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Multiplicity and transitoriness of chaotic events

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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. 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.
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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 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.
2
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
3
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.
4
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
5
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
6
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
7
(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
... The equation above is valid provided the oscillations are uniform, this is, as long as the distance between orbit turns in State Space keeps constant. In Chaotic oscillators the displacements are far from being uniform and the State Space is literally chaotic, in the most common sense of the word [5], [6]. ...
... displacements are far from being uniform and the State Space is literally chaotic, in the most common sense of the word [5], [6]. ...
... In Chaos Theory [5], [6] the computational detection of peaks and valleys, this is, the extremes of the x(t) curve, used in this experiment is tantamount to extracting the Poincaré Maps at 0° and at 180° respectively (see figure 4). Chaos theory does not work precisely with the logarithmic decrement, but the algorithm to extract the Poincaré Map is similar to that used in the present work to extract the oscillation extremes. ...
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A university-level educational Virtual Lab that in order to detect the damping an oscillator is experimenting applies the technique of Logarithmic Damping Decrement has been created. After input parameters (mass, elastic constant and viscosity of the medium) are entered by the user of the simulation module, this displays the corresponding curve of displacement vs time, x(t), on computer screen. Next the module allows the user to manually click with the mouse on the peaks (or valleys) of the displayed curve and, once the user has clicked 10 of these, the module computes the logarithmic decrement and from this, the experimental damping of the oscillator. The user may repeat this stage by clicking 10 valleys (or peaks) of the x(t) curve. With the aim on obtaining reference theoretical results, right after input data has been entered the module automatically detects the extreme displacements of the x(t) curve and it applies the logarithmic decrement algorithm to these data and from this the damping of the system is calculated. At the end of the simulation the virtual lab shows the theoretical as well as the experimental results so that the user can compare them. In this way, if the user has correctly clicked the extremes (peaks and/or valleys) of x(t), his experimental results –as it is expected-are verified as being very close to the expected result calculated by the virtual lab.
... Equation (1) is valid provided the oscillations are uniform; this is, as long as the distance between orbit turns in State Space keeps constant. By comparing with chaotic oscillators, in these the displacements are far from being uniform and the State Space is literally chaotic, in the most common sense of the word [5,6]. However, as it will be seen ahead, in order to automatically detect oscillation extremes, which are used to apply the LD, the Poincaré maps -a technique from Chaos Theory-may be used. ...
... Additionally from eq. computed, as well as the period T of these oscillations. Also at this stage the initial displacement and initial velocity are both calculated from eqs. (5) and (6), respectively. Next the simulation of eq. ...
... Once the simulation starts, the curve of displacements vs time x(t) as well as the state space are both gradually depicted on computer screen, as time goes by (see figures 3 and 5). Then by automatically extracting Poincaré maps at the maximum displacements (peaks and valleys) of x(t) are detected [5,6,11,12]. Next equation (10) is evaluated for every two successive maximum amplitudes ...
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This paper reports the development of a university-level EduVirtualLab (educational virtual lab) that in order to identify the control parameters (viscosity, initial phase, maximum amplitude) of an oscillation, applies the technique of Logarithmic Decrement, LD. Once the user of the virtual lab supplies the input data of the oscillation, the module displays the corresponding curve of displacement vs time, x(t), on computer screen. With the aim on obtaining reference theoretical results, the module automatically detects the extreme displacements (peaks and valleys) of the x(t) curve and it applies the LD algorithm to these data to calculate the damping of the system and, from this the control parameters of the oscillation are identified. Next the module allows the user to click with the mouse on the peaks (or valleys) of x(t) and, once the user has clicked 10 of these, the module computes the LD and from this, the experimental control parameters of the oscillation are obtained. The user may repeat this stage by clicking 10 valleys (or peaks) of x(t). At the end of the simulation, the virtual lab displays both, theoretical as well as experimental results, so that the user can compare them. In this way, if the user has correctly clicked the extremes (peaks and/or valleys) of x(t), his experimental results-as it is expected-are verified as being very close to the theoretical results calculated by the virtual lab. In order to automatically detect the extremes of x(t) this module extracts Poincaré Maps, a technique from Chaos theory.
... Consequently, each chaotic episode must have its own Lyapunov spectra and its own maximum Lyapunov exponent. It is worthwhile to remember at this point that not all chaotic systems display a period bifurcation cascade; however the oscillator dealt with in this research belongs to those systems that evolve towards chaos via a period bifurcation cascade [3,4,13,14,15]. ...
... This virtual lab numerically solves the differential equation of motion of the NLDFO, by means of the Runge-Kutta method. As it is already known [13][14][15], the NLDFO displays a multiplicity of chaotic events. However, from the computer simulation based experience of this researcher it is concluded that not all state space orbits -this is, not all oscillationsof the NLDFO are necessarily chaotic, most are not. ...
