in parametrically forced surface waves,’’ - Ciliberto, Gollub - 1985 |

15 |
At Least One Lyapunov Exponent Vanishes if the Trajectory of an Attractor does not Contain a Fixed Point,”
- Haken
- 1983
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Citation Context ...fined by the first three principal axes grows as 2 (x'+x2+x~)t, and so on. This property yields another definition of the spectrum of exponents: tWhile the existence of this limit has been questioned =-=[8, 9, 22]-=-, the fact is that the orbital divergence of any data set may be quantified. Even if the limit does not exist for the underlying system, or cannot be approached due to having finite amounts of noisy d... |

14 |
Low-dimensional chaos in a hydrodynamic system
- Brandstäter, Swift, et al.
- 1983
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Citation Context ...ng fails to help with exponent estimation if there is substantial contamination of the data at frequencies lower than the filter cutoff. In a simple study of multiperiodic data with added white noise =-=[3]-=- the estimated exponent returned (very nearly) to zero for a sufficient amount of filtering. It thus appears that in some cases external noise can be distinguished from chaos by this procedure.A. Wol... |

10 |
Dimension of strange attractors,
- Russell, Hanson, et al.
- 1980
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Citation Context ...between d r (a static property of an attracting set) and the Lyapunov290 ,4. 14/olf et aL / Determining Lyapunov exponents from a time series exponents appears to be satisfied for some model systems =-=[30]-=-. The calculation of dimension from this equation requires knowledge of all but the most negative Lyapunov exponents. 3. Calculation of Lyapunov spectra from differential equations Our algorithms for ... |

9 |
The dimension of chaotic attractors, Physica o 7
- Farmer, Yorke
- 1983
(Show Context)
Citation Context ...puted for each system with the code in appendix A. ~As defined in eq. (2). The Lyapunov spectrum is closely related to the fractional dimension of the associated strange attractor. There are a number =-=[19]-=- of different fractional-dimension-like quantities, including the fractal dimension, information dimension, and the correlation exponent; the difference between them is often small. It has been conjec... |

8 | Dimension of strange attractor: an experimental determination for the chaotic regime of two convective systems - Malraison, Atten, et al. |

6 |
A two-dimensional mapping with a strange attractor
- H&non
- 1976
(Show Context)
Citation Context ...odel systems considered in this paper and their Lyapunov spectra and dimensions as computed from the equations of motion Lyapunov Lyapunov System Parameter spectrum dimension* values (bits/s)t H~non: =-=[25]-=- ~1 = 0.603 X. +1 = 1 - aX;. + Yn { b = 1.4 h 2 = - 2.34 Y. + 1 = bX. = 0.3 (bits/iter.) 1.26 Rossler-chaos: [26] )( = - (Y + Z) [ a = 0.15 )k 1 = 0.13 )'= X+ aY I b = 0.20 ~2 =0.00 = b + Z(X- c) c = ... |

5 | Chaos in the Belousov-Zhabotinskii reaction - HUDSON, MANKIN - 1981 |

5 |
The Ergodic Theory of Axiom-A Flows",
- Bowen, Ruelle
- 1975
(Show Context)
Citation Context ...e of this curve (but not its absolute magnitude, see reference [3]) was independently verified by the calculation of the metric entropy h~-which is equal to X 1 if there is a single positive exponent =-=[38]-=-. nent using 1-D map analysis [2] as a comparison. Our algorithm gives a result in the plateau region of 0.0054 + 0.0005 bits/s, while the 1-D map estimation yiclds a result of 0.0049 + 0.0010 bits/s.... |

4 |
i n computing Lyapunov exponents from experimental data. I n
- Wolf, Swift, et al.
- 1984
(Show Context)
Citation Context ... and quality of the data as well as on the complexity of the dynamical system. We have tested our method on model dynamical systems with known spectra and applied it to experimental data for chemical =-=[2, 13]-=- and hydrodynamic [3] strange attractors. Although the work of characterizing chaotic data is still in its infancy, there have been many approaches to quantifying chaos, e.g., fractal power spectra [1... |

3 | Method for calculating a Lyapunov exponent - Wright - 1984 |

2 |
Observation of a strange attractor, Physica 8D
- Roux, Simoyi, et al.
- 1983
(Show Context)
Citation Context ... and quality of the data as well as on the complexity of the dynamical system. We have tested our method on model dynamical systems with known spectra and applied it to experimental data for chemical =-=[2, 13]-=- and hydrodynamic [3] strange attractors. Although the work of characterizing chaotic data is still in its infancy, there have been many approaches to quantifying chaos, e.g., fractal power spectra [1... |

2 |
Trajectory divergence for coupled relaxation oscillators: measurements and models
- Gollub, Romer, et al.
- 1980
(Show Context)
Citation Context ...ll be discussed later. We will describe a technique which for the first time yields estimates of the non-negative Lyapunov exponents from finite amounts of experimental data. A less general procedure =-=[6, 11-14]-=- for estimating only the dominant Lyapunov exponent in experimental systems has been used for some time. This technique is limited to systems where a welldefined one-dimensional (l-D) map can be recov... |

2 | Symbolic dynamics of noisy chaos, Physica 7D - Crutchfield, Packard - 1983 |

2 | The Dripping Faucet - Shaw - 1984 |

1 | the references in: H.L. Swinney, "Observations of Order and Chaos in Nonlinear Systems - See - 1983 |

1 | Dimension Measuremerits for CJeostrophic Turbulence - Guckenheimer, Buzyna - 1983 |

1 |
et al. / Determining Lyapunov exponents from a time series 317 "Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems; A Method for Computing All of Them," Meccanica 15
- Benettin, Galgani, et al.
- 1980
(Show Context)
Citation Context ...magnitude of the exponent reflecting the time scale on which system dynamics become unpredictable [10]. For systems whose equations of motion are explicitly known there is a straightforward technique =-=[8, 9]-=- for computing a complete Lyapunov spectrum. This method cannot be applied directly to experimental data for reasons that will be discussed later. We will describe a technique which for the first time... |

1 | Experiment on Chaotic Response of Forced Belousov-Zhabotinskii Reaction - Nagashima - 1982 |

1 |
Power of Chaos," Physica 3D
- Blacher, Perdang
- 1981
(Show Context)
Citation Context ...3] and hydrodynamic [3] strange attractors. Although the work of characterizing chaotic data is still in its infancy, there have been many approaches to quantifying chaos, e.g., fractal power spectra =-=[15]-=-, entropy [16-18, 3], and fractal dimension [proposed in ref. 19, used in ref. 3-5, 20, 21]. We have tested many of these algorithms on both model and experimental data, and despite the claims of thei... |

1 |
private communication
- RueUe
(Show Context)
Citation Context ...was uncommonly slow. +We are aware of an attempt to estimate Lyapunov spectra from experimental data through direct estimation of local Jacobian matrices and formation of the long time product matrix =-=[31]-=-. This calculation is essentially the same as ours (we avoid matrix notation by diagonalizing the system at each step) and has the same problems of sensitivity to external noise, and to the amount and... |

1 |
Quasiperiodicity in chemical dynamics
- Roux, Rossi
- 1984
(Show Context)
Citation Context ...stems shows chaos on large and small length scales respectively. In the Texas attractor [2] a), the separation between a single pair of points is shown for one orbital period. In the French attractor =-=[36]-=-; b), the separation between a pair of points is shown for two periods. Estimation of Lyapunov exponents is quite difficult for the latter system. perimental data. There are three somewhat related rea... |