What is a Fractal in Dynamics?
Terminology
A fractal is generally referred to as a geometric object possessing self-similarity, and while its examples and concepts are widely recognized, a universally accepted definition remains elusive1.
Easy Agreement
A complex geometric form possessing intricate structures at arbitrary small scales is called a fractal. Most fractals exhibit some degree of self-similarity2.
Difficult Agreement
A structure with complex patterns repeated over a wide range of scales, or having a fractal dimension that is a non-integer, is termed a fractal3. Types of fractal dimensions include:
Explanation
In fractal geometry, which concerns itself solely with fractals, it might be different; however, fractals associated as elements within a dynamical system are notoriously difficult to define clearly, much like many concepts of dynamics.
At least in dynamics, fractals have a strong relationship with the concept of chaos. Broadly speaking, chaos refers to systems whose orbits neither converge nor diverge but remain bounded, lack periodicity, and are sensitive to initial conditions, rendering them unpredictable. Yet, when examining the phase plane, one finds fractal structures, which makes sense considering the essence of chaos.
- Chaotic orbits never return to a previously traversed point, and for this to occur in a bounded space, there must be paths for the orbit to pass through, no matter how finely the space is divided.
- The boundary of a fractal is visually hard to conceptualize. Upon selecting a point near the boundary, it’s uncertain whether it lies inside or outside of the fractal, and slightly moving that point may result in an unknown transition between basins. When equating selecting that point to determining initial conditions, one may begin to perceive the correlation between them.