Chaotic Behaviour of the Earth System in the Anthropocene

Abstract

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.

Full text

Chaotic Behaviour of the Earth System in the Anthropocene
A. E. Bernardini,1, ∗O. Bertolami,1, †and F. Francisco1, ‡
1Departamento de F´ısica e Astronomia,
Faculdade de Ciˆencias da Universidade do Porto,
Rua do Campo Alegre 687, 4169-007, Porto, Portugal.
(Dated: April 22, 2022)
Abstract
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.
∗Electronic address: alexeb@ufscar.br; On leave of absence from Departamento de F´ısica, Universidade
Federal de S˜ao Carlos, PO Box 676, 13565-905, S˜ao Carlos, SP, Brasil.
†Electronic address: orfeu.bertolami@fc.up.pt; Also at Centro de F´ısica das Universidades do Minho e do
Porto, Rua do Campo Alegre s/n, 4169-007, Porto, Portugal.
‡Electronic address: frederico.francisco@fc.up.pt
1
arXiv:2204.08955v2 [astro-ph.EP] 21 Apr 2022

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 [1–3]. According to this approach, the relevant thermody-
namic 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, hTHi, i.e., ψ= (T−hTHi)/hTHi.
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 Anthro-
pocene 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,
ηand H, where ηis associated with the astronomical, geophysical and internal dynami-
cal drivers, and Hcorresponds 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) [6–8] in order to gauge the predictability of our
model. Finally, in section IV, we present our conclusions.
2

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, (ψ0,˙
ψ0),
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,
F(η, H ) = F0+a(η)ψ2+b(η)ψ4−γHψ, (1)
where F0is 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= (q1, . . . , qn), for the ES in our
description, this should include, not only the order parameter ψ, but also the natural and
human drivers, ηand 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,
L(q, ˙q) = µ
2˙
ψ2+ν
2˙η2−F0−a(η)ψ2−b(η)ψ4+γHψ, (2)
where µand νare 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 in ˙η2
can be safely dropped1.
This reduces the system to a single canonical coordinate, the order parameter ψ. Follow-
1Of course, we could have introduced a kinetic term for the human activities, ˙
H2. As explained in Ref. [2],
this term is associated with quite fast effects of the human activities on themselves, and even though this
feedback loop is present, we assume instead that His an external force and drop its kinetic term.
3

FIG. 1: Stability landscape of the ES in terms of ψand H. Figure from Ref. [2].
ing the Hamiltonian approach, we can identify the canonical conjugate momentum:
p=∂L
∂˙
ψ=µ˙
ψ, (3)
and the following Hamiltonian,
H(ψ, p) = p2
2µ+aψ2+bψ4−γH ψ, (4)
from which springs the Hamilton equations,
˙
ψ=∂H
∂p ,˙p=−∂H
∂ψ ,(5)
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, ηis fixed and His 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
4

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]:
H(t) = H0t. (6)
With this, the equations of motion are given by [2]:
µ¨
ψ=−2aψ +γH0t, (7)
and the departure from equilibrium, ˙
ψ(0) = 0, with ψ(0) = ψ0, by the analytical solution,
ψ(t) = ψ0cos(ωt) + αt, (8)
where ω=p2a/µ is an angular frequency, α=γH0/2a.
As discussed in Ref. [2], this solution corresponds to elliptical trajectories in the phase
space with moving foci of the form
Ψ2
ψ2
0
+˙
Ψ2
ψ2
0ω2= 1,(9)
where Ψ = ψ+αt.
These results provide a qualitative picture that holds even after introducing the cubic
term ∝ψ3in 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.
5

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 envi-
ronmental 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 [10–13].
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 [14–16].
In general, the existence and stability of equilibrium points for dynamical systems can be
stablished through the Lyapunov theorem. A critical point (ψc, pc) is said to be Lyapunov
stable if any trajectory starting at a given neighbourhood of that point remains within a
finite neighbourhood of (ψc, pc). In the continuous domain of the time variable, (ψc, pc) is
identified by solving the constraint ( ˙
ψ, ˙p) = (0,0), which results into pc= 0 and ψcas the
unique real solution of the cubic equation,
O(ψ) = 2aψ + 4bψ3=γH. (10)
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 Hgiven by the impact of the human activities on the PB
6

