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Lyapunov exponents of classical chaotic systems
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Numbers
system
$i$ 
$\lambda_i$
logistic map $x_{n+1}=4x_n(1-x_n)$
1:
0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
comment: $\lambda_1=\log 2$ for the logistic map at $r=4$ [4] [3].
tent map $x_{n+1}=2\min(x_n,1-x_n)$
1:
0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
comment: $\lambda_1=\log 2$ for the full tent map $x\mapsto 2\min(x,1-x)$ [5].
doubling map $x_{n+1}=2x_n\bmod 1$
1:
0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
comment: $\lambda_1=\log 2$ for the doubling map $x\mapsto 2x\bmod 1$ [6].
continued-fraction Gauss map $x_{n+1}=1/x_n-\lfloor1/x_n\rfloor$
1:
2.373138220831250905643445951894474241136713072944108671908508597305619264112508886600696602216973752
comment: $\lambda_1=\pi^2/(6\log2)$ for the continued-fraction Gauss map [7]; this is twice the Khinchin-Levy constant in the table of regular continued-fraction constants.
Arnold's cat map, matrix $\begin{pmatrix}2&1\\1&1\end{pmatrix}$
1:
0.9624236501192068949955178268487368462703686687713210393220363376803277352164435488240188582454469500
comment: The expanding eigenvalue of Arnold's cat-map matrix is $(3+\sqrt5)/2$ [8].
Arnold's cat map, matrix $\begin{pmatrix}2&1\\1&1\end{pmatrix}$
2:
-0.9624236501192068949955178268487368462703686687713210393220363376803277352164435488240188582454469500
comment: The contracting eigenvalue of Arnold's cat-map matrix is $(3-\sqrt5)/2$, so $\lambda_2=-\lambda_1$.
baker's map with stretch factor $2$, folded or unfolded
1:
0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
comment: The baker's map expands one coordinate by $2$ [9].
baker's map with stretch factor $2$, folded or unfolded
2:
-0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
comment: The baker's map contracts the other coordinate by $1/2$; the folded and unfolded versions have the same Lyapunov spectrum.
Henon map $x_{n+1}=1+y_n-1.4x_n^2$, $y_{n+1}=0.3x_n$
1:
0.41922 †
comment: Sprott lists the Henon map at $a=1.4$, $b=0.3$ [3].
Henon map $x_{n+1}=1+y_n-1.4x_n^2$, $y_{n+1}=0.3x_n$
2:
-1.62319 †
comment: For the Henon map at $a=1.4$, $b=0.3$, the exponent sum is $\log 0.3$.
Chirikov standard map $y_{n+1}=y_n+\sin x_n\bmod 2\pi$, $x_{n+1}=x_n+y_{n+1}\bmod 2\pi$
1:
0.10497 †
comment: Sprott lists the Chirikov standard map at $k=1$ [3].
Chirikov standard map $y_{n+1}=y_n+\sin x_n\bmod 2\pi$, $x_{n+1}=x_n+y_{n+1}\bmod 2\pi$
2:
-0.10497 †
comment: The Chirikov standard map is area-preserving, so $\lambda_2=-\lambda_1$.
Lorenz system $\dot x=10(y-x)$, $\dot y=-xz+28x-y$, $\dot z=xy-\frac83z$
1:
0.9056 †
comment: Sprott lists the Lorenz system at $(\sigma,\rho,\beta)=(10,28,8/3)$ [3].
Lorenz system $\dot x=10(y-x)$, $\dot y=-xz+28x-y$, $\dot z=xy-\frac83z$
2:
0
comment: An autonomous flow has a zero exponent in the flow direction.
Lorenz system $\dot x=10(y-x)$, $\dot y=-xz+28x-y$, $\dot z=xy-\frac83z$
3:
-14.5723 †
comment: For the Lorenz system at $(10,28,8/3)$, the divergence gives $\lambda_1+\lambda_2+\lambda_3=-41/3$.
Rossler system $\dot x=-y-z$, $\dot y=x+0.2y$, $\dot z=0.2+z(x-5.7)$
1:
0.0714 †
comment: Sprott lists the Rossler system at $a=b=0.2$, $c=5.7$ [3].
