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Chaotic Behaviour of the Earth System in the Anthropocene

It is shown that the Earth System (ES) can, due to the impact of human activities, behave in a chaotic fashion. Our arguments are based on the assumption that the ES can be described by a Landau-Ginzburg model, which on its own allows for predicting that the ES evolves, through regular trajectories in the phase space, towards a Hothouse Earth scenario for a finite amount of human-driven impact. Furthermore, we find that the equilibrium point for temperature fluctuations can exhibit bifurcations and a chaotic pattern if the human impact follows a logistic map.


I. INTRODUCTION
Recently, we have argued that the transition of the Earth System (ES) from the Holocene to other stable states was similar to a phase transition, admitting a description by the Landau-Ginzburg (LG) Theory [1][2][3]. According to this approach, the relevant thermodynamic variable required to specify the state of the ES is the free energy, F . The phase transition is expressed in terms of an order parameter, ψ, which corresponds to a reduced temperature relative to the Holocene average temperature, T H , i.e., ψ = (T − T H )/ T H .
Considering the great acceleration of the human activities visible from the second half of the 20th century onwards [4], the LG framework allows for obtaining the so-called Anthropocene equation, i.e., the evolution equation of the ES describing the transition from the Holocene to the Anthropocene conditions. Such a physical approach allows for determining the state and equilibrium conditions of the ES in terms of the driving physical variables, η and H, where η is associated with the astronomical, geophysical and internal dynamical drivers, and H corresponds to the human activities, introduced in the phase-transition model as an external field [2].
In this work we carry out a phase space analysis of the temperature field, ψ, as previously examined in Ref. [2], under the condition that the human activities follow a logistic map. As this assumption is consistent with the hypothesis that human driven changes are limited, the logistic map assumption does not affect our previous conclusions [2], namely that the recently discussed Hothouse Earth scenario [5] corresponds to an attractor of regular trajectories of the ES dynamical system. Naturally, the logistic map can lead to more complex behaviour as, depending on the intensity of the human impact, it might give origin to stability point bifurcations and a chaotic behaviour that precludes a prediction of the evolution of the ES.
The paper outline is as follows. In section II, we review the LG model proposal and discuss the Anthropocene equation (AE) as well as the dynamical system emerging from this description in the phase space. In section III we introduce the hypothesis of the logistic map for human activities and show that it can lead to stability point bifurcations and chaotic behaviour in the phase space. We also consider the interactions between different terms of the planetary boundaries (PB) [6][7][8] in order to gauge the predictability of our model. Finally, in section IV, we present our conclusions.

II. THE ANTHROPOCENE EQUATION PHASE-SPACE
Dynamical systems are described either through a Lagrangian function or equivalently through a Hamiltonian formulation, irrespectively of being classical, statistical or quantum.
In both cases, the phase space variables can be evaluated. The phase space of our model is fully specified through the variables (ψ,ψ), once that a set of initial conditions, (ψ 0 ,ψ 0 ), is assumed. The initial value problem can be established and solved through the evolution equation, which leads to a trajectory of the dynamical system in the phase space.
In Refs. [1,2] it has been proposed that transitions of the ES are like phase transitions which can be described by the LG Theory through the free energy function in terms of the order parameter ψ, where F 0 is a constant, the above mentioned set of natural parameters that affect the ES is denoted by η, with the natural effects modeled by a(η) and b(η), while the strength of the human activities, H, is set by γ.
Considering a general set of canonical coordinates, q = (q 1 , . . . , q n ), for the ES in our description, this should include, not only the order parameter ψ, but also the natural and human drivers, η and H, respectively. Therefore, we have q = (ψ, η, H) [2]. In this case, the Lagrangian function includes, besides the potential, which is identified with the free energy, a set of kinetic terms for the canonical coordinates, where µ and ν are constants.
As we focus our analysis on the Anthropocene, the ES is dominated by the effects of human activities. Hence, the effect of these are greater and faster than the longer time scales of natural features [2]. Therefore, in order to study the ES currently, the term inη 2 can be safely dropped 1 . This reduces the system to a single canonical coordinate, the order parameter ψ. Follow- ing the Hamiltonian approach, we can identify the canonical conjugate momentum: and the following Hamiltonian, from which springs the Hamilton equations, that provide the evolution equations of the dynamical system.
