## 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.

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

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

## Chaos

In the second half of the last century it was discovered that most (non-linear) dynamical systems exhibit chaotic behavior. This came as a surprise. It was thought that deterministic systems would behave in a predictable way and that chaotic behavior in such systems would be due to many different external influences on a system. It turned out, however, that non-linear systems which are fully deterministic and quite simple and straightforward in mathematical sense often show chaotic behavior. This means that a very small change in the initial conditions of the system may may lead to completely different end results thereby making the outcome completely unpredictable. This was surprising because it occurs in systems whose behavior is governed by well-understood physical principles described by exact mathematical formulae. The example that is often mentioned is the butterfly in Alaska which may cause a thunderstorm in Texas just by flapping its wings. Scientist may be surprised by this finding, however every pinball player knows that a very slight change in the velocity of the ball at some point along its trajectory (for instance by kicking the machine) may change the end result completely.

The examples contain two well-known examples of systems that exhibit chaotic behavior. Both are incorporated in the Demonstrations, so you just have to choose Demonstrations from Mathgrapher’s menu to see how you may calculate and analyze such systems. The first example is a simple iterative algorithm: the logistic map. The second example is a well known system of Ordinary Differential Equations (ODE’s): the Lorenz equations.