Mandelbrot set; are these trajectories chaotic? I am using the complex Mandelbrot set, with an exponent of 2 so that the iterative equation is z = z^2 + c. The escape threshold is 4.0, and the maximum number of iterations is 5000.
I find that all trajectories that belong in the set are cyclical, with the exception of a few trajectories that do not repeat during the iterations. Some of these acyclical trajectories' starting locations are:
-1, -0.25
-1, 0.25
0.25, -0.5
0.25, 0.5
Are these trajectories chaotic? Is it the acyclical nature of the trajectory that makes it chaotic?

Where the exponent is 4, there are also acyclical trajectories:

Keep in mind that the exponent does not need to be an integer, and does not need to be positive either.
 A: 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.
