Quadratic Polynomials
A very popular complex dynamical system is given by the family of quadratic polynomials, a special case of rational maps. The quadratic polynomials can be expressed as
where is a complex parameter.
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Filled Julia set for fc, c=1−φ where φ is the golden ratio
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Julia set for fc, c=(φ−2)+(φ−1)i =-0.4+0.6i
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Julia set for fc, c=0.285+0i
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Julia set for fc, c=0.285+0.01i
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Julia set for fc, c=0.45+0.1428i
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Julia set for fc, c=-0.70176-0.3842i
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Julia set for fc, c=-0.835-0.2321i
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Julia set for fc, c=-0.8+0.156i
The parameter plane of quadratic polynomials - that is, the plane of possible -values - gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all such that is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor space: in this case it is sometimes referred to as Fatou dust.
In many cases, the Julia set of looks like the Mandelbrot set in sufficiently small neighborhoods of . This is true, in particular, for so-called 'Misiurewicz' parameters, i.e. parameters for which the critical point is pre-periodic. For instance:
- At, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
- At, the tip of the long spiky tail, the Julia set is a straight line segment.
In other words the Julia sets are locally similar around Misiurewicz points.
Read more about this topic: Julia Set