Diagram: Bifurcations


Complex systems do not follow linear, predictable chains of cause and effect. Instead, system trajectories can diverge wildly into entirely different regimes. The moments in time when a system splits into different, but equally probable trajectories is referred to as a system bifurcation.

CAS are { open-dissipative-structures}}, Path Dependency, systems, with {{multiple-equilibriumand the ability to generate new order. Here, system inputs like heat, energy, food, etc., can traverse dissipative boundaries and ‘drive’ the system towards order: seemingly in violation of the second law of thermodynamics. As the intensity of such inputs increases, critical values are reached whereupon the system can move into different, but equally viable, potential states.

The ‘choice’ of these states is predicated upon an extreme {{sensitivity-to-initial conditions}} HANDLEBAR FAIL, with small changes in initial conditions potentially leading to large shifts in the system’s ultimate global behavior.  Further, as the amplitude of the relevant control parameter increases, the number of system states multiplies - eventually arriving at a ‘chaotic’ regime wherein all potential states are accessible. The total range of potential system states are equated with its Degrees of Freedom or Phase Space

Bifurcation diagrams (also called logistic maps) and Reimann Manifolds are used to help map the breadth and topology of Phase Space, while also illustrating how systems can move between multiple potential states and shift suddenly at critical points. These threshold moments, dubbed 'catastrophes'  by Rene Thom, bifurcations  by Mitchell Feigenbaum, and popularized as 'tipping points' by Malcolm Gladwell, coincide with moments when a system has the capacity to move into alternative regimes of newly available trajectories,(such as in Benard experiments when water molecules form 'roll' patterns at critical temperatures that can either flow left to right or right to left (see the video in the feed on the right)), or when system components suddenly acquire new features (such as when system components suddenly acquire new features (such as when water molecules turn to ice at a critical temperature).

A system bifurcation tree shows how layers of these changes  unfold - with the result that a broad range of possible trajectories have the potential to be traced over the course of time. 



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Benard Rolls

This video shows the emergence of Benard Convection rolls: a classic example of an emergent phenomena that also is subject to a system bifurcation. The direction of the rolls move is subject to a bifurcation point in the system - with an equal probability that the rolls move in one vs the opposite direction. The actual direction the rolls move cannot be predicted and is part of the non-linearity of the system.