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 focused on the often chaotic, i.e. unpredictable behavior of (non-linear) ODE’s. A well-known example is the Lorenz atractor, 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.
Several analytical tools are available for ODE’s such as: Time series | Power spectra | 2D and 3D projection | Phase portraits | Poincare section | Equilibrium points | Lyapunov exponents
Examples: Lorenz Equations | Hénon-Heiles potential | Rössler Equation | van der Pol oscillator | Duffing oscillator | Predator-Prey equation (Voltera)
Lyapunov exponents for ODE’s
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 behavior of quasi-periodic and chaotic behavior 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 neighboring 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 that case:
The set of Lyapunov exponents define a dimension: the Lyapunov or Kaplan-Yorke dimension.
Lyapunov exponents – Calculation.
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 step:
where Ri is the the R-matrix at the i-th orthonormalization step.
Linearization of the equations
Suppose we have a system of 2 coupled ODE’s
The eigenvalues of this matrix determine the type of equilibrium point and its stability properties. They are the roots of the equation
Lyapunov (Kaplan – Yorke) dimension
The Lyapunov dimension (or Kaplan-Yorke dimension) is defined as follows: