# Concepts from bifurcation theory

A major component of non-linear dynamics is the theory of bifurcations, these being points in the slow evolution of a system at which qualitative changes or even sudden jumps of behaviour can occur.

In the field of dissipative dynamics co-dimension-1 bifurcations are those events that can be 'typically' encountered under the slow sweep of a single control parameter. A climate model will often have (or be assumed to have) such a parameter under the quasi-static variation of which the climate is observed to gradually evolve on a 'slow' timescale. Slowly varying parameters are external influences that vary on geological timescales, for example, the obliquity of the Earth's orbit. Another common type of slowly varying parameter occurs if one models only a subsystem of the climate, for example, oceanic water circulation. Then the influence of an interacting subsystem (for example, freshwater forcing from melting ice sheets) acts as a parameter that changes slowly over time.

Table 3.2 Safe bifurcations. These include the supercritical forms of the local bifurcations and the less well-known global 'band merging'. The latter is governed by a saddle-node event on a chaotic attractor. Alternative names are given in parentheses.

Point to cycle Cycle to torus Cycle to cycle

Chaos to chaos

These bifurcations are characterised by the following features:

Subtle: continuous supercritical growth of new attractor path Safe: no fast jump or enlargement of the attracting set Determinate: single outcome even with small noise No hysteresis: path retraced on reversal of control sweep No basin change: basin boundary remote from attractors No intermittency: in the responses of the attractors

(a) Local supercritical bifurcations

1. Supercritical Hopf

2. Supercritical Neimark-Sacker (secondary Hopf)

3. Supercritical flip (period-doubling)

(b) Global bifurcations

### 4. Band merging

Table 3.3 Explosive bifurcations. These are less common global events, which occupy an intermediate position between the safe and dangerous forms. Alternative names are given in parentheses.

Point to cycle Cycle to torus Point to chaos Cycle to chaos Chaos to chaos Chaos to chaos

Catastrophic: global events, abrupt enlargement of attracting set Explosive: enlargement, but no jump to remote attractor Determinate: with single outcome even with small noise No hysteresis: paths retraced on reversal of control sweep No basin change: basin boundary remote from attractors Intermittency: lingering in old domain, flashes through the new

5. Flow explosion (omega explosion, SNIPER)

6. Map explosion (omega explosion, mode-locking)

7. Intermittency explosion: Flow

8. Intermittency explosion: Map (temporal intermittency)

9. Regular-saddle Explosion (interior crisis)

10. Chaotic-saddle Explosion (interior crisis)

These bifurcations are characterised by the following features:

An encounter with a bifurcation during this evolution will be of great interest and significance, and may give rise to a dynamic jump on a much faster timescale. A complete list of the (typical) co-dimension-1 bifurcations, to the knowledge of the authors at the time of writing, is given by Thompson & Stewart (2002). It is this list of local and global bifurcations that is used to populate Tables 3.2-3.5. The technical details and terminology of these tables need not concern the general

Table 3.4 Dangerous bifurcations. These include the ubiquitous folds where a path reaches a smooth maximum or minimum value of the control parameter, the subcritical local bifurcations, and some global events. They each trigger a sudden jump to a remote 'unknown' attractor. In climate studies these would be called tipping points, as indeed might other nonlinear phenomena. Alternative names are given in parentheses.

(a) Local saddle-node Bifurcations

11. Static fold (saddle-node of fixed point)

12. Cyclic fold (saddle-node of cycle)

(b) Local Subcritical Bifurcations

13. Subcritical Hopf

14. Subcritical Neimark-Sacker (secondary Hopf)

15. Subcritical flip (period-doubling)

(c) Global Bifurcations

16. Saddle connection (homoclinic connection)

17. Regular-saddle catastrophe (boundary crisis)

18. Chaotic-saddle catastrophe (boundary crisis)

These bifurcations are characterised by the following features:

Catastrophic: sudden disappearance of attractor Dangerous: sudden jump to new attractor (of any type) Indeterminacy: outcome can depend on global topology Hysteresis: path not reinstated on control reversal Basin: tends to zero (b), attractor hits edge of residual basin (a, c) No intermittency: but critical slowing in global events

From point From cycle

### From point From cycle From cycle

From cycle From chaos From chaos reader, but they do serve to show the vast range of bifurcational phenomena that can be expected even in the simplest nonlinear dynamical systems, and certainly in climate models as we see in Section 3.6.

