Commonly authors of literature dealing with chaos report a single and truncated chaotic event
occurring in the chaotic system they have investigated. This paper reports a multiplicity of
chaotic events detected in the non-linear damped and forced oscillator. In order to detect chaos
in this oscillator, a Virtual Lab (integrated and interactive computer program) was developed by
the author of this report. With this Virtual Lab many chaos simulations were executed and the
resulting Poincaré Maps for angles of 0° and 180° were extracted and filtered to avoid event
duplicity. It has been found that chaotic events do not last forever; they have a beginning and an
end, which means they are transitory. No numerical connection has been detected between the
natural frequency of a chaotic oscillator with that of the periodical applied force.
Keywords: Nonlinear, dynamics, chaos, computers, simulation, Runge-Kutta, Poincaré Maps, numerical
Multiplicidad y transitoriedad de los eventos caóticos
Comúnmente los autores de literatura sobre Caos reportan un solo evento caótico truncado que
acontece en los sistemas caóticos que ellos han investigado. Este documento reporta múltiples
eventos caóticos detectados en el oscilador no lineal amortiguado y forzado. Con la finalidad
de detectar Caos en este oscilador, el autor de este reporte desarrolló un Laboratorio Virtual
(software interactivo e integrado), con el cual se ejecutaron muchas simulaciones y también se
extrajeron y compararon Mapas de Poincaré para ángulos de 0° y 180°, a fin de evitar
duplicidad de eventos. Se encontró que los eventos caóticos no son imperecederos, ellos
tienen un inicio y un final, lo cual significa que ellos son transitorios. Tampoco se detectó
relación numérica alguna entre los valores de la frecuencia natural del oscilador caótico con
aquella de la fuerza periódica aplicada.
Palabras clave: Dinámica no lineal, Caos, simulación, computarizada, Runge-Kutta, Mapa de
Poincaré, Métodos numéricos
The most common –though not the unique- route to Chaos is that of period bifurcation, in which
the oscillating system vibrating with a single period, eventually changes and vibrates with two
periods (bifurcates), sometime later these two periods are replaced each by other two new
periods, and so on until chaos sets in and it becomes impossible to foresee the next period of
Usually the literature (research papers and books) dealing with chaos report a single and partial
cascade of period bifurcations, this is, they show only a unique and truncated cascade. This is
confusing because the newcomer to chaos land is urged to think that chaos once established is
infinite and that chaotic systems experiment only a single and unique chaotic event.
This investigation is about chaos in the nonlinear damped and forced oscillator. It has been
found that chaotic events in this system are finite, they do not last forever, it has also been
encountered that this oscillator displays many chaotic events. Even more, this research has not
found any connection between the natural frequency of the oscillator and the frequency of the
applied force. Very likely other systems prone to chaos behave the same way.
Real-life systems likely to experiment chaos are not prepared to vibrate the way a chaotic
system does, they usually collapse soon after chaos begins. Mechanical devices commonly
present in factories(1,2) collapse soon after the system changes the frequencies they are
supposed to oscillate with. The human heart is also an oscillator and it is suspected that when
it undergoes fibrillation, it has entered chaos. Obviously, the systems just mentioned are not
adequate to perform a chaos research, for this reason chaos must be investigated by means of
computerized simulation of mathematical models. Currently high speed computers enabled with
high resolution computer graphics are of great help in chaos research. Evidently the algorithms
developed by researchers play a critical role in these investigations.
A Virtual Lab(3,4) (interactive and integrated computer program) to investigate chaos in the
nonlinear damped and forced oscillator has been developed by the author of this report. This
Virtual Lab uses the Runge-Kutta method to numerically solve the differential equation of the
above mentioned vibrator. The program has been prepared to execute up to 30 million time
Interested readers who are not so fond of computer programming, may numerically solve the
differential equation dealt with in this report, by means of commercial computer softwares
(MathLab, Mathematica, Maple, etc) which are available in the market. It is worthwhile
mentioning that a very great amount of the literature on chaos is based on research performed
with these commercial softwares. Obviously these investigations are limited by what the
softwares are prepared to accomplish.
2. Brief introduction to Chaos research
This section contains a brief theoretical introduction to the techniques used in this research.
2.1 The State Space
The time evolution of a system can be visualized in a State Space(5,6,7), in the case of a
mechanical system this is a tridimensional plotting of displacement X(t) and velocity V(t) versus
time, as depicted in Fig. 1. The State Space of a Hamiltonian (energy preserving) system is
known as a Phase Space. The state space is a geometrical representation of the behavior of
In the simplest cases a state space is presented as the bi-dimensional projection of the
evolution of the system on the X-V plane, and if the system is not chaotic the shape of its state
space is smooth and understandable, but when the system is chaotic the shape of the state
space becomes extremely irregular as time elapses, hence in order to analyze the 3D-version of
the state space a Poincaré section (Fig.1) is used.
An Attractor is the orbit in state space the behavior of the system settles down to, or is
attracted to(5) in the long term.
Fig. 1 shows the state space of a damped oscillator, which eventually stops, it can be seen that
the state space orbit gradually approaches a point and once there, it remains there. Whatever
the initial amplitude of this oscillator, its orbits collapse to a point in state space, hence the
point is known as an Attractor. If the oscillator were not damped, energy would be conserved
and the shape of the orbit would be a loop (circle, ellipse), and this would be the attractor.
Chaotic attractors have a complicated geometry, they are associated to unpredictable motions
and they are fractals (their dimension is fractional). Fractal structures unveil more and more
details as they are more and more magnified.
2.3 Poincaré Maps
Commonly in order to visualize chaos in a given system, the Poincare sections(5,6) are extracted
and plotted. A Poincare section is the 2D plotting (see Fig. 1) of the points where the phase
space orbit intersects a surface (usually a plane) at a selected angle chosen from 0o to 360o on
the X-V plane. The objective of the Poincaré section is to detect any structure in the attractor, if
there is one.
