 As we've previously discussed, the qualitative dynamic behavior of nonlinear systems is largely defined by the positive and negative feedback loops that regulate their behavior. With negative feedback working to dampen down or constrain change to a linear progression, whilst positive feedback works to amplify change typically in a super linear fashion. As opposed to negative feedback where we get a gradual and often stable development over a prolonged period of time, what we might call normal or equilibrium state of development. Positive feedback development is fundamentally unsustainable because all systems in reality exist within an environment that will ultimately place a limit to this growth. From this we can see how exponential growth enabled by positive feedback loops is what we might call special. It can only exist for a relatively brief period of time. When we look around us we see the vast majority of things are in a stable configuration constrained by some negative feedback loop, whether this is the law of gravity, predator prey dynamics or just the economics of having to get out of bed every day and go to work. These special periods of positive feedback development are characteristic and key drivers of what we call phase transitions. A phase transition may be defined as some smooth small change in a quantitative input variable that results in a qualitative change in the system state. The transition of ice to steam is one example of a phase transition. At some critical temperature a small change in the system's input temperature value results in a systemic change in the substance after which it is governed by a new set of parameters with a new set of properties. For example we can talk about cracking ice but not water or we can talk about the viscosity of a liquid but not a gas as these are in different phases under different physical regimes and thus we describe them with respect to different parameters. Another example of a phase transition may be the change within a columnar bacteria that when we change the heat and nutrient input to the system we change the local interactions between the bacteria and get a new emergent structure to the colony. Although this change in input value may only be a linear progression it results in a qualitatively different pattern emerging on the macro level of the colony. It is not simply that a new order or structure has emerged but the actual rules that govern the system change and thus we use the word regime and talk about it as a regime shift as some small change in a parameter that affected the system on the local level leads to different emergent structures on the macro level then feeds back to define a different regime that the elements have to operate under. Another way of talking about this is within the language of bifurcation theory whereas with phase transitions we're talking about the qualitative change in the properties of the system. Bifurcation theory really talks about how a small change in a parameter can cause a topological change in a system's environment resulting in new attractor states emerging. A bifurcation means a branching. In this case we're talking about a point where the future trajectory of an element in a system divides or branches out as new attractor states emerge. From this critical point it can go in two different trajectories which are the product of these attractors. Each branch represents a trajectory into a new basin of attraction with a new regime and equilibrium. So to take a real world example of a bifurcation say you've been studying fine art as an undergraduate. This subject has for the past few years represented your basin of attraction that is to say your studies have cycled through its many different domains but never moved off into another totally different subject. But now that you've graduated you have the option to continue your studies in either sculpture or painting. You've now reached a bifurcation point as two new attractors have opened up in front of you. Some small event at this critical point could define your long-term trajectory into one of these two different basins of attraction. Lastly, as opposed to linear systems that may develop in an overall incremental fashion the exponential growth that nonlinear systems are capable of through feedback loops and phase transitions leads to a different overall pattern to their development what we might call punctuated equilibrium. Within this model of punctuated equilibrium the development of a nonlinear system is marked by a dynamic between positive and negative feedback. With negative feedback holding the system within a basin of attraction that represents a period of stable development. These stable periods are then punctuated by periods of positive feedback which take the system far from its equilibrium and into a phase transition as the fundamental topology of its attractor states change and bifurcate. Examples of this punctuated equilibrium might be the development of economies that go through periods of stable development then rapid change through economic crisis and recovery or ecosystems as they collapse due to some environmental change and then an ensuing period of rapid regrowth towards a new equilibrium of stable development again. The same punctuated equilibrium may be seen in the development of a human being as they go from childhood to adulthood to old age. Each period represents a stable basin of attraction with changes between each being marked by periods of rapid and defining change. To summarize then in this module we have discussed phase transitions and bifurcations as periods of qualitative and often rapid change in the dynamics of a nonlinear system state. We have tried to present an integrated picture of this that incorporates our previous discussion on feedback loops to show how nonlinear systems often don't develop in an incremental fashion but their development is marked by a model of punctuated equilibrium with contrasting periods of stability and rapid growth.