A broad classification of the co-dimension-1 attractor bifurcations of dissipative systems into safe, explosive and dangerous forms (Thompson et al., 1994) is illustrated in Tables 3.2-3.4 and Figure 3.2, while all are summarised in Table 3.5 together with notes on their precursors. It must be emphasised that these words are used in a technical sense. Even though in general the safe bifurcations are often literally safer than the dangerous bifurcations, in certain contexts this may not be the case. In particular, the safe bifurcations can still be in a literal sense very dangerous: as when a structural column breaks at a 'safe' buckling bifurcation!

Note carefully here that when talking about bifurcations we use the word 'local' to describe events that are essentially localised in phase space. Conversely we use the word 'global' to describe events that involve distant connections in phase space. With this warning, there should be no chance of confusion with our use, elsewhere, of the word 'global' in its common parlance as related to the Earth.

Table 3.5 List of all co-dimension-1 bifurcations of continuous dissipative dynamics, with notes on their precursors. Here S, E and D are used to signify the safe, explosive and dangerous events respectively. LDR is the local decay rate, measuring how rapidly the system returns to its steady state after a small perturbation. Being a linear feature, the LDR of a particular type of bifurcation is not influenced by the sub- or supercritical nature of the bifurcation.

### Precursors of Co-dimension-1 Bifurcations

Supercritical Hopf Supercritical Neimark Supercritical flip Band merging Flow explosion Map explosion Intermittency explosion: Intermittency explosion: Regular interior crisis Chaotic interior crisis Static fold Cyclic fold Subcritical Hopf Subcritical Neimark Subcritical flip Saddle connection Regular exterior crisis Chaotic exterior crisis

S: point to cycle S: cycle to torus S: cycle to cycle S: chaos to chaos E: point to cycle E: cycle to torus flow E: point to chaos map E: cycle to chaos E: chaos to chaos E: chaos to chaos D: from point D: from cycle D: from point D: from cycle D: from cycle D: from cycle D: from chaos D: from chaos

LDR ^ 0 linearly with control LDR ^ 0 linearly with control LDR ^ 0 linearly with control Separation decreases linearly Path folds. LDR ^ 0 linearly along path Path folds. LDR ^ 0 linearly along path LDR ^ 0 linearly with control LDR ^ 0 as trigger (fold, flip, Neimark) Lingering near impinging saddle cycle Lingering near impinging chaotic saddle Path folds. LDR ^ 0 linearly along path Path folds. LDR ^ 0 linearly along path LDR ^ 0 linearly with control LDR ^ 0 linearly with control LDR ^ 0 linearly with control Period of cycle tends to infinity Lingering near impinging saddle cycle Lingering near impinging accessible saddle

In Tables 3.2-3.4 we give the names of the bifurcations in the three categories, with alternative names given in parentheses. We then indicate the change in the type of attractor that is produced by the bifurcation, such as a point to a cycle, etc. Some of the attributes of each class (safe, explosive or dangerous) are then listed at the foot of each table. Among these attributes, the concept of a basin requires some comment here. In the multidimensional phase space of a dissipative dynamical system (described in Section 3.2) each attractor, or stable state, is surrounded by a region of starting points from which a displaced system would return to the attractor. The set of all these points constitutes the basin of attraction. If the system were displaced to, and then released from any point outside the basin, it would move to a different attractor (or perhaps to infinity). Basins also undergo changes and bifurcations, but for simplicity of exposition in this brief review we focus on the more common attractor bifurcations.

In Figure 3.2 we have schematically illustrated three bifurcations that are co-dimension-1 , meaning that they can be typically encountered under the variation of a single control parameter, which is here plotted horizontally in the left column.

