Interesting paper. I have only dealt with fluid dynamic instabilities in simple systems close to equilibrium , I wonder how large departures from equilibrium ( due to our ever increasing radiative forcing ) and the properties of non equilibrium thermodynamics can affect our ability to study abrupt changes in the earth system.

Climate change is wicked problem, and depending on the complexity (or wickedness) of such problems, solutions to such problems may be uncalculatable in reasonable timespan and thus require alternate approximate approaches/strategies such as those discussed in the following Wikipedia article focused on social policy planning:

https://en.wikipedia.org/wiki/Wicked_problemHowever, science has advanced sense C. West Churchman introduced the term "wick problem", and this post focuses on how chaos theory, strange attractors, information networks and energy landscapes can be used to create and calibrate alternate models for wick problems (including better understanding how systemic isolation leads to a lack of human willpower to effective tackle climate change)

The first linked article is entitled: "A mathematical view on personality"; and it introduces the concept of how attractors can be used to model the human psyche within a network (including the use of energy landscape concepts).

http://blogs.plos.org/neuro/2016/03/21/a-mathematical-view-on-personality-by-solve-saebo/Extract: "Interestingly, more diffuse properties of the human psyche, like personality (Van Eenwyk, 1997) and consciousness (Tononi, 2004) may, in fact, be connected to mathematical properties of networks, and in this post I will focus on what mathematics can teach us about these matters.

In mathematics there are complex models for information transfer across networks called attractor networks, and the neural network of our brain appears to be well approximated by these models

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Attractor networks are built from nodes (for example neurons) which typically are recurrently linked (loops) with edges (like synaptic connections), and the dynamics of the network tend to stabilize at least locally to certain patterns. These stable patterns are the attractors. For example, a memory stored in long time memory may be considered as a so-called point attractor, a subnetwork of strongly connected neurons.

The point attractors are low-energy states in an energy landscape with surrounding basins of attraction, much like hillsides surrounding the bottom of a valley, as shown in the figure below. {see the first attached image}

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Also other types of mathematical attractors exist, like line-, plane- and cyclic attractors, and these have been used to explain neural responses like eye-vision control and cyclic motor control, like walking and chewing (Eliasmith, 2005).

Common to these attractors are their stability and predictability, and this is good with regard to having stable memory and stable bodily control, but what about personality? Is personality also an attractor? Do we all have our basins of attraction, which pulls our personality towards stable behavior?

Probably yes, but if you think about it, personality is a more unpredictable property than memory and body control. We think we know someone, and the suddenly they behave in an unexpected manner. Still, the overall personality seems to be more or less stable. How can something be both stable and unpredictable at the same time?

Well there is another class of attractors that may occur in attractor networks. These are the strange (or chaotic) attractors, and they are exactly that, partly stable and partly unpredictable. We say they are bounded, but non-repeating.

A famous example is the Lorenz attractor discovered by Edward Lorenz while he was programming his “weather machine” where typical weather patterns appeared, but never repeated themselves. In the figure below {see the second attached image} the blue curve is pulled towards the red strange attractor state, and once it enters the attractor, it is bound to follow a certain pattern, though it never repeats itself.

The discovery of strange attractors led to the development of chaos theory and fractal geometry in mathematics. Many phenomena around us may develop smoothly in linear predictable fashions until a certain border is reached, at which point a chaotic state appears before a new order may be settled."

The second linked reference cites the development of a dissipative strange attractor that coexists with an invariant conservative torus that can be used to better model brain dynamics.

Artuor Tozzi and James F. Peters (2016), "TOWARDS EQUATIONS FOR BRAIN DYNAMICS AND THE CONCEPT OF EXTENDED CONNECTOME"

http://rxiv.org/pdf/1609.0045v1.pdfAbstract: "The brain is a system at the edge of chaos equipped with nonlinear dynamics and functional energetic landscapes. However, still doubts exist concerning the type of attractors or the trajectories followed by particles in the nervous phase space. Starting from an unusual system governed by differential equations in which a dissipative strange attractor coexists with an invariant conservative torus, we developed a 3D model of brain phase space which has the potential to be operationalized and assessed empirically. We achieved a system displaying both a torus and a strange attractor, depending just on the initial conditions. Further, the system generates a funnel-like attractor equipped with a fractal structure. Changes in three easily detectable brain phase parameters (log frequency, excitatory/inhibitory ratio and fractal slope) lead to modifications in funnel’s breadth or in torus/attractor superimposition: it explains a large repertoire of brain functions and activities, such as sensations/perceptions, memory and self-generated thoughts."

Extract: "Starting from the unusual Sprott’s system of ODEs, we built a system equipped with both a conservative torus and a dissipative strange attractor. When a moving particle starts its trajectory from a given position x,y,z in the 3D nervous phase space, we may predict whether it will fall in the torus or into the strange attractor. The funnel shape is fractal, and not just a simple fixed-point attractor. A narrower funnel means that the trajectory is constrained towards a small zone of the phase space. When the two structures are closely superimposed, we might hypothesize a state of phase transition at the edge of the chaos, equipped with high symmetry, in which it is difficult to evaluate every single initial position: a slightly change in the starting point could indeed lead to completely different outcomes. When the torus and the strange attractor are clearly splitted, a single starting point gives rise to a sharp outcome. It means that in the latter case, the two conformations are neatly separated, as if the system went out of phase transition and a symmetry breaking occurred."

The third linked reference develops the concept of an attractor network to better understand how to calibrate nonlinear dynamical networks.

Wang et al (2016), "A geometrical approach to control and controllability of nonlinear dynamical networks", Nature communications 7, Article No. 11323, doi: 10.1038/ncomms11323.

http://www.nature.com/articles/ncomms11323The fourth linked reference and the associated fifth linked article, discuss a new efficient Monte Carlo method that can be used to more efficiently find solutions to models of wick problems:

Stefano Martiniani et al. Structural analysis of high-dimensional basins of attraction, Physical Review E (2016). DOI: 10.1103/PhysRevE.94.031301

http://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.031301Abstract: "We propose an efficient Monte Carlo method for the computation of the volumes of high-dimensional bodies with arbitrary shape. We start with a region of known volume within the interior of the manifold and then use the multistate Bennett acceptance-ratio method to compute the dimensionless free-energy difference between a series of equilibrium simulations performed within this object. The method produces results that are in excellent agreement with thermodynamic integration, as well as a direct estimate of the associated statistical uncertainties. The histogram method also allows us to directly obtain an estimate of the interior radial probability density profile, thus yielding useful insight into the structural properties of such a high-dimensional body. We illustrate the method by analyzing the effect of structural disorder on the basins of attraction of mechanically stable packings of soft repulsive spheres."

The linked article is entitled: "New method for making effective calculations in 'high-dimensional space'".

http://phys.org/news/2016-10-method-effective-high-dimensional-space.htmlExtract: " Researchers have developed a new method for making effective calculations in "high-dimensional space" – and proved its worth by using it to solve a 93-dimensional problem.

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Those include, for example, trying to model the likely shape and impact of a decaying ecosystem, such as a developing area of deforestation, or the potential effect of different levels of demand on a power grid.

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"There is a very large class of problems that can be solved through the sort of approach that we have devised," Martiniani said. "It opens up a whole world of possibilities in the study of things like dynamical systems, chemical structure prediction, or artificial neural networks."

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The set of initial conditions leading to this stable state is called a "basin of attraction". The fundamental theory is that, if the volume of each basin of attraction can be calculated, then this begins to provide some sort of indication of the probability of a given state's occurrence."