KOCH CURVE : I was going through chaos lectures by peter smith [1] the other day and come to know the fractal structure Koch curve.These curves are of extremely importance in fractal mathematics and exhibit the characteristics of chaos, a challenging problem in modeling the natural phenomenon. Both ordered and random Koch curves are discussed presented here. The programming methodology of the Koch curve with the necessary derivations is also explained.
Stretching Via Geometric Progression
Geometric progression is a series in which any two consecutive numbers are when divided, we get the same number, called common ratio. In other words we get the next term by multiplying previous term with a common ratio. The fact that the next number is a common ratio multiple of previous number gives us a hint that it can be used for stretching because the series grows with increasing power of the ratio.
Lagrangian coherent structures : LCS are distinguished surfaces of trajectories in a dynamical systemthat exert a major influence on nearby trajectories over a time interval of interest.The type of this influence may vary, but it invariably creates a coherent trajectory pattern for which the underlying LCS serves as a theoretical centerpiece. In observations of tracer patterns in nature, one readily identifies coherent features, but it is often the underlying structure creating these features that is of intereset
Physical phenomena governed by LCSs include floating debris, oil spills, surface drifters and chlorophyll patterns in the ocean; clouds of volcanic ash and spores in the atmosphere;and coherent crowd patterns formed by humans and animals.
JULIET SET : In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets (Julia 'laces' and Fatou 'dusts') defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is 'regular', while on the Julia set its behavior is 'chaotic'. I made a program to print out the juliet set shapes for various inputs Z.
