Iterations

Calculate algebraic series such as e = 1+ 1/2! + 1/3! + …, a square wave, Fibonacci numbers. Study iterative maps, e.g. the (one-dimensional) Logistic map: (see below) or more complicated multi-dimensional maps. The logistic map is perhaps one of the simplest mathematical system showing many characteristics of the development of chaotic behavior.
Several analytical tools are valuable to study the results of Iterations and ODE’s such as:

Time series | Power spectra | 2D projections | Fixed points | Lyapunov exponents

A special window has been added in Mathgrapher v2 to allow detailed presentation of 2D orbits at the pixel level and to study the stability of the orbits (see the Examples and Demonstrations for the Henon map, Standard map and Mandelbrot and Julia sets.

Logistic map: Sensitivity to initial conditions | Projection in 2D | Power spectrum | Bifurcation diagram | Lyapunov exponents

Henon map: Definition | 2D orbit | Region of Stability

Mandelbrot and Julia sets: Definition | Mandelbrot: vary parameters | Julia: vary initial conditions

Maps – Lyapunov exponents – Definition and calculation

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Lyapunov exponents are very important and useful in the description of chaotical dynamical systems. They describe the exponential rate at which neighboring orbits diverge. They can be 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. An n-dimensional map has n Lyapunov exponents. Mathgrapher calculates all of them. The method used 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.).
The calculation of the exponents is easier than in the case of ODE’s. It involves linearization of the map at each point x(i) along the orbit, i.e. calculating the tangent map M(i), For an n-dimensional system this is a n x n matrix. The tangent map gives the amount of expansion or contraction of neighboring orbits near xi. After a few iterations the product M(k)M(k-1) ..M(1) is likely to grow rapidly and become ill-conditioned. So we have to apply orthonormalization at regular intervals as in the case of ODE’s. This is done by QR decomposition. The lyapunov exponents are calculated from the diagonal elements of R (see also ODE’s).
The simplest case is the one-dimensional map (see for example the logistic map).

For n-dimensional maps

For area conserving maps contraction in one direction is balanced by expansion in another direction, so we have

The set of Lyapunov exponents define a dimension: the Lyapunov or Kaplan-Yorke dimension