Nonlinearity: The Simple and Useful Side of Dynamical Analysis
April 8th, 2015
by Ted Carron
Appeared originally at Inside, vol.2, no. 4
The reform of economics requires widespread appreciation of dynamics – especially nonlinear dynamics. The maths of dynamics are admittedly intimidating, but dynamic system behaviours can be accessibly described without recourse to mathematical jargon.
In my recent ‘Inside’ article about dynamics and morality (“Love, High School, and Learning Dynamics Early”), I suggested that our view of the world would be made richer and more humane by an appreciation of the metaphors available from the ideas in nonlinear dynamics. In this article I will describe some those ideas in more detail – though still at a non-mathematical level. So as not to be too intimidating, I will break the topic down into to two articles. This first article will deal with basic dynamic concepts and different aspects of equilibrium. The second article will deal with cycles and chaos (now less romantically called ‘complexity’).
All the ideas I am going to talk about come from an area of dynamics known as ‘dynamical’ analysis (not misspelled – that really is a word). Dynamical analysis is the description of sometimes fiendishly complex dynamic behaviours in terms of a small number simple, ‘graphical’ components.
This is a crucial notion for dynamic economists because the behaviour of nonlinear dynamic systems varies hugely depending on initial conditions, so the full nature of the system cannot be understood just by watching it in action in a simulator. Dynamical analysis gives people who work with dynamic systems a language with which to identify and describe the overall nature of a dynamic system. Surprisingly, the language of dynamical analysis is very accessible and can be usefully understood with no maths at all.
Dynamical analyses are expressed graphically using phase diagrams which are a kind of graph where each axis represents a variable of the system. Typically phase diagrams have one, two, or three variables, and hence one, two, and three axes. Each location on the diagram represents a unique state. The coordinates of each point are the variable values for that state. The set of all possible system states is called the phase space.
Here is a phase diagram for a system with two variables (speed and temperature) – and therefore two axes. The red cross indicates the system state where ‘speed’ = 2 and ‘temperature’ = 3.
Evolution and Trajectories
A dynamic system is comprised of a number of controlling formulae – one for each variable. For each possible state the controlling formula determines which adjacent state the system will move to (‘evolve’ to) in the following instant. As soon as the system arrives at this new state it will immediately move (evolve) to the next state determined by those same system formulae and then the next and the next, and so on.
Successive infinitely small evolutions will trace out a path called a ‘trajectory’. Here is a trajectory in a two-variable system where the system moves from state (-4,-2) to state (3,1)
- trajectories are always continuous: they never have gaps; and
- trajectories can never cross each other because each point can lead to only one other point – this is called the uniqueness rule (Borrelli and Coleman (1987))
The simplest and most ubiquitous component of dynamical analysis is the “fixed point.” This is a system state where the system formulae indicate that the ‘next’ state is that same state. In other words, the system stops evolving and comes to rest. Dynamic systems can have zero, one, or many fixed points. Identification of all the fixed points of a system is a key activity in dynamical analysis.
Fixed points are further classified according to the behaviour of the system close to those points, as follows.
Stable Fixed Points
A stable fixed point can be thought of as an attractor: if the system is in a state near a stable fixed point, it will be drawn towards the fixed point. You can also think of a stable fixed point as an equilibrium state for its system.
Here is a one-dimensional phase diagram for a system comprised of a weight on a spring where the x-axis represents the spring’s extension.
If the spring is longer than its equilibrium length, it will contract. If it is shorter, it will expand, eventually coming to rest at its equilibrium length.
The solid black dot on the x-axis is the standard way of indicating a stable fixed point in a dynamic system. In this case the fixed point is at the extension value where the recovering force of the spring matches the force of gravity on the weight. The two arrows indicate two sample trajectories illustrating how the system state is ‘attracted’ to the stable fixed point.
‘Global’ and ‘Local’ Equilibria (stable fixed points.)
If every possible state in the system is ‘attracted’ towards a single equilibrium, then that equilibrium is global (e.g., a weight on the spring). This would describe the general equilibrium of neoclassical economics – if such a thing existed.
If a system has more than one stable equilibrium, then each equilibrium is described as a “local equilibrium.” In a system with multiple stable equilibria, the final resting state of the system depends on which equilibrium the system is most influenced by in its initial state. The possibility of multiple equilibria has long been theorized in development economics (Rosenstein-Rodan (1943)), where it is thought that countries below a certain level of development can find themselves in a lower equilibrium level of development than those already above the watershed.
Unstable Fixed Points
An unstable fixed point can be thought of as a repeller. A system will be at rest whilst it is exactly at an unstable fixed point but if the system is in a state close by to the fixed point it will evolve AWAY from it. A simple example of system with an unstable fixed point is a pencil balanced on its point: Theoretically there is some position where the pencil will remain balanced in this way, but that condition is infinitesimally small. In reality the pencil will fall away from – be repelled by – the theoretical point of balance.