Safe event Super-critical Hopf

q

(a)

-----------------

' jr Cycle

Point ^ -1

Merit

Stable path Unstable path
 <1 slightly larger disturbance ^ ^srnall disturbance

Figure 3.2 Schematic illustration of the three bifurcation types. On the left the control parameter, ¡i , is plotted horizontally and the response, q, vertically. The middle column shows the time series of a response to small disturbances if ¡i < ^ crit- On the right we show how the system drifts away from its previously stable steady state if ¡i > //„it- The different types of events are (from top to bottom) (a) safe, (b) explosive and (c) dangerous.

Figure 3.2 Schematic illustration of the three bifurcation types. On the left the control parameter, ¡i , is plotted horizontally and the response, q, vertically. The middle column shows the time series of a response to small disturbances if ¡i < ^ crit- On the right we show how the system drifts away from its previously stable steady state if ¡i > //„it- The different types of events are (from top to bottom) (a) safe, (b) explosive and (c) dangerous.

The response, q, is plotted vertically. To many people, the most common (safe) bifurcation is what is called the supercritical pitchfork or stable-symmetric point of bifurcation (Thompson & Hunt, 1973). This was first described by Euler (1744) in his classic analysis of the buckling of a slender elastic column, and is taught to engineering students as 'Euler buckling' in which the load carried by the column is the control parameter. Poincare (1885) explored a number of applications in astrophysics. In this event, the trivial primary equilibrium path on which the column has no lateral deflection (q = 0), becomes unstable at a critical point, C, where ^ = ^crit. Passing vertically though C, and then curving towards increasing is a stable secondary equilibrium path of deflected states, the so-called post-buckling path. The existence of (stable) equilibrium states at values of ¡> ¡crit is why we call the bifurcation a supercritical pitchfork. In contrast, many shell-like elastic structures exhibit a dangerous bifurcation with an (unstable) post-buckling path that curves towards decreasing values of the load, and is accordingly called a subcritical pitchfork. These two pitchforks are excellent examples of safe and dangerous bifurcations, but they do not appear in our lists because they are not co-dimension-1 events in generic systems. That the bifurcation of a column is not co-dimension-1 manifests itself by the fact that a perfectly straight column is not a typical object; any real column will have small imperfections, lack of straightness being the most obvious one. These imperfections round off the corners of the intersection of the primary and secondary paths (in the manner of the contours of a mountain pass), and destroy the bifurcation in the manner described by catastrophe theory (Poston & Stewart, 1978; Thompson, 1982). We shall see a subcritical pitchfork bifurcation in a schematic diagram of the THC response due to Rahmstorf (2000) in Figure 3.3 below. This is only observed in very simple (non-generic) models and is replaced by a fold in more elaborate ones.

It is because of this lack of typicality of the pitchforks, that we have chosen to illustrate the safe and dangerous bifurcations in Figure 3.2 by other (co-dimension-1) bifurcations. As a safe event, we show in Figure 3.2(a) the supercritical Hopf bifurcation. This has an equilibrium path increasing monotonically with ¡i whose point attractor loses its stability at C in an oscillating fashion, throwing off a path of stable limit cycles which grow towards increasing This occurs, for example, at the onset of vibrations in machining, and triggers the aerodynamic flutter of fins and ailerons in aircraft. Unlike the pitchfork, this picture is not qualitatively changed by small perturbations of the system.

As our explosive event, we show in Figure 3.2(b) the flow explosion involving a saddle-node fold on a limit cycle. Here the primary path of point attractors reaches a vertical tangent, and a large oscillation immediately ensues. As with the supercritical Hopf, all paths are re-followed on reversing the sweep of the control parameter there is no hysteresis.

