Section 1
Classical Deterministic Systems
Using the dynamics given in Equation (3.4), determine what the state of the system would be if you start with the state $[5,5,0,2,0,15]^T$.
For the matrix $M$ given in Equation (3.4), calculate $M^2, M^3$, and $M^6$. If all the marbles start at vertex 2 , where will all the marbles end up after 6 time steps?
What would happen if we relaxed the requirement that exactly one edge leaves each vertex, i.e., what would happen if we permitted any graph?
What would happen if we permitted not only 0's and 1's but also -1 in the adjacency matrix? Give an interpretation of this scenario in terms of marbles.
Consider the following graph representing city streets. Singleheaded arrows $(\longrightarrow)$ correspond to one-way streets and double-headed arrows $(\longleftrightarrow)$ correspond to two-way streets.(FIGURE CAN'T COPY)Imagine that it takes one time click to traverse an arrow. You may assume that everyone must move at every time click. If every corner starts with exactly one person, where will everyone be after one time click? After two time clicks? After four time clicks?