Here is a phase diagram showing an unstable fixed point (hollow point) between two stable fixed points.
Hopefully you can see that the system will end up at one of the local fixed points, depending on which side of the repeller it started.
We now have enough information to do a little dynamical analysis of our own – at least intuitively. Consider a pendulum with a stiff bar. What fixed point(s) does it have?
You can likely see that it is at a stable fixed point when it is hanging straight down: If we move the pendulum away from this state (point), it will end up (eventually) back resting at that point.
What might be less obvious is that the pendulum also has a second fixed point. That is when it balanced straight up. Of course, this is an unstable fixed point. Any slight movement away from the vertical will cause the system (pendulum) to move away from that state. So, in short, a pendulum has two fixed points: one stable (pointing straight down) and one unstable (pointing straight up).
Saddle Fixed Points
This is a fixed point which is an attractor from both sides in one pair of directions and a repeller on both sides of the other. Here is the phase diagram for a saddle point in a two-dimensional system.
The ‘stable manifold’ is simultaneously the watershed of repulsion and also the only line of attraction for the fixed point. If you study this for a moment, you can see that a saddle point is just a flavour of repeller. The only trajectories that are not repelled are those which start exactly on the dotted, infinitely thin stable manifold. All other trajectories are repelled.
Saddle points feature in the neoclassical DGSE price determination model. Strangely, however, the DGSE saddle point is characterized in the conventional treatment not as a nasty unstable repeller, but as a nice stable equilibrium. This transformation is achieved by presuming that the system is compelled to sit exactly on that stable manifold like that pencil balanced perfectly on its point. In order to justify this convenient unlikelihood, it is necessary to assume that economic agents – in aggregate – perfectly predict the future under the doctrine of the rational agency. (Burmeister et al. (1973))
Bifurcation & Parameter Drift
Okay, now things get really interesting. “Bifurcation” refers to the processes whereby fixed points are created, moved, and/or destroyed. This can happen as the result of a change in the value of a system parameter: When the parameter has values in a particular range a fixed point exists; when it is outside that range, the fixed point does not exist.
To understand the possible consequences of this, consider a system with two equilibria (stable fixed points) each side of a saddle point repeller.
Let us assume that the system starts at rest in the right-most stable equilibrium (shown by the red cross).
Everything is nice and peaceful. But over time a parameter variable changes (drifts) and gradually the saddle point and the right-most equilibrium move toward each other.
At this point, nothing changes. Whilst all equilibria are in existence, the state of a system which is stationary at a stable fixed point is unaffected by the drifting parameter.
But at some point the parameter variable drifts outside the crucial range, and the right-most stable equilibrium merges with the saddle node and disappears. Now all of a sudden the system is unstable and QUICKLY evolves to the only remaining stable fixed point. This sudden ‘movement’ is known in dynamical analysis (and often in the real world) as a ‘catastrophe’.
The ‘interesting’ thing about this situation is that even if the controlling parameter moves back into the range where the temporary equilibrium is reinstated, the system will remain at its new – post-catastrophe – equilibrium where it came to rest.
This is something like what goes on when an asset price bubble bursts. The drifting parameter is liquidity; in the good times, liquidity is high, driving prices towards an ever-higher equilibrium asset price. When liquidity is reduced far enough, the high equilibrium price evaporates, and prices move catastrophically down towards a much lower equilibrium price. (This is a barely forgivable over-simplification, but it gives a valid idea of the dynamic process going on.)
Greenhouse gases and world temperatures are also thought, by some, to behave like this. Currently, it is believed, we inhabit a temperature equilibrium which we have gotten used to over hundreds of years. It is possible that, if greenhouse gas levels rise sufficiently, that this historically stable equilibrium will disappear and we will move ‘catastrophically’ and irreversibly toward a hotter equilibrium. The process of sudden adaption to this new equilibrium will be famine and war. It is not the risk of heatstroke in the new equilibrium which scares scientists it is the suddenness of the ‘catastrophic’ movement from one equilibrium to another.
So that’s it. Hopefully from now on you will see attractors, repellers and catastrophes everywhere :) In the next piece I will talk about chaos – the poster child of nonlinear dynamics.
If you want a very, very accessible introduction to the maths of dynamical analysis I recommend Steven Strogatz’ Nonlinear Dynamics and Chaos (Westview Press). The book does contain a fair amount of maths, but also a lot of interesting accessible examples from engineering and biology. You might bleed a little from the ears, but you will end up with a better view of the world.
(Borrelli and Coleman 1987) Differential Equations: A modeling approach (Prentice-Hall, Englewood Cliffs, NJ)
(Rosenstein-Rodan 1943) Rosenstein-Rodan, P. “Problems of Industrialization of Eastern and Southeastern Europe,” Economic Journal 53, 202–211.
Burmeister, E., Caton, C., Dobell, A.R., Ross, S., (1973) “The ‘saddle-point’ property and the structure of dynamic heterogeneous capital good models”, Econometrica, 41, pp.79-95.