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Journal ArticleDOI

Determining Lyapunov exponents from a time series

01 Jul 1985-Physica D: Nonlinear Phenomena (Elsevier)-Vol. 16, Iss: 3, pp 285-317
TL;DR: In this article, the authors present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series, which provide a qualitative and quantitative characterization of dynamical behavior.
About: This article is published in Physica D: Nonlinear Phenomena.The article was published on 1985-07-01 and is currently open access. It has received 8128 citations till now. The article focuses on the topics: Lyapunov exponent & Lyapunov equation.

Summary (4 min read)

1. Introduction

  • This section includes table I which summarizes the model systems that are used in this paper.
  • Section 3 is a review of the calculation of the complete spectrum of exponents for systems in which the defining differential equations are known.
  • Which to their knowledge has not been published elsewhere.
  • In section 5 the authors describe the algorithms for estimating the two largest exponents.
  • The authors method is applied to model systems and experimental data in section 8, and the conclusions are given in section 9.

t--,oo t pc(O)'

  • Thus the Lyapunov exponents are related to the expanding or contracting nature of different directions in phase space.
  • Since the orientation of the ellipsoid changes continuously as it evolves, the directions associated with a given exponent vary in a complicated way through the attractor.
  • (See section 7. 1.) the sum of the first j exponents is defined by the long term exponential growth rate of a j-volume element.
  • The magnitudes of the Lyapunov exponents quantify an attractor's dynamics in information theoretic terms.
  • After this time the small initial uncertainty will essentially cover the entire attractor, reflecting 20 bits of new information that can be gained from an ad: ditional measurement of the system.

4. An approach to spectral estimation for experimental data

  • In principle the authors might attempt the estimation of negative exponents by going to higher-dimensional volume elements, but information about contracting phase space directions is often impossible to resolve.
  • In a system where fractal structure can be resolved, there is the difficulty that the volume elements involving negative exponent directions collapse exponentially fast, and are therefore numerically unstable for experimental data (see section 7.1).

5. Spectnd algorithm implementation

  • The authors have implemented several versions of their algorithms including simple "fixed evolution time" programs for ~'1 and X1 ÷ hE, "variable evolution time" programs for XI+~:, and "interactive" programs that are used on a graphics machinet.
  • This program is not sophisticated, but it is concise, easily understood, and useful for learning about their technique.
  • The authors can also provide a highly efficient data base management algorithm that can be used in any of their programs to eliminate the expensive process of exhaustive search for nearest neighbors.
  • The authors now discuss the fixed evolution time program for A t and the variable evolution time program for ~x + h2 in some detail.

6.1. Selection of embedding dimension and delay time

  • In their data [2] for the Belousov-Zhabotinskii chemical reaction (see fig.
  • In the first case the authors obtain a "pencil-like" region which obscures the folding region of the attractor.
  • This structure opens up and grows larger relative to the total extent of the attractor for the larger values of ~', which is clearly desirable for their algorithms.
  • A check of the stationarity of exponent estimates with t-is again recommended.

6.2. Evolution times between replacements

  • In the Lorenz attractor, the separatrix between the two lobes of the attractor is not a good place to find a replacement dement.
  • An element chosen here is likely to contain points that will almost immediately fly to opposite lobes, providing an enormous contribution to an exponent estimate.
  • This effect is certainly related to the chaotic nature of the attractor, but is not directly related to the values of the Lyapunov exponents.
  • This has the same effect as the catastrophes that can arise from too low a value of embedding dimension as discussed in section 6.1.
  • While the authors are not aware of any foolproof approach to detecting troublesome regions of attractors it may be possible for an exponent program to avoid catastrophic replacements.

6.3. Shorter lengths versus orientation errors

  • In this case their algorithm is closely related to obtaining the Lyapunov exponent of the map through a determination of its local slope profile [13] .
  • The transverse vector component plays the role of the chord whose image under the map provides a slope estimate.
  • This chord should obviously be no longer than the smallest resolvable structure in the 1-D map, a highly system-dependent quantity.
  • Since the underlying maps of commonly studied model and physical systems have not had much detailed structure on small length scales (consider the logistic equation, cusp maps, and the Belousov-Zhabotinskii map [2] ) the authors have somewhat arbitrarily decided to consider 5-10% of the transverse attractor extent as the maximum acceptable magnitude of a vector's transverse component.

