Chaos theory represents a multidisciplinary area of study that encompasses both scientific inquiry and mathematics. It examines the essential models and deterministic principles governing dynamical systems that exhibit extreme sensitivity to initial conditions. Previously, these systems were believed to exist in a state of complete randomness and disorder. However, chaos theory posits that beneath the surface of apparent randomness in chaotic complex systems lie underlying patterns, interconnections, continuous feedback mechanisms, repetitions, self-similarity, fractals, and self-organization. The butterfly effect, a fundamental concept in chaos theory, illustrates how a minor alteration in one aspect of a nonlinear system can result in significant variations later on, highlighting a fragile dependence on initial circumstances. This phenomenon is often illustrated by the metaphor that a butterfly flapping its wings in Brazil could potentially influence or avert a tornado in Texas by altering the conditions in its environment.

fractal

A geometric object with intricate structure at arbitrary tiny sizes is called a fractal in mathematics. Typically, the fractal dimension strictly exceeds the topological dimension. As seen by the Mandelbrot set's sequential magnifications, many fractals look similar at different scales. Self-similarity, often referred to as expanding symmetry or unfolding symmetry, is the display of similar patterns at progressively smaller scales; affine self-similarity is the shape that occurs when this replication is precisely the same at every scale, as in the Menger sponge. Measure theory is an area of mathematics that includes fractal geometry.