With these equations, the phase portrait of the dynamical system is obtained, and the corresponding orbits and attractors can be identified. The corresponding stability landscape of the temperature field, ψ, is depicted in Fig. 1, from which the Holocene minimum described in Ref. [1] can be identified by the center of the valley, for H = 0. Moving away from H = 0, we clearly see that the human intervention opens up a deeper hotter minimum that can be identified with the Hothouse Earth [5], which in fact corresponds to a dynamical attractor.
At the Anthropocene, η is fixed and H is treated as a parameter with a time dependence.
The ES dynamics is reflected in the phase space and, thus, on the position and strength of its attractors, as exemplified in Ref. [2]. In that case, for b 0, the cubic term can be dropped in the equations of motion, which now correspond to a harmonic oscillator with an external force, H(t). Then, before considering more realistic scenarios for the external force, the simple case of a linear time evolution can be described by [2]: With this, the equations of motion are given by [2]: and the departure from equilibrium,ψ(0) = 0, with ψ(0) = ψ 0 , by the analytical solution, where ω = 2a/µ is an angular frequency, α = γH 0 /2a.
As discussed in Ref. [2], this solution corresponds to elliptical trajectories in the phase space with moving foci of the form where Ψ = ψ + αt.
These results provide a qualitative picture that holds even after introducing the cubic term ∝ ψ 3 in the free energy Eq. (1). Its effect is to slightly deform the elliptic orbits and to slow its movement towards higher values of ψ. In fact, the equations of motion can be solved, for instance, numerically and the trajectory of the ES can be obtained for any set of initial conditions [2]. This analysis can be generalised for any time dependence of H(t) and, of course, the system is only stable if H(t) is bounded. An issue that will be considered below.

III. LOGISTIC MAP AND CHAOTIC BEHAVIOUR IN THE ANTHROPOCENE
The linear model for the evolution of human activities, Eq. (6), presented in the previous section is meant only as a simple example to illustrate the process to obtain the trajectory of the ES in its phase-space. The next natural step is to attempt a more realistic assumption regarding H.
A reasonable incremental step would be to consider an exponential evolution, following what has been the evolution of many of the socio-economic trends in the last century, in what has become known as The Great Acceleration [4].
However, it is certain that the growth in all human (and natural) activities is limited by the available resources of the ecosystem. Given the worldwide spread of humanity, the ecosystem to consider is global, but still limited. Even considering gains in efficiency brought by technology, it does not change the fact the resources and the capacity to replenish them is ultimately limited.
Considering resource limitation as a reasonable hypothesis, the most suitable function to describe the evolution of human activities is the logistic map. Indeed, the logistic map is widely used in problems of ecology and dynamics of population. Precisely due to environmental limitations, the rate of growth of a population is limited and bound to reach a saturation value corresponding to the equilibrium between a given species, its predators and the resources of the environment where it lives can provide [10][11][12][13].
The logistic map is a fairly simple although very rich model, exhibiting features such as stable attractors, bifurcations and chaotic behaviour depending of the values of a single parameter [14][15][16].
In general, the existence and stability of equilibrium points for dynamical systems can be It is now necessary to specify the human drivers, collectively denoted by H. As suggested in Ref. [2], a fruitful strategy is to consider the impact of the human activities in the context of the Planetary Boundaries (PB) Framework [6], in which the state of the ES is specified through a set of 9 parameters, such that the human impact can be measured in terms of the actual magnitude of these parameters as compared to their values at the Holocene. The later set of values is usually referred to as Safe Operating Space (SOS) [7].
The most general form of H given by the impact of the human activities on the PB parameters, h i , contains also the interactions among them [2]: with N = 9, if we are considering the PB Framework. A mathematically convenient analysis requires that [g ij ] be read as a 9 × 9 symmetric and non-degenerate matrix, with g ij = g ji and det[g ij ] = 0, respectively. For such a 9-variable PB Framework, the 9th variable could be associated with the "Technosphere". In particular, if h 1 > 0, h 9 < 0 and g 1 9 > 0, the destabilising effect of h 1 can be mitigated by Technosphere contributions [2], which would maintain the equilibrium point due to human activities closer to its Holocene value [1]. The second order coupling terms involving h i h j represent the interactions among the various effects of the human intervention on the ES. As discussed in Ref. [2], they can affect the equilibrium configuration and suggest some mitigation strategies depending on the sign of the matrix entries, g ij , and their strength [2]. This methodology was considered to show that the interaction term between the climate change variable (CO 2 concentration), h 1 , and the oceans acidity, h 2 , is non-vanishing and on the order of 10% of the value of the single contributions by themselves [3]. Finally, suppressed higher order interaction terms are sub-dominant and their importance has to be established empirically. Therefore, our analysis shall be restricted to second order contributions and, in fact, to a subset of planetary boundary parameters.
