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Many dynamical systems in physics, astronomy, chemistry, physiology, meteorology, economics, population
dynamics can be described by
Ordinary Differential Equations.
In the second half of the 20th century much attention has been focussed on the
i.e. unpredictable behaviour of (non-linear) ODE's. A well-known example is the
Lorenz atrractor, illustrating the "Butterfly effect": small causes can have large effects.
Mathgrapher uses an accurate Adams-Bashforth variable order, variable step predictor-corrector
algorithm to integrate systems of up to 20 coupled differential equations.
Lyapunov exponents are very important in the
description of chaotical dynamical systems. They describe the exponential
rate at which neighbouring orbits diverge. They are used to determine the
behaviour of quasi-periodic and chaotic behavious as well as the stability
of equilibrium points and periodic solutions. For orbits near equilibrium
points they are equal to the real parts of the eigenvalues at these points.
The number of Lyapunov exponents is equal to the dimension of the system of ODE's,
so a system of n ODE's is characterized by n Lyapunov exponents. Mathgrapher
calculates all of them.
Lyapunov exponents - definition
Lyapunov exponents give the exponential separation of neighbouring orbits.
Suppose we have a system of n coupled ODE's given by
calculation of the Lyapunov exponents involves linearization of the
differential equations and integration of all the n^2 components of the the tangent
map (or perturbation vectors) along with the n ODE's. The displacement vectors
grow rapidly and should be orthonormalized at regular intervals to prevent overflow.
For Hamiltonian systems the n-dimensional volume spanned by the displacement will be
deformed, but not changed due to Liouville's theorem. Since the Lyapunov exponents
give the exponential rates of change of the vectors spanning this volume we have in
The method used to calculate the Lyapunov exponents is the one described in
"Practical Numerical Algorithms for Chaotic Systems" by T.S. Parker and L.O. Chua (1989,
Springer-Verlag New York Inc.). A set of n independent perturbation (or displacement)
vectors is integrated simultaneously with the orbital coordinates.
The evolution of these perturbation vectors is described by the Variational
equation: i.e. the linearization
of the vector field along the trajectory. This is a set of n^2 linear equations describing
the evolution of each of the n components of the n perturbation vectors. The total number
of equations that needs to be integrated in order to obtain all the Lyapunov exponents is
therefore n+n^2. In a chaotic system the perturbation vectors increase exponentially.
To prevent overflow, these vectors are orthonormalized at regular time intervals.
This is done by QR decomposition of the tangent matrix. The Lyapunov exponents are
found from the diagonal elements of the R matrices formed at each orthormalization
where Ri is the the R-matrix at the i-th orthonormalization step.