6.4. The accumulation of orientation errors

  • If the authors can manage to evolve the vector for two orbits, the permissible initial orientation error is about 27 degrees, and so on.
  • The authors see that a given orientation error at replacement time shrinks to a value negligible compared to the next orientation error, provided that propagation times are long enough.
  • The ease of estimating the i th exponent depends on how small the quantity 2 (x'*~-x,)tr is.
  • Problems arise when successive exponents are very close or identical.

7.1. Probing small length scales

  • Even though infinitesimal length scales are not accessible, Lyapunov exponent estimation may still be quite feasible for many experimental systems.
  • The same problem arises in calculations of the fractal dimension of strange attractors.
  • In these calculations the authors hope that as the smallest accessible length scales are approached, scaling converges to the zero length scale limit.
  • Similarly, provided that chaos production is nearly the same on infinitesimal and the smallest accessible length scales, their calculations on the small scales may provide accurate results.
  • A successful calculation requires that one has enough data to approach the appropriate length scales, ignores anything on the length scale of the noise, and has an attractor with a macroscopic stretching/folding mechanism.

7.2. Noise

  • It is ironic that measurement noise is not a problem unless large amounts of data are available to define the attractor.
  • Noise is only detectable when the point density is high enough to provide replacements near the noise length scale.
  • This suggests a simple approach to the noise problem, simply avoiding principal axis vectors whose magnitude is smaller than some threshold value the authors select.
  • Avoiding noisy length scales is not a trivial matter, as noise may not be of constant amplitude throughout an attractor and the noise length scale may be difficult to determine.
  • 6d the authors confirm that a straightforward small distancg cutoff works in the case of the Rossler attractor by showing stationarity of the estimated exponent over a broad range of cutoff values.

7.3. Low pass filtering

  • Filtering dramatically altered the shape of the reconstructed attractor, but the estimated values of hi differed by at most a few percent for reasonable cutoff frequencies.
  • A similar calculation for the Rossler attractor indicated that the low-frequency cutoff could be moved all the way down to the attractor's sole large spectral feature before the exponent estimate showed any noticeable effect.
  • 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.
  • Unfiltered experimental data for the Belousov-Zhabotinskii reaction [2] ; b) the same data, low-pass filtered with a filter cutoff of 0.046 Replacement frequencies in the region of stationarity for these results were approximately 0.005 Hz. c) the data are over-filtered at 0.023 H_z. h 1 differs by only 20% from the exponent estimate for unfiltered data.

N i~=l~ki od_j

  • The authors have ignored several prefactors that might modify these estimates by at most an order of magnitude.
  • From these values eq. ( 24) predicts to first order that between = 10 d and -30d points are necessary to fill out a d-dimensional attractor, independent of how many nonnegative exponents the authors are calculating.
  • Note that lengthening the time series not only allows more time for convergence but also provides more replacement candidates at each replacement step.
  • This would provide the same sampling resolution for the slope profile of the map (see ref. 13 ) in each dimension.
  • The last and simplest point the authors consider is the required number of points per orbit, P.

8. Results

  • The authors emphasize that no explicit use was made of the differential equations defining the model systems, except to produce a dynamical observable (the x-coordinate time series) which was then treated as experimental data.
  • For the equations that define each system see table I.
  • The quoted uncertainty values for each system were calculated either from the known values of the exponents or from the variation of their results with changes in parameters.

9. Conclusions

  • The structure of this program is very similar to the program for hi: locate the two nearest neighbors of the first delay coordinate point, determine the initial area defined by this triple, enter the main program loop for repeated evolution and replacement.
  • Triples are evolved EVOLV steps through the attractor and replacement is performed.
  • The authors approach to triple replacement is a two step process; first the authors find a small set of points which, together with the fiducial trajectory, define small separation vectors and lie close to the required two-dimensional subspace.
  • The authors then determine which of all of the possible pairs of these points will make the best replacement triple.
  • The relative importance of replacement lengths, skewness, orientation changes, etcetera, are weighted by the user chosen factors.