Rossler system $\dot x=-y-z$, $\dot y=x+0.2y$, $\dot z=0.2+z(x-5.7)$
2:
0
comment: An autonomous flow has a zero exponent in the flow direction.
Rossler system $\dot x=-y-z$, $\dot y=x+0.2y$, $\dot z=0.2+z(x-5.7)$
3:
-5.3943 †
comment: Sprott gives the negative Rossler exponent with four significant digits [3].
Ueda oscillator $\dot x=y$, $\dot y=-x^3-0.05y+7.5\sin z$, $\dot z=1$
1:
0.1034 †
comment: Sprott lists the Ueda oscillator at $B=7.5$, $k=0.05$ [3].
Ueda oscillator $\dot x=y$, $\dot y=-x^3-0.05y+7.5\sin z$, $\dot z=1$
2:
0
comment: An autonomous flow has a zero exponent in the flow direction.
Ueda oscillator $\dot x=y$, $\dot y=-x^3-0.05y+7.5\sin z$, $\dot z=1$
3:
-0.1534 †
comment: For the Ueda oscillator in Sprott's normalisation, $\lambda_1+\lambda_2+\lambda_3=-k=-0.05$.
Sprott's simplest quadratic flow $\dot x=y$, $\dot y=z$, $\dot z=-2.017z+y^2-x$
1:
0.0551 †
comment: Sprott lists the simplest quadratic flow at $A=2.017$ [3].
Sprott's simplest quadratic flow $\dot x=y$, $\dot y=z$, $\dot z=-2.017z+y^2-x$
2:
0
comment: An autonomous flow has a zero exponent in the flow direction.
Sprott's simplest quadratic flow $\dot x=y$, $\dot y=z$, $\dot z=-2.017z+y^2-x$
3:
-2.0721 †
comment: For Sprott's simplest quadratic flow, the divergence gives $\lambda_1+\lambda_2+\lambda_3=-A$.
Sprott's simplest piecewise-linear flow $\dot x=y$, $\dot y=z$, $\dot z=-0.6z-y-|x|+1$
1:
0.0362 †
comment: Sprott lists the simplest piecewise-linear flow at $A=0.6$ [3].
Sprott's simplest piecewise-linear flow $\dot x=y$, $\dot y=z$, $\dot z=-0.6z-y-|x|+1$
2:
0
comment: An autonomous flow has a zero exponent in the flow direction.
Sprott's simplest piecewise-linear flow $\dot x=y$, $\dot y=z$, $\dot z=-0.6z-y-|x|+1$
3:
-0.6362 †
comment: For the piecewise-linear flow, the divergence gives $\lambda_1+\lambda_2+\lambda_3=-A$ away from the switching surface.
random Fibonacci recurrence $f_n=f_{n-1}\pm f_{n-2}$
1:
0.1239755988033 +/- 5e-14
comment: This is $\log V$, where $V$ is Viswanath's constant $=1.1319882487943\ldots$ [11] [1]. The fourteen digits of $V$ that are not in doubt determine $\log V$ to about $4\times10^{-14}$, and no further.
Definition
For each listed chaotic map, flow or random recurrence, this table gives the Lyapunov exponent $\lambda_i$ [2], ordered $\lambda_1\geq\lambda_2\geq\cdots$, in natural-log units per iteration for maps, per recurrence step for the random recurrence, and per unit time for flows.
Parameters
system
—   system (the map, flow or random recurrence is one of the systems listed here, with the displayed parameter values)
$i$
—   Lyapunov exponent number ($1\leq i\leq d$, where $d$ is the dimension of the listed map or flow; the random Fibonacci recurrence has only $i=1$)
Formulas
(1)
For a one-dimensional map $x_{n+1}=f(x_n)$, $\lambda(x_0)=\lim_{n\to\infty}\frac1n \sum_{j=0}^{n-1}\log|f'(x_j)|$ when the limit exists [2].
(2)
For the logistic, tent and doubling maps, $\lambda_1=\log2$. For the continued-fraction Gauss map, $\lambda_1=\frac{\pi^2}{6\log2}$, where $\pi$ is $\pi$. For Arnold's cat map, $\lambda_1=-\lambda_2=\log\frac{3+\sqrt5}{2}$. For the baker's map, $\lambda_1=-\lambda_2=\log2$.