... Since it is also known that chaotic events are finite [13][14][15], the maximum Lyapunov exponent is extracted by processing the data of the full chaotic event, from beginning to end. ...
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... In Chaotic oscillators the displacements are far from being uniform and the State Space is literally chaotic, in the most common sense of the word [3]. ...
... In Chaos Theory [3] the above mentioned computational detection of peaks and valleys is tantamount to extracting the Poincaré Maps at 0° and at 180° respectively. ...
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... La ecuación (1) es válida siempre que las oscilaciones sean uniformes, esto es, siempre y cuando la distancia entre vueltas de la órbita en el Espacio de Estado se mantenga constante. Comparando con osciladores caóticos, en estos los desplazamientos están lejos de ser uniformes y el espacio de estado es literalmente caótico, en el sentido más común de la palabra [5,6]. Sin embargo, como se verá más adelante, con el fin de detectar los extremos de la oscilación, los cuales son usados para aplicar el DL, los Mapas de Poincaré -una técnica de la teoría del Caos-puede ser exitosamente usada. ...
... Una vez que se inicia la simulación, la curva de desplazamiento vs tiempo x(t), así como el espacio de estado son gradualmente graficados en la pantalla de la computadora, a medida que transcurre el tiempo (ver figs 3 y 5). Luego automáticamente se detectan los extremos de x(t), extrayendo mapas de Poincaré a que corresponden a los picos y valles de x(t), respectivamente [5,6,11,12]. A continuación, la ecuación (10) se evalúa para cada dos amplitudes máximas sucesivas de la oscilación y el promedio de λ se calcula y, se utiliza junto con la ecuación (15) para obtener el valor de la amortiguación G del sistema y la frecuencia ω de las oscilaciones amortiguadas. ...
Technical Report
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... Ever since it is known that chaotic happenings in this system are multiple [7]- [9], several chaotic events were studied, it was observed that all of them display similar behavior. This report contains the results obtained from one of these events, which was randomly selected and it is in this paper referred to as NLDFO (3) and, it lasts four-million time steps. ...
... In order to quantify the level of chaos, the maximum Lyapunov exponent was obtained from each experiment. Seeing that chaotic events are finite [7]- [9], in order to compute the maximum Lyapunov exponent [10]- [13] of each happening the complete chaotic phenomenon was scanned. The shortest among the chaotic events investigated by this researcher lasts one million time steps, as a consequence, obtaining the whole spectrum of the Lyapunov exponents becomes extremely difficult and, it has not been incorporated into this research. ...
... Ever since it is known that chaotic happenings in this system are multiple [7]- [9], several chaotic events were studied, it was observed that all of them display similar behavior. This report contains the results obtained from one of these events, which was randomly selected and it is in this paper referred to as NLDFO (3) and, it lasts four-million time steps. ...
... In order to quantify the level of chaos, the maximum Lyapunov exponent was obtained from each experiment. Seeing that chaotic events are finite [7]- [9], in order to compute the maximum Lyapunov exponent [10]- [13] of each happening the complete chaotic phenomenon was scanned. The shortest among the chaotic events investigated by this researcher lasts one million time steps, as a consequence, obtaining the whole spectrum of the Lyapunov exponents becomes extremely difficult and, it has not been incorporated into this research. ...
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It is intuitively expected that the vibrations of an oscillator be hindered when the viscosity of the medium is increased. The higher the viscosity, the higher the hindering effect. This paper reports the detailed way the viscosity of the environment affects the chaotic vibrations of the nonlinear damped and forced oscillator (NLDFO), which transitions towards chaos through a cascade of period bifurcations. It has been found that when the damping of the environment is increased, the mentioned bifurcation cascades become weaker, implying lower chaos intensity, which is corroborated by the corresponding decreasing values of the Lyapunov exponents and by the descent of the Kolmogorov entropy. It is also informed that the value of the extreme displacement as well as that of extreme velocity during a chaotic event decrease when the damping is increased, which is interpreted as a contraction of the orbit in state space, additionally the effect of the damping on the Return Maps of extreme displacements is described. Furthermore this paper gives account of evidence of serial chaos, which is obtained by extending during a very long time the computer simulation after a chaotic event.
... In order to detect the maximum Lyapunov exponent of a chaotic event this investigator uses an initial separation [4] of = 10 −8 between the supposed chaotic orbit and the test orbit [4,7] and since the extent of the chaotic event is known [12][13][14], the search is executed along the entire event. ...
