Climate tipping as a noisy bifurcation: a predictive technique

Instantaneous Basin loss at a Fold
Before After

Introduction
Focus on the Earth, or a relevant sub-system (Lenton).
Regard it as a nonlinear dissipative dynamical system.
Ignore discontinuities and memory effects.
We have a large but finite set of ODEs and phase space.
This large complex system has activity at many scales.

Effective Noise
Small fast action is noise to the overall dynamics (OD)
Models of the OD might need added random noise
Bifurcations of the OD may underlie climate tipping
…………………………………………………..………………………….
Control Parameters
We may have many slowly-varying control parameters, µi
But they can subsumed into a single µ (eg. slow time)
This limits the relevant bifurcations to those with co-dimension (CD) = 1
We now explain the co-dimension concept, before moving on to classify the CD = 1 bifurcations

Unfolding Euler’s Pitchfork
A real column has imperfections.
With P it does not reach pitchfork, C.
Catastrophe Theory shows that only
one extra control is needed to hit C.
One such control is the side load, R.
R = R* cancels out the imperfections.
Needing 2 controls to be observable
we say a pitchfork has co-dimension 2.
A climate tip from a single slow evolution must be co-dimension 1.

Co-Dimension 1 Bifurcations
(we shall be listing all 18)
Bifurcations can be classified as:
(a) Safe Bifurcations
(b) Explosive Bifurcations
(c) Dangerous Bifurcations

Safe and
dangerous
forms of the
Hopf
bifurcation
click

SAFE

EXPLOSIVE

Example of an Explosive Event
Flow-explosion transforms point attractor to a cycle
Equilibrium path has a regular saddle-node fold.
Saddle outset flows around a closed loop to the node.
A stable cycle is created.
Initial period is infinite (critical slowing).
Precursor: same as static fold.

DANGEROUS

BASINS (1)

BASINS (2)

Precursors of our 18 bifurcations

INDETERMINATE JUMP

Concluding Remarks
Bifurcation concepts for climate studies:
Co-dimension-one events in dissipative systems.
Safe, explosive and dangerous forms.
Hysteresis and basin boundary structure
Slowing of transients prior to an instability.