parameters, hi, contains also the interactions among them [2]:
H=
N
X
i=1
hi+
N
X
i,j=1
gijhihj+. . . , (11)
with N= 9, if we are considering the PB Framework. A mathematically convenient analysis
requires that [gij] be read as a 9 ×9 symmetric and non-degenerate matrix, with gij =gji
and det[gij]6= 0, respectively. For such a 9-variable PB Framework, the 9th variable could
be associated with the “Technosphere”. In particular, if h1>0, h9<0 and g1 9 >0, the
destabilising effect of h1can 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 hihjrepresent 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, gij, and their strength [2]. This methodology was considered to show
that the interaction term between the climate change variable (CO2concentration), h1,
and the oceans acidity, h2, 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 hiterms do not depend on the
temperature. However, it is most likely, in physical terms, that hi=hi(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
hψi ∼ H1
3[1]. In this case, considering the beginning of the Anthropocene [9] about 70
years ago, and assuming that the growth of Hremains linear [2], if the effect of the human
action does lead to an increase of 1 K since then, one should expect in 2050 a temperature
increase of 3
√2=1.26 K. Hence, a quite general conclusion is that the critical point of the
dynamical system corresponds to an ES trajectory evolving towards a minimum of the free
energy where the temperature is greater than hTHi.
In fact, a global temperature, T, greater than the Holocene averaged temperature, hTHi,
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
7

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
Hdescribing 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, hi, iteratively evolves according to the recurrence relation
of a logistic map,
h1(n+1) =r h1(n)(1 −α−1h1(n)).(12)
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,
˙
h1=ν h1(1 −κ−1h1),(13)
where the constant νdefines the growth rate and κits carrying capacity.
Instabilities and chaotic patterns emerge from the conversion of a dynamical (time) vari-
able 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:
˙
h17→ h1(n+1) −h1(n),
ν7→ r−1,
κ7→ α(1 −r−1),
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. 2is well-known as it shows all possible behaviours of the
equilibrium points, h1≡hEq , as a function of r:i) for rbetween 0 and 1, the impact on
hEq vanishes, independent of the initial conditions; ii) for rbetween 1 and 2, hEq quickly
approaches the value r−1/r, independent of the initial conditions; iii) for rbetween 2 and
3, hEq 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 hEq, which approaches permanent
8

FIG. 2: Bifurcation diagram for the logistic map (α= 1).
oscillations between two equilibrium values given by hEq
±=r+ 1 ±p(r−3)(r+ 1)/2r;
v) for rbetween ∼3.4495 and ∼3.5441 the bifurcations are doubly degenerated, the lengths
of the rparameter 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 hEq. In what follows we shall see that this logistic map has a direct impact on the
equilibrium temperature predictions, ψEq.
From Eq. (10), with H∼h1, a preliminary conclusion is that the equilibrium point, ψEq ,
can be drastically affected by the h1growth rate r, since an equivalent map for ψEq 7→ ψEq
n
can be identified by
ψEq
n7→ ψEq
n+1 =ψEq
n+1(ψEq
n) (14)
9

such that, through the operator Oidentified from Eq. (10),
ψEq
n+1(ψEq
n) =O−1H(h1(n+1) )
=O−1Hr h1(n)1−α−1h1(n)
=O−1Hr H −1O[ψE q
n]
1−α−1H−1O[ψEq
n],(15)
which exhibits a similar bifurcation diagram as depicted in Fig. 3. It is interesting to notice
FIG. 3: Bifurcation diagram for ψEq for O(ψ) = h1, with a=b= 1.
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 h1and h2, and assume, in particular, that the
second parameter is conditioned by the first one.
Parametrising the coupling between h1and h2by an interaction term given by g12 h1h2,
such a reduced ES can be described by a normalised human activity function, ˜
H, written as
˜
H=H
HT
=h1+h2+g12 h1h2
h1T+h2T+g12 h1Th2T
,(16)
where constants h1(2)Twere introduced to establish a consistent comparative analysis.
For the hypothesis of a linear correlation between h1and h2, that is, h2=λh1, one has
˜
H=h1+ωh2
1
1 + ω,with ω=g12 λ
1 + λ,(17)
10

and from Eq. (18), the map for ψEq , is given by
ψEq
n+1(ψEq
n) = O−1r
1 + ωxn(1 −xn) + rωx2
n(1 −xn)2(18)
with
xn=1
2ω1+4ω(1 + ω)O[ψEq
n]1/2−1,(19)
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, ψEq, not affecting the above mentioned
islands of equilibrium, which correspond to the emerging blanks within the chaotic pattern.
FIG. 4: Bifurcation diagram for ψEq in case of a linear coupling between, parametrised by O(ψ) =
˜
H= (h1+ωh2
1)/(1 + ω). Results are for ω= 0.2 (red), 0 (black) and −0.2 (blue), with a=b= 1.
They are presented in a zoom in view in the second plot.
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
11

