Author: Tom

Phase plots are the key to understanding life, the universe and the dynamics of everything.

Well, maybe that’s a bit of an overstatement. But they do nicely explain tipping points and bifurcations, which explain a heck of a lot (as I’ll eventually get to).

Fortunately, phase plots for simple systems are easy to work with. Consider a one-dimensional (first-order) system, like the stock and flow in my bathtub posts.

In Vensim lingo, you’d write this out as,

Stock = INTEG( Flow, Initial Stock )

Flow = ... {some function of the Stock and maybe other stuff}

In typical mathematical notation, you might write it as a differential equation, like

x' = f(x)

where x is the stock and x’ (dx/dt) is the flow.

This system (or vector field) has a one dimensional phase space – i.e. a line – because you can completely characterize the state of the system by the value of its single stock.

Fortunately, paper is two dimensional, so we can use the second dimension to juxtapose the flow with the stock (x’ with x), producing a phase plot that helps us get some intuition into the behavior of this stock-flow system. Here’s an example:

Pure accumulation

In this case, the flow is always above the x-axis, i.e. always positive, so the stock can only go up. The flow is constant, irrespective of the stock level, so there’s no feedback and the stock’s slope is constant.

Left: flow vs. stock. Right: resulting behavior of the stock over time.

Exponential growth

Adding feedback makes things more interesting.

In this simplest-possible first order positive feedback loop, the flow is proportional to the stock, so the stock-flow relationship is a rising line (left frame). There’s a trivial equilibrium (or fixed point) at stock = flow = 0, but it’s unstable, so it’s indicated with a hollow circle. An arrowhead indicates the direction of motion in the phase plot.

The resulting behavior is exponential growth (right frame). The bigger the stock gets, the steeper its slope gets.

Exponential decay

Negative feedback just inverts this case. The flow is below 0 when the stock is positive, and the system moves toward the origin instead of away from it.

The equilibrium at 0 is now stable, so it has a solid circle.

Linear systems like those above can have only one equilibrium. Geometrically, this is because the line of stock-flow proportionality can only cross 0 (the x axis) once. Mathematically, it’s because a system with a single state can have only one eigenvalue/eigenvector pair. Things get more interesting when the system is nonlinear.

S-shaped (logistic) growth

Here, the flow crosses zero twice, so there are two fixed points. The one at 0 is unstable, so as long as the stock is initially >0, it will rise to the stable equilibrium at 1.

(Note that there’s no reason to constrain the axes to the 0-1 unit line; it’s just a graphical convenience here.)

Tipping point

A phase diagram for a nonlinear model can have as many zero-crossings as you like. My forest cover toy model has five. A system can then have multiple equilibria. A pair of stable equilibria bracketing an unstable equilibrium creates a tipping point.

In this arrangement, the stable fixed points at 0 and 1 constitute basins of attraction that draw in any trajectories of the stock that lie in their half of the unit line. The unstable point at 0.5 is the fence between the basins, i.e. the tipping point. Any trajectory starting with the stock near 0.5 is drawn to one of the extremes. While stock=0.5 is theoretically possible permanently, real systems always have noise that will trigger the runaway.

If the stock starts out near 1, it will stay there fairly robustly, because feedback will restore that state from any excursion. But if some intervention or noise pushes the stock below 0.5, feedback will then draw it toward 0. Once there, it will be fairly robustly stuck again. This behavior can be surprising and disturbing if 1=good and 0=bad.

This is the very thing that happens in project fire fighting, for example. The 64 trillion dollar question is whether tipping point dynamics create perilous boundaries in the earth system, e.g., climate.

Not all systems are quite this simple. In particular, a stock is often associated with multiple flows. But it’s often helpful to look at first order subsystems of complex models in this way. For example, Jeroen Struben and John Sterman make good use of the phase plot to explore the dynamics of willingness (W) to purchase alternative fuel vehicles. They decompose the net flow of W (red) into multiple components that create a tipping point:

You can look at higher-order systems in the same way, though the pictures get messier (but prettier). You still preserve the attractive feature of this approach: by just looking at the topology of fixed points (or similar higher-dimensional sets), you can learn a lot about system behavior without doing any calculations.

