I searched Google for "magnetic fractal", and found the answer on the first hit. It quotes the Fractint documentation (I can't resist mentioning that Fractint is the grand-daddy of freeware fractal-generating software for personal computers -- it had its first release in 1988, and is still being maintained!):
These fractals use formulae derived from the study of hierarchical lattices, in the context of magnetic renormalisation transformations. This kinda stuff is useful in an area of theoretical physics that deals with magnetic phase-transitions (predicting at which temperatures a given substance will be magnetic, or non-magnetic). In an attempt to clarify the results obtained for Real temperatures (the kind that you and I can feel), the study moved into the realm of Complex Numbers, aiming to spot Real phase-transitions by finding the intersections of lines representing Complex phase-transitions with the Real Axis. The first people to try this were two physicists called Yang and Lee, who found the situation a bit more complex than first expected, as the phase boundaries for Complex temperatures are (surprise!) fractals.
The formulas for the two fractals are also given there. They are
$$z \mapsto \left(\frac{z^2 + (c-1)}{2z + (c-2)}\right)^2$$
for magnet 1, and
$$z \mapsto \left(\frac{z^3 + 3(c-1)z + (c-1)(c-2)}{3z^2 + 3(c-2)z + (c-1)(c-2) + 1}\right)^2$$
for magnet 2.
I'm by no means knowledgeable on this subject, but I've been looking at some of Robert Devaney's papers, which I came across via tetration.org. Looking at Devaney's images, I'd guess that the reason why these fractals have the beautiful Sierpinski-gasket-like structures, while the standard quadratic Julia and Mandelbrot sets don't, is that each of the formulas defining these fractals is a rational function, that is, a ratio of two polynomials, rather than a single polynomial. I believe the field that studies these things is called complex dynamics. My knowledge doesn't extend to how rational functions with poles give rise to Julia sets with gaskets, but you could try looking for the answer to that in some of Devaney's papers, or in the book Iteration of Rational Functions by Alan F. Beardon which is cited in the Wikipedia article on Julia sets.
As noted by Sheldon, the good starting point must be a critical point. There is indeed a theorem that says that if you have an attracting cycle, then at least one critical point must belong to its attraction basin.
Roughly, the idea of the proof is as follows : around an attracting fixed point, there is a linearizing coordinate $\phi$ such that $f(\phi(z))=\lambda \phi(z)$. That coordinate is defined only in a neighborhood of the attracting fixed point, however using the functional equation it satisfies it is possible to prolonge it until you meet a critical point.
Now if you have a cycle instead of a fixed point, just replace $f$ by $f^p$ to get the same result.
The Mandelbrot set is of mathematical interest because in complex dynamics, the global behaviour of the dynamics is generally ruled by the dynamics of the critical points. Thus knowing the dynamics of the critical points give you information on all of the dynamics. In the simplest family $z^2+c$, there is only 1 critical point (0) and so it is natural to look at what happens to its dynamics depending on $c$.
If you want to generalize the notion of Mandelbrot set for, say, cubic polynomials $z^3 + az+b$, you would have to look at the behaviour of two critical points, and so not only would you get a set in $\mathbb{C}^2$, you would also need to make a choice in your definition : are you looking at parameters where both critical points are attracted to a cycle, or one of them, or none ?
In your case, there is only one critical point, so your set is a reasonable analogue of the Mandelbrot set.
EDIT : note that the definition of the Mandelbrot set does not use attracting cycles, but depends on whether or not the critical point goes to infinity. It is conjectured (it's one of the most important conjecture in the field) that the interior of the Mandelbrot set is exactly composed of parameters for which the critical point is attracted to a cycle. However it is well known that in the boundary of the Mandelbrot set, you have no attracting cycles.
EDIT 2 : One of the most interesting features of the Mandelbrot set is that its boundary is exactly the locus of bifurcation, i.e. the set of parameters for which the behaviour of the dynamics changes drastically. If you choose any holomorphic family of holomorphic maps $f_\lambda$, you can also define the locus of bifurcation for this family. It has been proved that this set is either empty or contains copies of the Mandelbrot set.
Best Answer
You have a few points of confusion that make this question (and your previous ones here and on StackOverflow) difficult to answer. Here is some clarification.
The Mandelbrot set lives in the context of complex dynamics, i.e. the study of the iteration of analytic functions mapping the complex plane to itself. A huge majority of MSE questions on the Mandelbrot set should tagged [complex-dynamics], rather than [fractals].
Given a complex analytic function, it makes no sense to refer to a single trajectory as chaotic. It might make sense to refer to the dynamics of the function itself as chaotic on some closed subset of the plane. Bob Devaney is generally credited with writing down the exact criteria that a function should satisfy to be considered chaotic a on a set.
The fundamental set associated with the chaotic dynamics of a complex analytic function is the Julia set. For a polynomial, the Julia set can be characterized as the closure of the set of repelling periodic points of that polynomial. To be clear, the Julia set of a function lives in the dynamical plane for that function - i.e. the plane where the iteration happens.
The Mandelbrot set specifically involves the iteration of functions chosen from the family $$ f_c(z) = z^2 + c, $$ where $c$ is a complex parameter. Specifically, the Mandelbrot set is defined to be the set of all complex numbers $c$ such that the Julia set of $f_c$ is connected or equivalently, such that the orbit of the critical point $z_0=0$ under iteration of $f_c$ remains bounded. In particular, the Mandelbrot set provides a general classification of the types of dynamics that arise for quadratic functions.
Note that the the Mandelbrot set lives in the parameter ($c$) plane - not in the dynamical ($z$) plane; in particular, trajectories don't live in the Mandelbrot set.
Here's a picture that might help clarify:
On the left, we see the Mandelbrot set with the single point $z_0=-0.9+0.15i$ highlighted in yellow. The corresponding Julia set appears on the right, together with an orbit that starts at the green dot and is ultimately attracted to a cycle of period two indicated by the two red dots.
If we change the point, we change the dynamical picture. For example, here's the picture for your point $c=0.25+0.5i$:
You can play with these sorts of images on this web page.
Judging from your pictures, I would guess that you are asking about the nature of the critical orbits (i.e. the orbits from the point $z_0=0$) of the functions $f_c$ for various choices of $c$, like $c=0.25+0.5i$. My second picture above shows how I would illustrate the orbit on the Julia set for that particular value of $c$. The dynamics of $f_c$ for each of the points you list happen to be parabolic, which is a bit tricky to deal with. There have been several discussions on this site involving parabolic dynamics.