Complex squaring map

The complex squaring map, a polynomial mapping of degree two, is a simple and accessible demonstration of chaos in dynamical systems. It can be constructed by performing the following steps:

(1) Choose any complex number on the unit circle whose argument (complex angle) is not a rational fraction of π,

(2) Repeatedly square that number.

This repetition (iteration) produces a sequence of complex numbers that can be described by their complex angle alone. Any choice of starting angle that satisfies (1) above will produce an extremely complicated sequence of angles, that belies the simplicity of the steps. In fact, it can be shown that the sequence will be chaotic, i.e. it is sensitive to the detailed choice of starting angle.

Chaos and the complex squaring map

The informal reason why the iteration is chaotic is that (a) the angle doubles on every iteration and doubling grows very quickly as the angle becomes ever larger, but (b) the angle must be limited to lie on the circle, i.e. the range zero to 2π (radians). Thus, when the angle exceeds 2π, it must wrap to the remainder on division by 2π. Now, π is a transcendental number and the fractional part has no known pattern. Thus, even if the initial angle is chosen such that it has a finite number of digits and is known with complete accuracy, the remainder on division by 2π inherits the lack of pattern in the fractional part of π. The iteration therefore "reads out" the patternless detail of the fractional part of π, and this is in fact the origin of the complexity of the angle sequence.

More formally, the iteration can be written as:

$\qquad z_{n+1}=z_{n}^{2}$

where $z_{n}$ is the resulting sequence of complex numbers obtained by iterating the steps above, and $z_{0}$ represents the initial starting number. We can solve this iteration exactly:

$\qquad z_{n}=z_{0}^{2^{n}}$

Starting with angle θ, we can write the initial term as $z_{0}={\exp }(i\theta )$ so that $z_{n}={\exp }(i2^{n}\theta )$. This makes the successive doubling of the angle clear. Also, the relation $z_{n}={\cos }(2^{n}\theta )+i{\sin }(2^{n}\theta )$ makes the wrapping of the angle onto the circle obvious. The doubling-and-wrapping is the same origin of chaos as the stretching-and-folding of the logistic map, for example, and leads to exponential divergence of close starting angles (see Lyapunov exponents). If the initial angle is expanded in a binary representation, then the iteration can be clearly seen to be equivalent to the shift map.

See also

• Chaos theory
• List of chaotic maps
• Logistic function
• Lyapunov stability for iterated systems
• Complex quadratic map