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Physica 16D (1985)285-317
North-Holland, Amsterdam
DETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIES
Alan WOLF~-, Jack B. SWIFT, Harry L. SWINNEY and John A. VASTANO
Department of Physics, University of Texas, Austin, Texas 78712, USA
Received 18 October 1984
We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time
series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to
the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov
exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be
applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor.
The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii
reaction and Couette-Taylor flow.
Contents
1. Introduction
2. The Lyapunov spectrum defined
3. Calculation of Lyapunov spectra from differential equations
4. An approach to spectral estimation for experimental data
5. Spectral algorithm implementation*
6. Implementation details*
7. Data requirements and noise*
8. Results
9. Conclusions
Appendices*
A. Lyapunov spectrum program for systems of differential
equations
B. Fixed evolution time program for ~'1
1. Introduction
Convincing evidence for deterministic chaos has
come from a variety of recent experiments [1-6]
on dissipative nonlinear systems; therefore, the
question of detecting and quantifying chaos has
become an important one. Here we consider the
spectrum of Lyapunov exponents [7-10], which
has proven to be the most useful dynamical di-
agnostic for chaotic systems. Lyapunov exponents
are the average exponential rates of divergence or
tPresent address: The Cooper Union, School of Engineering,
N.Y., NY 10003, USA.
*The reader may wish to skip the starred sections at a first
reading.
convergence of nearby orbits in phase space. Since
nearby orbits correspond to nearly identical states,
exponential orbital divergence means that systems
whose initial differences we may not be able to
resolve will soon behave quite differently-predic-
tive ability is rapidly lost. 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 ex-
plicitly known there is a straightforward technique
[8, 9] for computing a complete Lyapunov spec-
trum. This method cannot be applied directly to
experimental data for reasons that will be dis-
cussed later. We will describe a technique which
for the first time yields estimates of the non-nega-
tive Lyapunov exponents from finite amounts of
experimental data.
A less general procedure [6, 11-14] for estimat-
ing only the dominant Lyapunov exponent in ex-
perimental systems has been used for some time.
This technique is limited to systems where a well-
defined one-dimensional (l-D) map can be re-
covered. The technique is numerically unstable
and the literature contains several examples of its
improper application to experimental data. A dis-
cussion of the 1-D map calculation may be found
0167-2789/85/$03.30 © Elsevier Science Publishers
(North-H01!and Physics Publishing Division)

286
A. Wolf et al. / Determining Lyapunov exponents from a time series
in ref. 13. In ref. 2 we presented an unusually
robust 1-D map exponent calculation for experi-
mental data obtained from a chemical reaction.
Experimental data inevitably contain external
noise due to environmental fluctuations and limited
experimental resolution. In the limit of an infinite
amount of noise-free data our approach would
yield Lyapunov exponents by definition. Our abil-
ity to obtain good spectral estimates from experi-
mental data depends on the quantity 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 ap-
proaches to quantifying chaos, e.g., fractal power
spectra [15], entropy [16-18, 3], and fractal dimen-
sion [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 their proponents we have found that
these approaches often fail to characterize chaotic
data. In particular, parameter independence, the
amount of data required, and the stability of re-
suits with respect to external noise have rarely
been examined thoroughly.
The spectrum of Lyapunov exponents will be
defined and discussed in section 2. This section
includes table I which summarizes the model sys-
tems that are used in this paper. Section 3 is a
review of the calculation of the complete spectrum
of exponents for systems in which the defining
differential equations are known. Appendix A con-
tains Fortran code for this calculation, which to
our knowledge has not been published elsewhere.
In section 4, an outline of our approach to estimat-
ing the non-negative portion of the Lyapunov
exponent spectrum is presented. In section 5 we
describe the algorithms for estimating the two
largest exponents. A Fortran program for de-
termining the largest exponent is contained in
appendix B. Our algorithm requires input parame-
ters whose selection is discussed in section 6. Sec-
tion 7 concerns sources of error in the calculations
and the quality and quantity of data required for
accurate exponent estimation. Our method is ap-
plied to model systems and experimental data in
section 8, and the conclusions are given in
section 9.
2. The Lyapunov spectrum defined
We now define [8, 9] the spectrum of Lyapunov
exponents in the manner most relevant to spectral
calculations. Given a continuous dynamical sys-
tem in an n-dimensional phase space, we monitor
the long-term evolution of an
infinitesimal
n-sphere
of initial conditions; the sphere will become an
n-ellipsoid due to the locally deforming nature of
the flow. The ith one-dimensional Lyapunov expo-
nent is then defined in terms of the length of the
ellipsoidal principal axis
pi(t):
h~ = lim 1 log 2
pc(t)
t--,oo t pc(O)'
(1)
where the )h are ordered from largest to smallestt.
Thus the Lyapunov exponents are related to the
expanding or contracting nature of different direc-
tions in phase space. Since the orientation of the
ellipsoid changes continuously as it evolves, the
directions associated with a given exponent vary in
a complicated way through the attractor. One can-
not, therefore, speak of a well-defined direction
associated with a given exponent.
Notice that the linear extent of the ellipsoid
grows as 2 htt, the area defined by the first two
principal axes grows as 2 (x~*x2)t, the volume de-
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 data, Lyapunov exponent estimates could still provide
a useful characterization of a given data set. (See section 7.1.)

