Bifurcations
The moment at which a system may go two ways. The undercarriage of what we sometimes call a 'tipping moment'.
A bifurcation usually stands for a branching point: a moment in the dynamics of a system at which it could go in two (or more) directions. Which way it eventually goes depends on a combination of factors, several of which we cannot always survey well for the system.
The distinction between a phase transition (see there) and a bifurcation is subtle, but to our minds not unimportant. A phase transition usually refers to the change itself — water becoming ice, a depression entering remission. A bifurcation rather refers to the point at which that change might point somewhere, with several possible directions.
Imagine a river flowing down a small slope. At a certain moment a boulder comes into the path. The water must go around it — left or right. Which side it takes depends on small differences: a millimetre more or less, a twig in the way, the exact shape of the bed. The boulder is a bifurcation. Afterwards two streamlets run on, each with their own dynamics.
Bifurcations in a human being
In the life of a person we often recognise bifurcation points only afterwards. A choice that with the knowledge of then seemed small turns out later to have been the hinge. One who says, "if I had not chosen that course of study", or "if I had not met her that evening", recognises a bifurcation. The system could have gone another way.
In a psychiatric context the idea appears among other places in research on transitions between states — from working to sick leave, from sluggish recovery to sudden gain, from stable to relapse. In time series these bifurcations sometimes appear before they are clinically visible: variance increases, autocorrelations shift, the system 'hesitates' between two possible states.
Bifurcations have 'control parameters'
In the mathematics of complex systems, every bifurcation has a control parameter: a quantity whose change determines whether the system approaches a branching point. The most famous example is the logistic equation, a simple formula that, depending on one parameter, shows calm, periodic or chaotic behaviour.
For human beings, control parameters are not so neatly named. But the idea remains useful: there are often a few underlying factors — sleep, rhythm, social connectedness, physical health, hormonal state — that determine how stable someone is on her path and how close she is to a branching. Our interventions can sometimes also be seen as attempts to influence these control parameters, rather than the symptom directly.
A small caution
It is tempting to call every important moment in someone's life a bifurcation. That makes the concept meaningless. A bifurcation is technically the point at which a dynamical system splits into two or more qualitatively distinct states. To recognise this in people, we depend on measurements, not only on stories.
For the consulting room the concept offers another advantage: it recalls that change does not always proceed smoothly. Sometimes someone walks for a long time on the edge of a knife, and then it tips — usually in the direction in which the system has invested most energy, but not always.