(3)
For a map or flow with constant Jacobian determinant, the sum of the Lyapunov exponents is the logarithm of the absolute value of that determinant per iterate or unit time. Hence the Henon sum is $\log0.3$, the Chirikov, Arnold-cat and baker sums are $0$, the Lorenz sum is $-(\sigma+1+\beta)$, the Ueda sum is $-k$, and the two Sprott-flow sums are $-A$.
(4)
For the random Fibonacci recurrence, $\lambda_1=\log V$, where $V$ is Viswanath's constant [1].
Comments
(5)
The logarithm is the natural logarithm. For maps, one time unit is one iterate of the map. For flows, one time unit is the time variable in the displayed differential equations. The rows for autonomous flows include the zero exponent in the flow direction.
(6)
The nonzero Henon, Chirikov, Lorenz, Rossler, Ueda and Sprott-flow rows are the estimates printed by Sprott [3]; Sprott states that the least significant digit is only a best estimate. The exact map rows use the closed forms in (2).
(7)
For the measure-preserving examples with the standard invariant measure, the Kolmogorov-Sinai entropy is the sum of the positive Lyapunov exponents. Thus the logistic, tent, doubling and baker maps have entropy $\log2$, Arnold's cat map has entropy $\log\frac{3+\sqrt5}{2}$, and the continued-fraction Gauss map has entropy $\pi^2/(6\log2)$.
(8)
The random Fibonacci row is the Lyapunov exponent of the random matrix product for $f_n=f_{n-1}\pm f_{n-2}$ with independent fair signs [10]. Its exponential is Viswanath's constant [11].
Programs
(P1)
Sage
R = RealBallField(400)
log2 = R(2).log()
log2                                      # logistic, tent, doubling
R.pi()^2 / (6 * log2)                    # continued-fraction Gauss map
((R(3) + R(5).sqrt()) / 2).log()         # Arnold's cat map
-log2                                    # second baker exponent
References
[1]
D. Viswanath, Random Fibonacci sequences and the number 1.13198824..., Mathematics of Computation 69 (2000), 1131-1155. (doi)
Links
Similar tables
Feigenbaum constants —   the logistic map at $r=4$ is the endpoint of the period-doubling route governed by the Feigenbaum constants
Hausdorff dimension of some fractals —   Sprott's source gives Kaplan-Yorke dimensions for the same attractors whose Lyapunov exponents are listed here
Entropy constants of lattice models —   both tables store entropy constants, for dynamical systems here and for lattice models there
Constants of the regular continued fraction —   the continued-fraction Gauss map has Lyapunov exponent $2\beta$, where $\beta$ is the Khinchin-Levy constant
Golden ratio —   Arnold's cat-map exponent is $\log\varphi^2$
Viswanath's constant —   the random Fibonacci row here is $\log V$; that table holds $V$ itself, which is how the literature quotes it
Data properties
Entries are of type: real number
How well the digits are known: heuristic
Table is complete: no (it holds the exact spectra of six standard maps, Sprott's printed spectra for seven classical numerical examples, and the random Fibonacci exponent derived from Viswanath's constant)
How they were obtained:

The logistic, tent, doubling, continued-fraction Gauss, Arnold-cat and baker-map rows were computed from (2) in real ball arithmetic with $64$ guard bits beyond the $100$ written digits. The Henon, Chirikov, Lorenz, Rossler, Ueda and Sprott-flow rows were transcribed from Sprott's table [3].

more

Each printed decimal is stored with the precision it claims under NumberDB's decimal convention, so $0.41922$ denotes the interval with the last digit uncertain. The zero flow-direction rows are exact zeros. The random Fibonacci row was computed as $\log V$ from $V=1.1319882487943\ldots$ [11] and stored with radius $10^{-12}$, wider than the uncertainty from the last published digit of $V$. The exact closed forms were compared with direct derivative and matrix calculations. The transcribed rows were compared with Sprott's printed values, and the sums in (3) were checked on the stored intervals for the Henon, Chirikov, Lorenz, Ueda and Sprott-flow rows.



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