Gabriel Schreiber & Roberto Umansky
The mathematical theory of bifurcation originated in the seminal work of Henri Poincaré on systems of non-linear differential equations. The term bifurcation was coined by Poincaré to designate the emergence of several solutions from a given solution. Whenever the solution to an equation, or system of equations, changes qualitatively at a fixed value of a parameter, called a critical value, the phenomenon is called a bifurcation. The point in the parameter space where such an event occurs is defined a bifurcation point. From a bifurcation point several stable or unstable solution branches emerge. Successive bifurcations lead to an irregular and unpredictable time evolution of deterministic nonlinear systems, which is designated chaos. The unique character of chaotic dynamics is their sensitivity to initial conditions as described by Poincaré:
“It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon” (397).
If prediction becomes impossible, it is evident that a chaotic system can resemble a stochastic system (a system subject to random external forces). However, the source of the irregularity is quite different. For chaos, the irregularity is part of the intrinsic dynamics of the system, rather than unpredictable outside influences. Chaos enables determinism and unpredictability to coexist in the same system. Moreover, surprisingly, a very well defined universal route, which leads from order to chaos, was discovered by Mitchell Feigenbaum. There are abrupt qualitative changes: ordered successive bifurca- tions, which mark a universally ordered transition from order to chaos: Feigenbaum’s universality. The idea of bifurcation is central to contemporary physical theories of irreversible, far-from equilibrium thermodynamics. The contributions of Prigogine’s Brussels School (Prigogine, Prigogine & Stengers) in this regard are of prime importance, showing that bifurcations under far-from-equilibrium conditions constitute the natural mechanism of evolution and of acquisition of complexity.
With a sense of derealization, it may seem strange to the reader that this text has fallen into a journal on literature. What has it to do with literature? What has it to do with Borges? We may be reminded by the metaphysicians of Tlön who “Judge that metaphysics is a branch of the literature of fantasy”. Borges once claimed that the basic devises of all fantastic literature are only four in number: the work within a work, the contamination of reality by dream, the voyage in time, and the double (Irby xviii).