General fractal topography: an open mathematical framework to characterize and model mono-scale-invariances

In this work, we reported there are two kinds of independent complexities in mono-scale-invariance, namely to be behavioral complexity determined by fractal behavior and original one wrapped in scaling object. Quantitative characterization of the complexities is fundamentally important for mechanism exploration and essential understanding of nonlinear systems. Recently, a new concept of fractal topography (\(\mathrm {\Omega }\)) was emerged to define the fractal behavior in self-similarities by scaling lacunarity (P) and scaling coverage (F) as \(\mathrm {\Omega }(P, F)\). It is, however, a “special fractal topography” because fractals are rarely deterministic and always appear stochastic, heterogeneous, and even anisotropic. For that, we reviewed a novel concept of “general fractal topography”, \({\varvec{\Omega }}(\mathrm {{\mathbf {P}}}, \mathrm {{\mathbf {F}}})\). In \({\varvec{\Omega }}\), \(\mathrm {{\mathbf {P}}}\) is generalized to a set accounting for direction-dependent scaling behaviors of the scaling object G, while \(\mathrm {{\mathbf {F}}}\) is extended to consider stochastic and heterogeneous effects. And then, we decomposed G into \(\mathrm {{\mathbf {G}}}\left( G_+, G_-\right) \) to ease type controlling and measurement quantification, with \(G_+\) wrapping the original complexity while \(G_-\) enclosing behavioral complexity. Together with \({\varvec{\Omega }}\) and \(\mathrm {{\mathbf {G}}}\), a mathematical model \(F_\mathrm {3S}\left( {\varvec{\Omega }}, \mathrm {{\mathbf {G}}}\right) \) was then established to unify the definition of deterministic or statistical, self-similar or self-affine, single- or multi-phase properties. For demonstration, algorithms are developed to model natural scale-invariances. Our investigations indicated that the general fractal topography is an open mathematic framework which can admit most complex scaling objects and fractal behaviors.