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 h1and
h2:
Scenario 1 – Coupling between h1and h2evolving with the same growth rate, r, and different
saturation points, α= 1 and α > 1, respectively.
Considering two unconstrained dynamical variables, h1and h2, this case corresponds to
a couple of logistic maps identified by
h17→ h1(n+1) =r h1(n)(1 −h1(n)); (20)
h27→ h2(n+1) =r h2(n)(1 −α−1h2(n)).(21)
Hence, for the human activity function expressed by Eq. (16), the map for ψEq as function
of ris depicted in Fig. 4, for g12 = 0.1 and −0.1.
In particular, the analysis of the continuous domain relating h1to h2, shows that
dh1
dh2
=ν h1(1 −h1)
ν h2(1 −κ−1h2)=h1(1 −h1)
h2(1 −κ−1h2),(22)
which leads to the following constraint,
h2=c h1
h1(cκ−1−1) + 1.(23)
For an integration constant, c, set equal to κ, one has h2=κ h1, a linear relation in a
continuous domain that can be mapped into a discrete domain as
h1(n)7→ h2(n)=α h1(n);
h1(n+1) 7→ h2(n+1) =α h1(n+1).(24)
From the linear map Eq. (24) one notices that
h17→ h1(n+1) =r h1(n)(1 −h1(n)),(25)
consistently returns
h27→ h2(n+1) =r h2(n)(1 −α−1h2(n)).(26)
12

FIG. 5: Maps for ψEq as function of rfor the Scenario 1, for g12 = 0.1. Results are for α= 1 (first
row), 1.25 (second row), 5 (third row), and 10 (forth row). Second column depicts the results in
zoom view for g12 = 0.1 (black) and −0.1 (red). The case g12 = 0 corresponds to the intermediate
configuration.
That is, the linear correspondence between h1and h2is just a particular solution of the
Scenario 1.
Scenario 2 – Coupling between h1and h2evolving with the same saturation points (set equal
to unity) with different growth rates, rand αr (α < 1).
Considering two unconstrained dynamical variables, h1and h2, it ensues two distinct
13

logistic maps:
h17→ h1(n+1) =r h1(n)(1 −h1(n));
h27→ h2(n+1) =r α h2(n)(1 −h2(n)).(27)
Hence, for the human activity function again expressed by Eq. (16), then the map for
ψEq as function of ris depicted in Fig. 6, for g12 = 0.1 and −0.1.
As in the previous case, an analysis of the continuous domain relating h1to h2, shows
that
dh1
dh2
=ν h1(1 −h1)
ν κ h2(1 −h2)=h1(1 −h1)
κ h2(1 −h2),(28)
which leads to the following constraint,
h2=hκ
1
c(1 −h1)κ+hκ
1
,(29)
where cis an arbitrary integration constant. The above result shows that Scenarios 1 and
2are topologically distinct.
From Figs. 5and 6one 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 rintervals from which equilibrium points, ψEq, 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, hi, from the logistic map parametrisation it
follows that:
hi(n+1) =rαihi(n)(1 −βihi(n)),(30)
which are distinct from each other just by their growth rates and saturation points, rαiand
βi, i.e. by denoting
hi7→ xn+1 =rαixn(1 −βixn),(31)
14

FIG. 6: Maps for ψEq as function of rfor the Scenario 2, for g12 = 0.1. Results are for α= 2 (first
row), 1 (second row), 0.8 (third row), and 0.2 (forth row). Second column depicts the results in
zoom view for g12 = 0.1 (blue) and −0.1 (red). The case g12 = 0 corresponds to the intermediate
configuration.
15

the Hbehaviour can be reflected by a truncated version of Eq. (11),
Hn+1 =
9
X
i=1
hi(n+1) +
9
X
i,j=1
gijhi(n+1)hj(n+1)
=
9
X
i=1
rαixn(1 −βixn)
+
9
X
i,j=1
gijr2αiαjxnxn(1 −βixn)(1 −βjxn)
=A xn−B x2
n−C x3
n+D x4
n,(32)
with
A=r
9
X
i=1
αi>0,
B=r
9
X
i=1
αiβi−r2
9
X
i,j=1
gijαiαj,
C=r2
9
X
i,j=1
gijαiαj(βi+βj),
D=r2
9
X
i,j=1
gijαiαjβiβj,
that is a polynomial map of forth order driven by four phenomenological parameters,
A, B, C, and D, which can also be constrained by Max{H}=HT, 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 ψEq 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 activ-
ities 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
16

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].
Acknowledgements
The work of AEB is supported by the Brazilian Agency FAPESP (Grant No. 20/01976-5).
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