Notice that while the state space is a 3D plotting, the Poincaré section is a 2D one, hence the
latter is easier to analyze than the former.
2.4 Experimental detection of the oscillation period
It is worthwhile recalling that in the case of a regular oscillator the oscillation period may be
experimentally obtained as the distance in the amplitude versus time plotting, between any two
consecutive points with the same phase. Obviously, chaotic oscillators are far from regular,
however this criterion at the time of inspecting the period is maintained and, in the chaos argot
researchers speak of a period doubling cascade.
3. Analyzing Chaos simulations
Chaos computer simulations are accomplished by numerically solving the differential equations
modeling the systems in which chaos is to be investigated. When using the Runge-Kutta
method to solve these equations, two time series are generated(3,4), one being displacement x(t)
and the other velocity, v(t), both as a function of time. With the x(t) and v(t) time series a State
Space is generated. Since directly analyzing the state space is rather difficult, a Poincare
Section at some selected angle is extracted and plotted for analysis(3,4).
4. The Chaos research being reported
This research is focused in the chaotic oscillations of a periodically forced nonlinear vibrator
immersed in a medium of variable damping, which may be modeled by the mechanical system
shown in Fig 2, and which is mathematically represented by the differential equation
Where is the natural frequency of the oscillator and is the frequency of the applied
At simulation time the damping b of the system was kept constant and the applied force was
continuously varied, this is equivalent to maintaining the applied force constant and varying the
The Virtual Lab used in this investigation is prepared to show on screen a vibrator oscillating
according to the simulation evolution and simultaneously depicting the state space. This feature
helps to understand the orbits appearing in state space, which not always are so easy to follow,
because for example, sometimes the chaotic oscillator makes weird kicks instead of completing
an orbit, some other times the oscillator makes unexpected stops and changes in its motion
The Virtual Lab(3,4) is enabled to detect the Poincare section at any angle. However it has been
experimentally found that the state space orbit not necessarily hits a Poincare section at every
angle, it has been observed that for some angles the Poincare sections have very few hits or
they are completely empty. With the aim on maintaining a unique frame of reference for all
simulations, this research was focused on the Poincare sections at 0o and at 180°. Previous
research had determined that chaotic events display rather rich Poincare sections at these two
It has been encountered that chaotic events do not last forever, they are finite, having a
beginning and an end; evidence of this may be appreciated in Figs. 3, 4 and 5, where the
complete chaotic events are displayed. Also it has been found that the studied system may
display many chaotic events. Actually some 25 chaotic events were detected, but after
screening them some of them were discarded due to parameter similitudes and all were
reduced to the five shown in Fig. 5, where to save space only Poincare Maps at 0° are shown.
Fig. 3 displays the Poincaré Maps for a chaotic event along 30 million time steps. Fig. 4 displays
these maps for a simulation along 4.7 million time steps. Recalling that the Poincaré Map allows
visualizing portions of the state space, it is evident that the phase space orbit is not uniform, the
X and V behavior of the system continuously changes as time elapses.
Figs. 3 and 4 displays two period bifurcation cascades for the nonlinear damped and forced
oscillator, both are Poincaré Maps, the upper corresponds to 0o and that at the bottom is at
180°. Obviously with higher-resolution computer graphics more details may be appreciated. It is
evident that chaotic events are transitory (they have a beginning and an end), it is also evident
that the system abandons chaos with the same smoothness it entered.
For every identified chaotic event the natural frequency of the oscillator was compared with that
of the applied force and, no common feature was observed, this is, no connection between
them was detected.
Concerning the relationship between the frequencies associated to the chaos events depicted in
Fig. 5, the relation
does not show any regularity; their values (from top to bottom) are the
In these simulations when the system abandons chaos the oscillation amplitude increases, if not
immediately, after some short time, this is because in this simulation the applied force is
periodic and every time higher. The same behavior is observed at the beginning of the
simulation, before the system begins to bifurcate its period to enter chaos.
Many chaotic happenings were detected in the nonlinear damped and forced oscillator; these
were later classified as belonging to five different events. This means that there is a multiplicity
of chaotic events. If investigation is continued it is highly probable that more chaotic events will
be detected in this system.
It has been found that chaotic events are transitory, they do not last forever, this is they have a
beginning and an end; additionally it has been observed that the system leaves chaotic events
with the same smoothness it started them. When entering a chaotic event a bifurcation of the
period is observed, then each of these bifurcations bifurcates again and again, and then the
system bursts into chaos. When abandoning chaos the opposite effect is observed, this is, the
system collapses the period cascade by pairs until finally it finishes with a single period. This
means that the transition towards chaos is as smooth as the transition out of it.
(1) Machinery breakdown: a Chaos-Based Explanation. Montenegro Joo J, Industrial
Data, Vol. 12, No 1, 2009
(2) A pragmatic introduction to Chaos Theory for engineers, Montenegro Joo J, Industrial
Data, Vol. 12, No 2, 2009
(3) A Virtual Lab to study the Dynamics of the Damped Non-Linear Simple Pendulum,
Montenegro Joo J, RIF, UNMSM, Vol. 11, No 1, 2008
(4) Transition to chaos in the damped & forced non-linear oscillator. Montenegro Joo J,
RIF., UNMSM, Vol. 12, No 1, 2009
(5) Crutchfield J P, Farmer D, Packard N H, Shaw R S, Chaos, pre-print
(6) Chaos and Nonlinear Dynamics, Hilborn R C, Oxford University press, 2001
(7) Gonzalez-Miranda, J M, Synchronization and Control of Chaos, Imperial College Press,