ARRAY BASED DYNAMICAL SYSTEMS Dynamical Systems are things
ARRAY BASED DYNAMICAL SYSTEMS
• Dynamical Systems are things that change in time, e. g. the Earth moving around the sun, the stock market, the weather, etc. These are examples of continuous systems since time changes continuously. For motion problems, space is also continuous. Some systems though are discrete in that time changes in discrete time steps t = 0, 1, 2, …. .
A “DISCRETE” DYNAMICAL SYSTEM • Discrete lattice of identical sites (cells) in 1 -D, 2 -D, …. which can be easily handled using an array. • Each site takes on values from a finite set (e. g. {0, 1}). • Values of cells evolve in discrete time steps t = 0, 1, 2, …. . • The value of a cell is updated in each time step by a local rule depending on the neighboring cells.
A SYSTEM WITH 7 CELLS
TIME EVOLUTION • We can think of time evolution as a mapping that sends each state to the state it evolves to in one time step. • In many systems this map is “linear” and is represented by a matrix. • Iterating time evolution is simply matrix multiplication. Furthermore, the matrix encodes tremendous amounts of information regarding the system including ways to find the different cycle lengths and numbers of each cycle.
THE MATRIX HAS A “NICE” FORM • Due to the symmetry of the update rules, the associated matrices are “circulant” meaning each column is a simple shift of the previous one.
DESIRED SOFTWARE SYSTEM • User inputs the size of the array, (modular) alphabet, and update rule • The software generates the associated circulant matrix (easy) • Powers of the matrix will be computed and certain “zero sets” will be computed. • From here the “building blocks” of all cycles can be constructed which also reveals the numbers of cycles.
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