Non linear mappings

Mappings as discussed here take the x and y coordinates of a point in the 2D plane and calculate a new set of coordinates according to certain functions. Even simple mapping functions can lead to very complex behaviour when one feeds the output back as new input for the mapping and repeats the process many times (i.e. iterates the mapping). The points either go off to infinity, repeat themselves, trace out lines, or fill certain areas in the plane. See below for instructions on the applet, the Standard Map and the Henon Map.

This was my very first Java program, written around 1998. It is not perfect, but I learned about multithreading and it does what it's supposed to do

NOTE: Java must be enabled in your browser, and settings must allow "Active Content"

Instructions

Click into the white area above. This selects the x,y coordinates of the initial point. This point is then iterated using the mapping equations selected in the pop-up menu above the iteration area (the applet is CPU intensive, so don´t use it while burning CD´s or doing other resource intensive activities). At any time, you can click at another point in the mapping plane, continuing the iteration with the new starting location.

Alternatively, click the Start button. This starts iterating from x=0, y=0 at the center.

The Stop button halts the iteration.

It is possible to zoom into an area of the mapping plane by clicking and dragging the mouse. This shows a rectangle that will be blown up to the full size of the mapping area. This can be done repeatedly. Note, however, that since most of the points calculated will be outside the drawing area, the iteration will appear to become slow.

The Reset button stops the iteration, clears the mapping area, and resets the boundaries of the plane to their initial values.

For the use of the Stochasticity parameter, see below in the explanations for the Standard Map and the Henon map. If you change the value, you need to hit return for the new value to be applied.

The Standard Map

The mapping equations for the Standard Map are:

xn+1 = (xn + yn+1) mod 2π
yn+1 = [yn + k sin(xn)] mod 2π

The Standard Map is a generic description for a physical system with periodic motion which is perturbed by an external oscillating force. Examples are a pendulum which is periodically pushed, or an electron in a magnetic field (which forces it into a circular or helical orbit), influenced by an oscillating electrostatic field. The x and y coordinates represent the so-called action and angle variables of such physical systems, which decsribe the orbit of the electron (or the tip of the pendulum) in phase space. The Standard Map is a conservative mapping, meaning that a volume in phase space does not contract or expand along its orbit (its shape, however, may be distorted dramatically). This corresponds to closed physical systems in which energy is conserved.

k is the stochasticty parameter. Try values from 0 to around 3.0 (remember to hit return after entering k in the text field, otherwise the new value will not be registered). For k=0, the system corresponds for example to the ideal pendulum, exhibiting a completly regular motion. For k>0, the pendulum is disturbed by a periodic driving force. This has the dramatic effect of producing stochastic orbits for certain initial conditions (i.e. points in phase space). The standard treatment of physics is to deal with the perturbation by expanding the driving force using a Taylor series and to keep only the linear part, with the understanding that higher order contributions are sufficiently small to be neglected. However, this treatment completely misses the qualitative aspects of the driven pendulum. With increasing k increasingly large fractions of phase space become chaotic. Chaotic orbits extend over a two-dimensional area in the above mapping plane. In addition, there are periodic orbits, indicated by a finite number of points on the mapping plane, and quasi-periodic orbits, which produce one or more closed lines (loops) in the mapping plane. Note that for conservative mappings, regular and chaotic phase space regions are arranged on many scales in a self-similar fashion, giving a strong indication that this structure is fractal.

Further info on the Standard Map can be found in MathWorld and in an extensive article in Scholarpedia.
Another applet for the Standard Map is here.
A reference textbook for chaotic motion in general and this and many other maps is A.J.Lichtenberg, M.A.Libermann: Regular and Stochastic Motion, Springer Verlag 1982.

The Henon Map

This is a classical example of a dissipative mapping with a strange attractor. Dissipative systems are characterized by an energy flow into and out of the system, like in the case of a driven pendulum with friction. All orbits end up on the attractor orbit, regardless of their initial conditions. The equations are

xn+1 = yn + 1 - 1.4x2
yn+1 = 0.3 k xn

The zoom feature of the applet can be used to explore the feathery nature of the attractor.

The stochasticity parameter k should be between 0 and 1.0493. Larger values result in unbounded motion.

Further info on the Henon mapping can be found again in MathWorld and Wikipedia