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Abstract

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

methods.

Multiplicidad y transitoriedad de los eventos caóticos

Resumen

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

1. Introduction

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

oscillation.

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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

steps (iterations).

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

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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

the system.

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.

2.2 Attractors

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.

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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

periodic force

.

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

damping.

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

direction.

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

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research had determined that chaotic events display rather rich Poincare sections at these two

angles.

5. Results.-

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

following:

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.

6. Conclusions.-

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

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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.

7. References.-

(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

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(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,

2004