COPASI can calculate Lyapunov exponents along a trajectory, as well as the average divergence of the system. These exponents are determined for the reduced system (see Interpretation of the Model), so the maximum number of exponents corresponds to the number of independent variables. If you request fewer exponents, COPASI will compute the largest ones.
COPASI implements a well-established algorithm [Shimada79, Benettin80] that was originally implemented in FORTRAN by Wolf et al. [Wolf85]. This algorithm integrates one reference trajectory and, simultaneously, N difference trajectories (where N is the number of requested exponents) in a system linearized around the reference path. Integration is performed over a short “orthonormalization interval.” After each interval, the difference vectors are reorthonormalized. The exponents for each interval are calculated based on how the difference trajectories diverge or converge relative to the reference trajectory. This process is repeated, and the local exponents are averaged over the entire trajectory.
If requested, the divergence is calculated as the average trace of the Jacobian matrix. For consistency, the divergence is evaluated over the same orthonormalization intervals used for the Lyapunov exponents, which allows for local comparison between divergence and exponents.
If you are only interested in the final result for the Lyapunov exponents and the average divergence, you can use COPASI’s default report or view the results in the GUI. To create customized report templates containing the Lyapunov exponents and divergence, these options are found under “Results.” For access to the “local” estimates for each orthonormalization interval, you need to define a plot or report manually. In this version of COPASI, use the “expert” feature in the object selection dialog to reach the exponents, which appear in the “Lyapunov Exponents” branch under “Task List.” After each reorthonormalization interval, output can provide up to the ten largest exponents—both the local value from the last interval and the running average across all previous intervals. Similarly, the divergence can be output for the last interval or as an overall average.
The Jacobian used for both Lyapunov exponents and divergence computations is approximated using finite differences. Integration of the reference and difference trajectories uses the LSODA integrator [Hindmarsh83].
Orthonormalization interval:
This is the time interval after which orthonormalization of the difference
trajectories occurs. This setting is critical for accurate Lyapunov exponent
calculation. Smaller values usually yield more accurate results but require
longer calculation times, as numerical integration is restarted more often
for shorter intervals. To check if this parameter is adequate, compare the
sum of the exponents to the divergence of the system; these should match if
all exponents are calculated. Since divergence is robust, a mismatch usually
indicates a need for a shorter interval. Note that this parameter most
strongly affects the accuracy of the largest-magnitude exponents. Because
large positive exponents are rare, the most negative exponents are
particularly sensitive. If you do not need exact values for strongly negative
exponents, you can select a larger interval for faster computation. The
default is 1.0.
Overall time:
This parameter specifies the total duration for the calculation. Integration
will repeat in steps determined by the “orthonormalization interval” until
the overall time is reached. This value is also important for exponent
accuracy. Since COPASI cannot estimate how quickly the exponents converge,
no universally safe default can be provided for this parameter. As an
indicator, if the system does not reach steady state, one exponent should be
zero. If this does not occur, the overall time is probably too short to allow
the exponents to converge. The default is 1000.
Other options relevant for the calculation are related to the LSODA numerical integrator. These are described in the section on deterministic simulation.