A. Wolf et aL / Determining Lyapunov exponents from a time series
287
the sum of the first j exponents is defined by the
long term exponential growth rate of a j-volume
element. This alternate definition will provide the
basis of our spectral technique for experimental
data.
Any continuous time-dependent dynamical sys-
tem without a fixed point will have at least one
zero exponent [22], corresponding to the slowly
changing magnitude of a principal axis tangent to
the flow. Axes that are on the average expanding
(contracting) correspond to positive (negative) ex-
ponents. The sum of the Lyapunov exponents is
the time-averaged divergence of the phase space
velocity; hence any dissipative dynamical system
will have at least one negative exponent, the sum
of all of the exponents is negative, and the post-
transient motion of trajectories will occur on a
zero volume limit set, an attractor.
The exponential expansion indicated by a posi-
tive Lyapunov exponent is incompatible with mo-
tion on a bounded attractor unless some sort of
folding
process merges widely separated trajecto-
ries. Each positive exponent reflects a "direction"
in which the system experiences the repeated
stretching and folding that decorrelates nearby
states on the attractor. Therefore, the long-term
behavior of an initial condition that is specified
with
any
uncertainty cannot be predicted; this is
chaos. An attractor for a dissipatiVe system with
one or more positive Lyapunov exponents is said
to be "strange" or "chaotic".
The signs of the Lyapunov exponents provide a
qualitative picture of a system's dynamics. One-
dimensional maps are characterized by a single
Lyapunov exponent which is positive for chaos,
zero for a marginally stable orbit, and negative for
a periodic orbit. In a three-dimensional continuous
dissipative dynamical system the only possible
spectra, and the attractors they describe, are as
follows: (+,0,-), a strange attractor; (0,0,-), a
two-toms; (0, -, -), a limit cycle; and (-, -, -),
a fixed point. Fig. 1 illustrates the expanding,
"slower than exponential," and contracting char-
acter 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 cannot be expected to accu-
rately characterize the positive, zero, and negative
exponents; nevertheless, the three distinct types of
behavior are clear. In a continuous four-dimen-
sional dissipative 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 hyper-
chaos 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 Lyapunov exponents
quantify
an attractor's dynamics in information
theoretic terms. The exponents measure the rate at
which system processes create or destroy informa-
tion [10]; thus the exponents are expressed in bits
of information/s or bits/orbit for a continuous
system and bits/iteration for a discrete system.
For example, in the Lorenz attractor the positive
exponent has a magnitude of 2.16 bits/s (for the
parameter values shown in table I). Hence if an
initial point were specified with an accuracy of one
part per million (20 bits), the future behavior
could not be predicted after about 9 s [20 bits/(2.16
bits/s)], corresponding to about 20 orbits. After
this time the small initial uncertainty will essen-
tially cover the entire attractor, reflecting 20 bits of
new information that can be gained from an ad:
ditional measurement of the system. This new
information arises from scales smaller than our
initial uncertainty and results in an inability to
specify the state of the system except to say that it
is somewhere on the attractor. This process is
sometimes called an information gain- reflecting
new information from the heat bath, and some-
times is called an information loss-bits shifted
out of a phase space variable
"register"
when bits
from the heat bath are shifted in.
The average rate at which information con-
tained in transients is lost can be determined from

