Numerical Calculation of Largest Lyapunov
Exponent
Department of Physics, University of
Wisconsin, Madison, WI 53706, USA
October 15, 1997
(Revised January 8, 2015)
The usual test for chaos is calculation of the
largest Lyapunov exponent. A positive largest Lyapunov
exponent indicates chaos. When one has access to the
equations generating the chaos, this is relatively easy to
do. When one only has access to an experimental data record,
such a calculation is difficult to impossible, and that case will
not be considered here. The general idea is to follow two
nearby orbits and to calculate their average logarithmic rate of
separation. Whenever they get too far apart, one of the
orbits has to be moved back to the vicinity of the other along the
line of separation. A conservative procedure is to do this
at each iteration. The complete procedure is as follows:
- Start with any initial condition in the basin of
attraction.
Even better would be to start with a point known to be on the
attractor, in which case step 2 can be omitted.
- Iterate until the orbit is on the attractor.
This requires some judgement or prior knowledge of the system
under study. For most systems, it is safe just to iterate a
few hundred times and assume that is sufficient. It usually
will be, and in any case, the error incurred by being slightly off
the attractor is usually not large unless you happen to be very
close to a bifurcation point.
- Select (almost any) nearby point (separated by d_{0}).
An appropriate choice of d_{0} is one that is on
the order of the square root of the precision of the floating
point numbers that are being used. For example, in (8-byte)
double-precision (minimum recommended for such calculations),
variables have a 52-bit mantissa, and the precision is thus 2^{-52}
= 2.22 x 10^{-16}. Therefore a value of d_{0}
= 10^{-8} will usually suffice.
- Advance both orbits one iteration and calculate the new
separation d_{1}.
The separation is calculated from the sum of the squares of the
differences in each variable. So for a 2-dimensional system
with variables x and y, the separation would be d
= [(x_{a} - x_{b})^{2} + (y_{a}
- y_{b})^{2}]^{1/2}, where the
subscripts (a and b) denote the two orbits
respectively.
- Evaluate log |d_{1}/d_{0}|
in any convenient base.
By convention, the natural logarithm (base-e) is usually
used, but for maps, the Lyapunov exponent is often quoted in bits
per iteration, in which case you would need to use base-2.
(Note that log_{2}x = 1.4427 log_{e} x).
You may get run-time errors when evaluating the logarithm if d_{1}
becomes so small as to be indistinguishable from zero. In
such a case, try using a larger value of d_{0}.
If this doesn't suffice, you may have to ignore values where this
happens, but in doing so, your calculation of the Lyapunov
exponent will be somewhat in error.
- Readjust one orbit so its separation is d_{0}
and is in the same direction as d_{1}.
This is probably the most difficult and error-prone step. As
an example (in 2-dimensions), suppose orbit b is the one
to be adjusted and its value after one iteration is (x_{b1},
y_{b1}). It would then be reinitialized to x_{b0}
= x_{a1} + d_{0}(x_{b1}
- x_{a1}) / d_{1} and y_{b0}
= y_{a1} + d_{0}(y_{b1}
- y_{a1}) / d_{1}.
- Repeat steps 4-6 many times and calculate the average of
step 5.
You might wish to discard the first few values you obtain to be
sure the orbits have oriented themselves along the direction of
maximum expansion. It is also a good idea to calculate a
running average as an indication of whether the values have
settled down to a unique number and to get an indication of the
reliability of the calculation. Sometimes, the result
converges rather slowly, but a few thousand iterates of a map
usually suffices to obtain an estimate accurate to about two
significant digits. It is a good idea to verify that your
result is independent of initial conditions, the value of d_{0},
and the number of iterations included in the average. You
may also want to test for unbounded orbits, since you will
probably get numerical errors and the Lyapunov exponent will not
be meaningful in such a case.
Sample software that implements this procedure while searching for
chaotic solutions in general 2-D quadratic maps is available in
(DOS) BASIC source and executable code. Sample software that
calculates the Lyapunov exponent (-0.5 to 0.5) for the Hénon Map X_{n}_{+1}
= 1 - CX_{n}^{2} + BX_{n}_{-1}
for B = 0.3 and 0 < C < 2 is also available in
(DOS) BASIC source and executable code. Output from that
program is shown below:
If the system consists of ordinary differential equations (a
flow) instead of difference equations (a map), the procedure is
the same except that the resulting exponent is divided by the
iteration step size so that it has units of inverse seconds
instead of inverse iterations. You will typically need
millions of iterations of the differential equations to get a
result good to better than about two significant digits. An
example for the Lorenz attractor is available.
See also the code for calculating the
whole spectrum of Lyapunov exponents.
Sometimes you can get the whole spectrum of exponents using the
above method, for example when the system is a two dimensional
chaotic map or a three dimensional chaotic flow if you know the
rate of state space contraction averaged along the orbit (the
dissipation) which is the sum of the Lyapunov exponents (easily
calculated from the trace of the Jacobian matrix averaged along
the orbit for a flow or from the average determinant of the
Jacobian matrix for a map) and using the fact that one exponent
must be zero for a continuous flow.
To estimate the uncertainty in your calculated Lyapunov exponent,
you can repeat the calculation for many different initial
conditions (within the basin of attraction) and perturbation
directions. For a chaotic system, the initial condition need only
be changed slightly since orbits quickly become uncorrelated due
to the sensitive dependence on initial conditions. You can then
calculate a mean and standard deviation of the calculated values
so as to avoid the all too common mistake of quoting more digits
than are significant.
Ref: J. C. Sprott, Chaos and Time-Series
Analysis (Oxford University Press, 2003), pp.116-117.
Back to Sprott's Technical Notes