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Leveraging Deterministic Chaos to Mitigate Combinatorial Explosions
Published in Larry B. Rainey, Mo Jamshidi, Engineering Emergence, 2018
Many real-world phenomena are self-similar (i.e., a smaller section of the whole can be scaled iteratively to represent the whole). Examples include trees, broccoli, cauliflower, and their respective branches; planets with multiple moons that can scale to a solar system with multiple planets; coastlines that can scale both upward and downward; and many other phenomena. Many phenomena can be represented by recursive generating functions, or Iterated Function Systems (IFS), which typically generate fractals [5,19,20]. The IFS generates a fractal shape by constructing many copies of itself and overlaying these to produce the fractal pattern. The copies are affine transformations of the original shape, and are typically contractive-affine, or a reduced size copy of the original. This IFS is the result of a set of contractive affine elements, by use of what is called the Hutchinson operator, which converges to a unique attractor [22]. The overlays of these affine transformations are mathematically the union of these functionally transformed shapes, which create a fractal topology.
On the Application of Fractal Interpolation Functions within the Reliability Engineering Framework
Published in Ioannis S. Triantafyllou, Mangey Ram, Statistical Modeling of Reliability Structures and Industrial Processes, 2023
Polychronis Manousopoulos, Vasileios Drakopoulos
Fractal interpolation, as defined in [4], provides an efficient way of interpolating data that exhibit an irregular and non-smooth structure, possibly presenting details at different scales or some degree of self-similarity. In contrast to traditional interpolation techniques which use smooth functions such as polynomials and produce smooth interpolants, fractal interpolation based on the theory of iterated function systems [5] is successful, for example, in modelling projections of physical objects such as coastlines and plants, or experimental data of non-integral dimension.
I
Published in Phillip A. Laplante, Dictionary of Computer Science, Engineering, and Technology, 2017
iterated function system a finite collection of affine mappings in the plane which are combinations of translations, scalings, and rotations. Each mapping has a defined probability and should be contractive, that is, scalings are less than 1. Iterated function systems can be used for the generation of fractal objects and image compression.
Topological pressure for an iterated function system
Published in Dynamical Systems, 2021
Let be a family of continuous maps on X. The iterated function system (or system for short) is the action of the semigroup generated by , on compact metric space X. If , then it is a classical dynamical system. We use the standard notation: . Iterated function systems are used for the construction of deterministic fractals and have numerous applications.