288
A. Wolf et al. / Determining Lyapunov exponents from a time series
° o °°°.°•°•°°
". . t "." :....
".•
. . ..-. .... .. "•.. .... , :.~ "'.•'. •...
. ...,--.:-:.::-:.........
. .,:'..~..--..:~::.-.:..:'..:..:.. .. .. ,~ .,.:.. :'.."
-...•.. start •.'~"." .-
"" ":'" :"::"'" N~ "
~" ""
~'"" """
".':':"'( (,~'."~,m
~" " ":'"'"'""
. -'.:..""%V4";':'"
"'- ".'... " ....
~ " .
- ..,..:/~ . : .... "... ..........
i IIIIIl[lll
time-~
..°•.°.°°
-.. o,-:... -..
. ..... .. ,:,,.::. ".•. •.
(b) .
..',.<'."~::.:.':.. "... "-.. ,...~.'~:'-:,~:..~ "r.. :-:
. ~ ..... .:'~.?'-,',;~ "x -~il~ I - " . ..
" "~,~'.:~'~",-~,~'~xxx ~.~'- .'.• .:
" :::,.!...":"""-...-
"
". ":::,.f,~,_,,,~.~ -,.-, ,.,. -...: ......
$:L~. .... ~-"~.'...__'••.~.. ..:-.:..." .... ::.:..
;
:
.-'.~...~.;.;.,.... .... ...• ............
, " ".:,v':.:~ .:" "~.'. "'"... . . " " :
~start
[l,,m,,,,l,,,,,llli,,lllli,,dll
time
--~
.
..-.-..... :; ..- ---~..- : ..
... ;;.~.....~.::........... s,,,,
~.~,.~;,..':..::.~:"~-:.v
:'"
",',...:~.'.-2~'W~'".~-... ".".'.. "'::...':"'.:
--" ". "-
: "
., , " •~,'•••• .• •"
.........
time
Fig. 1. The short term evolution of the separation vector between three carefully chosen pairs of nearby points is shown for the
Lorenz attractor, a) An expanding direction (~1 > 0); b) a "slower than exponential" direction (~'2 = 0); C) a contracting direction
(X3 < 0).
the negative exponents• The asymptotic decay of a
perturbation to the attractor is governed by the
least negative exponent, which should therefore be
the easiest of the negative exponents to estimatet.
tWe have been quite successful with an algorithm for de-
termiuing the dominant (smallest magnitude) negative expo-
nent from pseudo-experimental data (a single time series ex-
tracted from the solution of a model system and treated as an
experimental observable) for systems that are nearly integer-
dimensional. Unfortunately, our approach, which involves mea-
suring the mean decay rate of many induced perturbations of
the dynamical system, is unlikely to work on many experimen-
tal systems. There are several fundamental problems with the
calculation of negative exponents from experimental data, but
For the Lorenz attractor the negative exponent is
so large that a perturbed orbit typically becomes
indistinguishable from the attractor, by "eye", in
less than one mean orbital period (see fig. 1).
of greatest importance is that
post-transient
data may not
contain resolvable negative exponent information and
per-
turbed
data must refl~t properties of the unperturbed system,
that is, perturbations must only change the state of the system
(current values of the dynamical variables). The response of a
physical system to a non-delta function perturbation is difficult
to interpret, as an orbit separating from the attractor may
reflect either a locally repelling region of the attractor (a
positive contribution to the negative exponent) or the finite rise
time of the perturbation.

A. Wolf et al. / Determining Lyapunov exponents from a time series
289
Table I
The model 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.)
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
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 - bZ b =
4.0 ;k 3 = - 32.4
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
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
1.26
2.01
2.07
3.005
3.64
tA mean orbital period is well defined for Rossler chaos (6.07 seconds) and for hyperchaos (5.16 seconds) for the parameter values
used here. For the Lorenz attractor a characteristic time (see footnote- section 3) is about 0.5 seconds. Spectra were computed 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 at-
tractor. There are a number [19] of different frac-
tional-dimension-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 conjectured relation between d r (a
static
property of an attracting set) and the Lyapunov

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3,164 citations

Journal ArticleDOI
TL;DR: A new method for calculating the largest Lyapunov exponent from an experimental time series is presented that is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level.