Of course, the above considerations also assume that the h i terms do not depend on the temperature. However, it is most likely, in physical terms, that h i = h i (ψ), meaning in fact that the human action does affect the free energy of the ES [2]. The actual evolution of the temperature can be estimated as the associated equilibrium state which evolves as ψ ∼ H In fact, a global temperature, T , greater than the Holocene averaged temperature, T H , could lead to a chain failure of the main regulatory ecosystems of the ES, for which tipping point features have already been detected [5]. Therefore, predicting the behaviour of the ES accurately in order to engender mechanisms that force their associated phase space trajectories to remain close to the Holocene minimum is essential.
In such a context, we notice that the stability pattern can drastically change if the function H describing the human intervention receives contributions from the PB parameters that are constrained by some kind of discretised logistic growth rate. Let us than assume that at least one of the PB parameters, h i , iteratively evolves according to the recurrence relation of a logistic map, where i = 1 has been arbitrarily chosen.
The logistic map is strictly related to a logistic function which describes the growth rates and their corresponding level of saturation through the evaluation of the logistic equation, where the constant ν defines the growth rate and κ its carrying capacity.
Instabilities and chaotic patterns emerge from the conversion of a dynamical (time) variable continuous domain into a discrete domain, where the typical time scale between the steps of the discrete chain is a year, setting the map correspondence to: The logistic map, even for α = 1, as depicted in Fig. 2, provides a good illustration of how equilibrium point bifurcations and chaotic behaviour can arise from a simple set of non-linear dynamical equations.
The interpretation of Fig. 2 is well-known as it shows all possible behaviours of the equilibrium points, h 1 ≡ h Eq , as a function of r: i) for r between 0 and 1, the impact on h Eq vanishes, independent of the initial conditions; ii) for r between 1 and 2, h Eq quickly approaches the value r − 1/r, independent of the initial conditions; iii) for r between 2 and 3, h Eq eventually approaches the same value of r − 1/r, but it slightly fluctuates around that value for r 3; iv) from r = 3 to r = 1 + √ 6 ≈ 3.4495, a bifurcation pattern arises, drastically changing the dynamical evolution of h Eq , which approaches permanent oscillations between two equilibrium values given by h Eq ± = r + 1 ± (r − 3)(r + 1) /2r; v) for r between ∼ 3.4495 and ∼ 3.5441 the bifurcations are doubly degenerated, the lengths of the r parameter intervals that duplicate the number of equilibrium points decreases rapidly producing a kind of period-doubling cascade with the ratio between the lengths of two successive bifurcation intervals approaching the Feigenbaum constant δ ∼ 4.6692 [17]. That is the chaotic dependence on the boundary value problem of the corresponding dynamical variable h Eq . In what follows we shall see that this logistic map has a direct impact on the equilibrium temperature predictions, ψ Eq . From Eq. (10), with H ∼ h 1 , a preliminary conclusion is that the equilibrium point, ψ Eq , can be drastically affected by the h 1 growth rate r, since an equivalent map for ψ Eq → ψ Eq n can be identified by ψ Eq n → ψ Eq n+1 = ψ Eq n+1 (ψ Eq n ) (14) such that, through the operator O identified from Eq. (10), which exhibits a similar bifurcation diagram as depicted in Fig. 3. It is interesting to notice the presence of some islands of unstable equilibrium domains, where the bifurcation pattern can be macroscopically identified.
Aiming to get from the above scenario a more realistic phenomenological description, we consider only a couple of parameters, say h 1 and h 2 , and assume, in particular, that the second parameter is conditioned by the first one.
Parametrising the coupling between h 1 and h 2 by an interaction term given by g 12 h 1 h 2 , such a reduced ES can be described by a normalised human activity function,H, written as where constants h 1(2)T were introduced to establish a consistent comparative analysis.