Finally, as our dangerous event in Figure 3.2(c), we have chosen the simple static fold (otherwise known as a saddle-node bifurcation), which is actually the most common bifurcation encountered in scientific applications: and we shall be discussing one for the THC in Section 3.6.1. Such a fold is in fact generated when a perturbation rounds off the (untypical) subcritical pitchfork, revealing a sharp imperfection sensitivity notorious in the buckling of thin aerospace shell structures (Thompson & Hunt, 1984). In the fold, an equilibrium path of stable point attractors being followed under increasing ^ folds smoothly backwards as an unstable path towards decreasing ^ as shown. Approaching the turning point at /xciit there is a gradual loss of attracting strength, with the LDR of transient motions (see Section 3.4) passing directly through zero with progress along the arc-length of the path. This makes its variation with ^ parabolic, but this fine distinction seems to have little significance in the climate tipping studies of Sections 3.6-3.7. Luckily, in these studies, the early decrease of LDR is usually identified long before any path curvature is apparent. As ^ is increased through ^crit the system finds itself with no equilibrium state nearby, so there is inevitably a fast dynamic jump to a remote attractor of any type. On reversing the control sweep, the system will stay on this remote attractor, laying the foundation for a possible hysteresis cycle.

We see immediately from these bifurcations that it is primarily the dangerous forms that will correspond to, and underlie, the climate tipping points that concern us here. (Though if, for example, we adopt Lenton's relatively relaxed definition of a tipping point based on time-horizons (see Section 3.5), even a safe bifurcation might be the underlying trigger.) Understanding the bifurcational aspects will be particularly helpful in a situation where some quasi-stationary dynamics can be viewed as an equilibrium path of a mainly deterministic system, which may nevertheless be stochastically perturbed by noise. We should note that the dangerous bifurcations are often indeterminate in the sense that the remote attractor to which the system jumps often depends with infinite sensitivity on the precise manner in which the bifurcation is realised. This arises (quite commonly and typically) when the bifurcation point is located exactly on a fractal basin boundary (McDonald et al., 1985; Thompson, 1992, 1996). In a model, repeated runs from slightly varied starting conditions would be needed to explore all the possible outcomes.

Table 3.5 lists the precursors of the bifurcations from Tables 3.2-3.4 that one would typically use to determine if a bifurcation is nearby in a (mostly) deterministic system. One perturbs the observed steady state by a small 'kick'. As the steady state is still stable the system relaxes back to the steady state. This relaxation decays exponentially proportional to exp(Xt) where t is the time and X (a negative quantity in this context) is the critical eigenvalue of the destabilizing mode (Thompson & Stewart, 2002). The local decay rate, LDR (called k in Section 3.4), is the negative of X.

Defined in this way, a positive LDR tending to zero quantifies the 'slowing of transients' as we head towards an instability. We see that the vast majority (though not all) of the typical events display the useful precursor that the LDR vanishes at the bifurcation (although the decay is in some cases oscillatory). Under light stochastic noise, the variance of the critical mode will correspondingly exhibit a divergence proportional to the reciprocal of the LDR. The LDR precursor certainly holds, with monotonic decay, for the static fold which is what we shall be looking at in Section 3.6.1 in the collapse of the North Atlantic thermohaline circulation. The fact, noted in Table 3.5, that close to the bifurcation some LDRs vary linearly with the control, while some vary linearly along the (folding) path is a fine distinction that may not be useful or observable in climate studies.

The outline of the co-dimension-1 bifurcations that we have just presented applies to dynamical flows which are generated by continuous systems where time changes smoothly as in the real world, and as in those computer models that are governed by differential equations. There are closely analogous theories and classifications for the bifurcations of dynamical maps governed (for example) by iterated systems, where time changes in finite steps. It is these analogous theories that will be needed when dealing with experimental data sets from ice cores, etc., as we shall show in the following section. Meanwhile the theory for discrete time data, has direct relevance to the possibility of tipping points in parts of the biosphere where time is often best thought of in generations or seasons; in some populations, such as insects, one generation disappears before the next is born.

The equivalent concept that we shall need for analysing discrete-time data is as follows. The method used in our examples from the recent literature (in Sections 3.6 and 3.7) is to search for an underlying linearised deterministic map of the form yn+1 = cyn which governs the critical slowing mode of the transients. This equation represents exponential decay when the eigenvalue of the mapping, c, is less than 1, but exponential growth when c is greater than 1. So corresponding to LDR dropping to zero, we shall be expecting c to increase towards unity.

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