A while back, Bruce Skarin asked for an explanation of the bifurcations in a nuclear core model. I can’t explain that model well enough to be meaningful, but I thought it might be useful to explain the concept of bifurcations more generally.

A bifurcation is a change in the structure of a model that brings about a qualitative change in behavior. Qualitative doesn’t just mean big; it means different. So, a change in interest rates that bankrupts a country in a week instead of a century is not a bifurcation, because the behavior is exponential growth either way. A qualitative change in behavior is what we often talk about in system dynamics as a change in behavior mode, e.g. a change from exponential decay to oscillation.

This is closely related to differences in topology. In topology, the earth and a marble are qualitatively the same, because they’re both spheres. Scale doesn’t matter. A rugby ball and a basketball are also topologically the same, because you can deform one into the other without tearing.

On the other hand, you can’t deform a ball into a donut, because there’s no way to get the hole. So, a bifurcation on a ball is akin to pinching it until the sides meet, tearing out the middle, and stitching together the resulting edges. That’s qualitative.

Just as we can distinguish a ball from a donut from a pretzel by the arrangement of holes, we can recognize bifurcations by their effect on the arrangement of fixed points or other invariant sets in the state space of a system. Fixed points are just locations in state space at which the behavior of a system maps a point to itself – that is, they’re equilbria. More generally, an invariant set might be a an orbit (a limit cycle in two dimensions) or a chaotic attractor (in three).

A lot of parameter changes in a system will just move the fixed points around a bit, or deform them, without changing their number, type or relationship to each other. This changes the quantitative outcome, possibly by a lot, but it doesn’t change the qualitative behavior mode.

In a bifurcation, the population of fixed points and invariant sets actually changes. Fixed points can split into multiple points, change in stability, collide and annihilate one another, spawn orbits, and so on. Of course, for many of these things to exist or coexist, the system has to be nonlinear.

My favorite example is the supercritical pitchfork bifurcation. As a bifurcation parameter varies, a single stable fixed point (the handle of the pitchfork) abruptly splits into three (the tines): a pair of stable points, with an unstable point in the middle. This creates a tipping point: around the unstable fixed point, small changes in initial conditions cause the system to shoot off to one or the other stable fixed points.

Similarly, a Hopf bifurcation emerges when a fixed point changes in stability and a periodic orbit emerges around it. Periodic orbits often experience period doubling, in which the system takes two orbits to return to its initial state, and repeated period doubling is a route to chaos.

I’ve posted some model illustrating these and others here.

A bifurcation typically arises from a parameter change. You’ll often see diagrams that illustrate behavior or the location of fixed points with respect to some bifurcation parameter, which is just a model constant that’s varied over some range to reveal the qualitative changes. Some bifurcations need multiple coordinated changes to occur.

Of course, a constant parameter in one conception of a model might be an endogenous state in another – on a longer time horizon, for example. You can also think of a structure change (adding a feedback loop) as a parameter change, where the parameter is 0 (loop is off) or 1 (loop is on).

Bifurcations provide one intuitive explanation for the old SD contention that structure is more important than parameters. The structure of a system will often have a more significant effect on the kinds of fixed points or sets that can exist than the details of the parameters. (Of course, this is tricky, because it’s true, except when it’s not.  Sensitive parameters may exist, and in nonlinear systems, hard-to-find sensitive combinations may exist. Also, sensitivity may exist for reasons other than bifurcation.)

Why does this matter? For decision makers, it’s important because it’s easy to get comfortable with stable operation of a system in one regime, and then to be surprised when the rules suddenly change in response to some unnoticed or unmanaged change of state or parameters. For the nuclear reactor operator, stability is paramount, and it would be more than a little disturbing for limit cycles to emerge following a Hopf bifurcation induced by some change in operating parameters.

More on this later.