2,942 citations

Journal ArticleDOI
TL;DR: COPASI is presented, a platform-independent and user-friendly biochemical simulator that offers several unique features, and numerical issues with these features are discussed; in particular, the criteria to switch between stochastic and deterministic simulation methods, hybrid deterministic-stochastic methods, and the importance of random number generator numerical resolution in Stochastic simulation.
Abstract: Motivation: Simulation and modeling is becoming a standard approach to understand complex biochemical processes. Therefore, there is a big need for software tools that allow access to diverse simulation and modeling methods as well as support for the usage of these methods. Results: Here, we present COPASI, a platform-independent and user-friendly biochemical simulator that offers several unique features. We discuss numerical issues with these features; in particular, the criteria to switch between stochastic and deterministic simulation methods, hybrid deterministic--stochastic methods, and the importance of random number generator numerical resolution in stochastic simulation. Availability: The complete software is available in binary (executable) for MS Windows, OS X, Linux (Intel) and Sun Solaris (SPARC), as well as the full source code under an open source license from http://www.copasi.org. Contact: mendes@vbi.vt.edu

2,351 citations


Cites background from "Determining Lyapunov exponents from..."

  • ...Since the users are provided with an interface following...

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References
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Journal ArticleDOI
TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into consider­ably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.

16,554 citations


"Determining Lyapunov exponents from..." refers background in this paper

  • ...1 illustrates the expanding, "slower than exponential," and contracting character of the flow for a three,dimensional system, the Lorenz model [23]....

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Book ChapterDOI
01 Jan 1981

9,756 citations


"Determining Lyapunov exponents from..." refers background or methods in this paper

  • ...We should obtain an embedding if m is chosen to be greater than twice the dimension of the underlying attractor [34]....

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  • ...Given the time series x(t), an m-dimensional phase portrait is reconstructed with delay coordinates [2, 33, 34], i....

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  • ...In principle, when using delay coordinates to reconstruct an attractor, an embedding [34] of the original attractor is obtained for any sufficiently large m and almost any choice of time delay ~-, but in practice accurate exponent estimation requires some care in choosing these two parameters....

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  • ...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....

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Journal ArticleDOI
TL;DR: In this article, a measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable, and the relation of this measure to fractal dimension and information-theoretic entropy is discussed.
Abstract: A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and information-theoretic entropy is discussed.

4,323 citations

Journal ArticleDOI
15 Jul 1977-Science
TL;DR: First-order nonlinear differential-delay equations describing physiological control systems displaying a broad diversity of dynamical behavior including limit cycle oscillations, with a variety of wave forms, and apparently aperiodic or "chaotic" solutions are studied.
Abstract: First-order nonlinear differential-delay equations describing physiological control systems are studied. The equations display a broad diversity of dynamical behavior including limit cycle oscillations, with a variety of wave forms, and apparently aperiodic or "chaotic" solutions. These results are discussed in relation to dynamical respiratory and hematopoietic diseases.

3,839 citations


Additional excerpts

  • ...Mackey-Glass: [27]...

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Journal ArticleDOI
TL;DR: In this paper, the existence of low-dimensional chaotic dynamical systems describing turbulent fluid flow was determined experimentally by reconstructing phase-space pictures from the observation of a single coordinate of any dissipative dynamical system and determining the dimensionality of the system's attractor.
Abstract: It is shown how the existence of low-dimensional chaotic dynamical systems describing turbulent fluid flow might be determined experimentally. Techniques are outlined for reconstructing phase-space pictures from the observation of a single coordinate of any dissipative dynamical system, and for determining the dimensionality of the system's attractor. These techniques are applied to a well-known simple three-dimensional chaotic dynamical system.

3,628 citations


"Determining Lyapunov exponents from..." refers background or methods in this paper

  • ...Given the time series x(t), an m-dimensional phase portrait is reconstructed with delay coordinates [2, 33, 34], i....

    [...]

  • ...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....

    [...]