Bifurcations in Time

"In all fiction, when a man is faced with alternatives he chooses one at the expense of the others. In the almost unfathomable Ts'ui Pen, he chooses—simultaneously—all of them. He thus creates various futures, various times which start others that will in their turn branch out and bifurcate in other times." (Jorge Luis Borges, The Garden of Forking Paths).

In this short story, Yu Tsun's ancestor wrote a maze-like novel. Rather than events having one outcome, various outcomes would occur simultaneously, leading to various futures. Theoretically, Ts'ui Pen considered all possibilities for the future and for the past as having occurred.  This philosophy of time can be better understood in comparison to Bifurcation diagrams in biological modeling.

Take for example an equation to model population size with respect to time, the discrete logistic equation: 

x_n+1 = rx_n(1-x_n)

x_n is the population size at time n. So the equation compares the population size at time n+1 to the size at time n. In the equation, "r" is a constant that describes the growth rate of the population. Depending on this parameter "r", the population will exhibit different growth patterns. A bifurcation diagram compares the value of the parameter "r" with the long-term behavior of the population size.

When "r" is small (less than about 3), the population will converge to a certain size. When r is greater than 3 but less than 3.5, the population will cycle between 2 population values. This is the "split" we see in the diagram (called a bifurcation point). When "r" is larger than this value, we begin to see more chaotic and less predictable behavior in the population size.

Ts'ui Pen novel could be seen as a sort of bifurcation diagram. He considers many possibilities of what could occur at each point in time. The "bifurcation points" in his novel are the events that change the course of things. The greater number of changes that Ts'ui Pen considered, the greater number of future possibilities he could see. The events that occur each moment determine the directions things will go. 

Just as a tsunami could throw off our model for population size, life will have many unexpected turns. Ts'ui Pen tried to demonstrate these different possibilities in his novel, but even then he did not cover all the possibilities. The future truly does hold an infinite number of possibilities.