For the hypothesis of a linear correlation between h 1 and h 2 , that is, h 2 = λh 1 , one has and from Eq. (18), the map for ψ Eq , is given by for which the bifurcation diagram is depicted in Fig. 4. We point out that the coupling, ω, just modulates the amplitude of the critical point, ψ Eq , not affecting the above mentioned islands of equilibrium, which correspond to the emerging blanks within the chaotic pattern. It is worth pointing out that the presence of bifurcations and chaotic pattern in the behaviour of ES prevents the capability to make predictions and to control the behaviour of the ES. One could say that these features are to be expected given the complexity of the ES and its interactions, however, our model allows for depicting as well as quantifying the approach to this complex behaviour via the tracking of the actual value of the parameter r, which, in principle, can be extracted from data about the human activity [18].
The above results admit two implementations depending on the coupling between h 1 and h 2 : Scenario 1 -Coupling between h 1 and h 2 evolving with the same growth rate, r, and different saturation points, α = 1 and α > 1, respectively.
Considering two unconstrained dynamical variables, h 1 and h 2 , this case corresponds to a couple of logistic maps identified by Hence, for the human activity function expressed by Eq. (16), the map for ψ Eq as function of r is depicted in Fig. 4, for g 12 = 0.1 and −0.1.
In particular, the analysis of the continuous domain relating h 1 to h 2 , shows that which leads to the following constraint, For an integration constant, c, set equal to κ, one has h 2 = κ h 1 , a linear relation in a continuous domain that can be mapped into a discrete domain as From the linear map Eq. (24) one notices that consistently returns That is, the linear correspondence between h 1 and h 2 is just a particular solution of the Scenario 1.
Scenario 2 -Coupling between h 1 and h 2 evolving with the same saturation points (set equal to unity) with different growth rates, r and αr (α < 1).
Considering two unconstrained dynamical variables, h 1 and h 2 , it ensues two distinct logistic maps: Hence, for the human activity function again expressed by Eq. (16), then the map for ψ Eq as function of r is depicted in Fig. 6, for g 12 = 0.1 and −0.1.
As in the previous case, an analysis of the continuous domain relating h 1 to h 2 , shows that which leads to the following constraint, where c is an arbitrary integration constant. The above result shows that Scenarios 1 and 2 are topologically distinct.
From Figs. 5 and 6 one notices that Scenarios 1 and 2 exhibit bifurcation patterns with paired discontinuities at zeroth and first order derivatives, respectively. This drastically affects the chaotic pattern. However, the r intervals from which equilibrium points, ψ Eq , are accountable, can be evaluated and are still clearly identifiable in both cases.

General phenomenological approach
We show here that the above results will ensue from a more general setup. Assuming a larger set of human activity contributions, h i , from the logistic map parametrisation it follows that: which are distinct from each other just by their growth rates and saturation points, rα i and β i , i.e. by denoting the H behaviour can be reflected by a truncated version of Eq. (11), with that is a polynomial map of forth order driven by four phenomenological parameters, A, B, C, and D, which can also be constrained by M ax{H} = H T , so that there remains just three free parameters. Despite its complexity, it is expected that this model, likewise those discussed above, affects the equilibrium point ψ Eq in a similar way.

IV. DISCUSSION AND OUTLOOK
In this work we have used a description of the ES in terms of the LG Theory of phase transitions and studied how it behaves if some of the variables accounting for human activities follow a behaviour that can be modelled by a logistic map. We have explored how it leads to a quite rich lore of possible trajectories of the ES in the Anthropocene, including regular and predictable trajectories, but also bifurcations and chaotic behaviour. Also, as previously discussed [2,3], the interaction among different PB parameters was investigated.
Dividing the human activities into its multiple components, we have studied a case with only two of those components following logistic maps and interacting with each other. Even for this simple case, we observed the emergence of chaotic behaviour in the equilibrium points of the ES. This leads to potentially important consequences, if at least some components of the human activities actually follow logistic maps, which is a quite reasonable hypothesis, given the physical limitations of the planet-wide system we live in.
With these quite plausible assumptions, a scenario where it may be impossible to predict the future equilibrium state of the ES in face of the human activities, becomes a distinct possibility that might be accounted for. The implications for designing managing strategies for the ES such as discussed, for instance in Refs. [19,20], may turn out to be quite dramatic as they might critically impair predictions and decisions making. Indeed, this work sets up a general phenomenological approach that can be used to verify and to further study of the dynamic behaviour of the ES. Furthermore, it shows that a phenomenological analysis based on data about human activities along the lines described above can indicate the nature of the evolution of the ES regarding its stability and our ability to predict its behaviour [18].