Non-linear and Non-smooth Dynamics in a
Sustainable Development System
Gerard Olivar
Department of Electrical and Electronics Engineering
and Computer Sciences
Universidad Nacional de Colombia sede Manizales
Cra 27 No. 6460, Manizales, Colombia
Email: golivart@unal.edu.co
Jorge Amador
Department of Electrical and Electronics Engineering
and Computer Sciences
Universidad Nacional de Colombia sede Manizales
Cra 27 No. 6460, Manizales, Colombia
Email: jaamadorm@unal.edu.co
Abstract— This paper presents the dynamic interaction be-
tween natural resource and population, and also introduce tech-
nological progress as an extra dynamic equation. Hopf, saddle-
node, and two-parameter bifurcation were found in the planar
system, and chaotic behaviour appeared in the 3D system. Non-
smooth phenomena such as sliding and stable pseudo equilibrium
appeared when smooth systems were replaced by non-smooth
system.
I. I NTRODUCTION
The concept of sustainable development implies world-
wide responsibility and shift to more sustainable lifestyles
and patterns of consumptions and production to obtain the
harmony among society, economy, and nature. This concepts
incorporates current and future global environmental concerns
and must consider the role of ethics and values in determining
choices affecting local and global environmental conditions.
Nowadays sustainable development is extensively described
and studied, but very few works are dedicated to mathematical
modelling techniques and numerical simulation. These two
tools can provide a point of entry into systems’s evolution
supplying some reasonable correspondence between models
and reality, and control actions can be designed in order to
address the dynamics to the sustainable development. The
existing models are mainly linear and have very poor numeri-
cal simulation and no stability analysis of equilibrium points.
Due to the high variable interaction among social, economical,
and environmental issues it is advisable to develop non-linear
models where non-linear phenomena such as bifurcation and
chaos can appear; or non-smooth phenomena such as sliding,
pseudo equilibria, and non-smooth bifurcation when non-linear
models are replaced by non-smooth models.
This paper is organized as follows. Section II describes a
planar non-linear dynamic system between population and nat-
ural resource including stability of equilibria and importance
of initial conditions. Section III presents existence of saddle-
node and Hopf bifurcations, and section IV presents the exis-
tence of two-parameter bifurcations. Section V introduces the
dynamic of technological change with the presence of chaotic
behaviour. Section VI describes the 2D and 3D systems when
they are defined by a discontinuous vector field presenting
pseudo equilibrium and sliding. Finally conclusions and future
work is presented in section VII.
II. RESOURCE STOCK AND POPULATION DYNAMICS
The most common non-linear models that describe the
interaction between population growth and the exploitation of
natural resources are related to the Lotka-Volterra predator-
prey model with population as the predator and natural re-
sources as the prey. Let us consider the system proposed by
[1] where resource dynamics can be defined as the growth
of the natural resource in absence of human habitation =
(/ − 1) (1 − /) minus the harvest rate =
where and are the carrying capacity and the threshold
where resource tends to extinction respectively, is the
regeneration rate of the resource, is the technology available
to exploit the resource, and is a parameter of preference. On
the other hand, population depends on total income supplied
by two economic activities: production of wood and corn, so
=
(
(1 − )
−1
+
)
where and represent
technology for the agricultural sector, and are caloric unit
of wood and corn respectively. A positive population growth
is obtained when total income is higher than a minimum level
of income needed to survive , i.e. poverty line. The final
set of coordinated differential equations is system (1). More
details about the meaning of parameters can be found in [1].
{
˙
=
[
− 1
1 −
−
]
;
˙
=
(
(1 − )
−1
+ −
)
.
(1)
A. Stability of equilibrium points
Equilibrium points are found when equations in system (1)
are simultaneously equal to zero presenting up to six steady
states. Four equilibria are trivial and will always exist ( 1, 2,
3, and 4); internal equilibria can also exist when positive
solutions exist ( 5 and 6). If internal equilibria exist 4
will be locally stable; 5 will always be a saddle while 6
will have a variable stability depending on parameter values;
other equilibria are unstable. Detailed computation of these
equilibrium points will be found in a future paper. Here, we
only state the results for shortness.
A useful tool is to calculate nullclines whose interception
correspond to the internal equilibria as it is shown in Figure
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