Multifractal system

A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed.

In a multifractal system $s$, the behavior around any point is described by a local power law:

$s(vec\{x\}+vec\{a\})-s(vec\{x\})\; sim\; a^\{h(vec\{x\})\}$

The exponent $h(vec\{x\})$ is called the singularity exponent, as it describes the local degree of singularity or regularity around the point $vec\{x\}$.

The ensemble formed by all the points that share the same singularity exponent is called the singularity manifold of exponent h, and is a fractal set of fractal dimension D(h). The curve D(h) vs. h is called the singularity spectrum and fully describes the (statistical) distribution of the variable $s$.

Multifractal systems are common in nature, especially Geophysics. They include Fully Developed Turbulence, Stock market time series, real world scenes, the Sun’s magnetic field time series, heartbeat dynamics, human gait, and natural luminosity time series. Embriogenesis is also multifractal system which represents a new type of physics, named fractal mechanics.

See also

• Multifractal Model of Asset Returns (MMAR)
• Multifractal Random Walk model (MRW)
• Fractional Brownian motion
• Mandelbrot cascade, continuous cascade and lognormal cascade