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It is very well known that the most common test to declare an event as chaotic is the detection of its maximum Lyapunov exponent, if this happens to be positive, then the event is considered chaotic. Additionally, the higher the maximum Lyapunov exponent, the higher the intensity of the chaotic event. On the other hand, negative Lyapunov exponents stand for the contractions of the orbit and the number of these exponents represents the number of shrinks of state space. This paper introduces the flowchart of a computational procedure to detect the maximum Lyapunov exponent as well as some other associated indicators of a chaotic event, like the maximum and minimum values of this exponent, and the Kolmogorov entropy, all in a single run. Presenting the computation of the above mentioned magnitudes as a flowchart has the advantage that it is easier to understand and to get acquainted with, so that interested readers may easily adapt the procedure to their own needs and rapidly express it in the computer programming language of their preference. Since there is no universal and absolute way to verify the Lyapunov exponent of a chaotic event, in order to confirm the effectiveness of the procedure here reported, the behavior of magnitudes like the maximum Lyapunov exponent, the average Lyapunov exponent and the Kolmogorov entropy of a chaotic event, for different values of the environment damping, have been assessed and, it is found that the results satisfy the physics expectative.
... In order to detect the maximum Lyapunov exponent of a chaotic event this investigator uses an initial separation [4] of between the supposed chaotic orbit and the test orbit [4,7] and since the extent of the chaotic event is known [12][13][14], the search is executed along the entire event. ...
Technical Report
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This document presents the flowchart of a computational algorithm to evaluate the maximum Lyapunov exponent as well as some other associated indicators of a chaotic event, like the maximum and minimum values of this exponent, and the Kolmogorov entropy. The maximum Lyapunov exponent is used in chaos research to declare or not an event as chaotic and, if it is chaotic, to indicate its degree of intensity. Presenting the computation of the above mentioned magnitudes as a flowchart has the advantage that it is easier to understand and to get acquainted with, so that interested readers may easily adapt the procedure to their own needs and rapidly express it in the computer programming language of their preference. Since there is no universal and absolute way to verify the Lyapunov exponent of a chaotic event, in order to confirm the effectiveness of the procedure here reported, the behavior of magnitudes like the maximum Lyapunov exponent, the average Lyapunov exponent and the Kolmogorov entropy of a chaotic event, for different values of the environment damping, have been assessed and, it is found that the results satisfy the physics expectative.
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La Teoría del Caos es un tema de la Física, que día a día incrementa su posicionamiento entre los asuntos de la vida diaria, pues a medida que se sabe más sobre caos, se reconoce más y más su presencia en muchas áreas. Se han reportado indicios de comportamiento caótico en el corazón, en el cerebro, en la tasa de cambio del dólar-USA, etc. Teniendo en cuenta que una imagen dice más que mil palabras, el autor ha incluido muchas figuras, a fin de lograr una fácil y rápida comprensión de los temas en el libro, el cual se inicia con una breve descripción del panorama actual del caos, a continuación se muestran los fundamentos del mismo, luego se presentan las herramientas que se usan en la investigación de caos mediante simulación computarizada, posteriormente se muestran los resultados de investigaciones realizadas por el autor, los cuales han sido publicados en revistas especializadas. Se analiza el comportamiento caótico de la tasa de cambio del dólar-USA. Este libro está dirigido a físicos, matemáticos, ingenieros y economistas, interesados en Teoría del Caos. Personas de otras áreas podrían aprovechar las descripciones y análisis no especializados que se incluyen en el libro.
Article
Full-text available
A Virtual Lab to study the Transition to Chaos in second order non-linear differential equations has been developed and successfully applied to the search for chaotic behavior in the damped and forced non-linear oscillator. This simulation and visualization software evaluates the equation under investigation at up to one million time-steps, generating in real-time and on the screen, plots like amplitude of oscillation, phase diagram, amplitude oscillation peaks and an animation of an oscillator governed by the problem equation. In this way the investigator not only gets important behavior graphs but he or she also gets a physical visualization of the system under investigation. Visualizing an animation of the system under study is an enormous help because it is not always easy to interpret behavior graphs. Resumen Se ha creado un Laboratorio Virtual para estudiar la transición al caos en ecuaciones diferenciales no-lineales de segundo orden. Este software ha sido exitosamente aplicado a la investigación de la transición al caos en un oscilador no-lineal, amortiguado y forzado. El software, de simulación y visualización, evalúa la ecuación que se esta estudiando, en hasta un millón de puntos, generando en tiempo-real y en la pantalla del monitor, graficas de amplitud de oscilación, diagrama de fase y picos de amplitudes, produce además una animación del oscilador que se estudia. De esta forma, el investigador obtiene no solo importantes graficas de comportamiento del sistema que investiga, sino que también puede visualizarlo. La visualización de una animación del sistema que se estudia, es una enorme ayuda, pues no es siempre fácil interpretar graficas de comportamiento.
Synchronization and Control of Chaos
  • J Gonzalez-Miranda
Gonzalez-Miranda, J M, Synchronization and Control of Chaos, Imperial College Press, 2004
pre-print (6) Chaos and Nonlinear Dynamics
  • J P Crutchfield
  • D Farmer
  • N H Packard
  • R S Shaw
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