Analysis of Chaotic Dynamics  1. Spacemodel
Formal Metadata
Title 
Analysis of Chaotic Dynamics  1. Spacemodel

Alternative Title 
Analyse chaotischer Schwingungen  1. Zustandsraum

Author 

License 
CC Attribution  NonCommercial  NoDerivatives 3.0 Germany:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and noncommercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
Identifiers 

IWF Signature 
C 1739

Publisher 

Release Date 
1990

Language 
English

Producer 

Production Year 
1989

Technical Metadata
IWF Technical Data 
Film, 16 mm, LT, 249 m ; F, 23 min

Content Metadata
Subject Area  
Abstract 
Unbalanced rotors exhibit a characteristic mechanical oscillation. Using nonlinear Duffing equations the dynamics are recreated in a computer model with springing and damping depending on the excitation amplitude. The curve is supplemented by synthesizer acoustics. Analysis using phase curve, trajectories, Poincaré sections, the 3DModel of a strange attractor and Uedadiagram.

Keywords  dynamics, non linear dynamics, mechanical dynamics, forced dynamics, chaotic strange Attractor phase diagram, two dimensional phase diagram, three dimensional trajectory orbit UedaDiagram nonlinear Duffing equation Poincaré section rotors, imbalance 
00:00
Chaos theory
Dynamical system
Symbolic dynamics
Different (Kate Ryan album)
Oscillation
Mathematical analysis
00:35
Logical constant
Dynamical system
Observational study
Variety (linguistics)
Multiplication sign
Model theory
Mathematical analysis
Oscillation
Analytic set
Algebraic structure
Parameter (computer programming)
Mass
Mathematical model
Chaos theory
Frequency
Group representation
Spring (hydrology)
Damping
Nichtlineares Gleichungssystem
Duffing equation
Initial value problem
Physical system
Group representation
02:51
Logical constant
Dependent and independent variables
Logical constant
Mass
Parameter (computer programming)
Parameter (computer programming)
Mass
Oscillation
Frequency
Coefficient
Spring (hydrology)
Frequency
Combinatory logic
Steady state (chemistry)
Coefficient
Physical system
04:10
Frequency
Frequency
Oscillation
2 (number)
Oscillation
04:56
Point (geometry)
Trajectory
Group action
Dynamical system
Existence
State of matter
Multiplication sign
Parameter (computer programming)
Mathematical model
Order of magnitude
2 (number)
Frequency
Mechanism design
Mathematics
Velocity
Different (Kate Ryan album)
Analogy
Bifurcation theory
Energy level
Spacetime
Diagram
Nichtlineares Gleichungssystem
Stationary state
Physical system
Stability theory
Dot product
Dependent and independent variables
Process (computing)
Quantum state
Closed set
Graph (mathematics)
State of matter
Curve
Oscillation
Twodimensional space
Sequence
Oscillation
Orbit
Chaos theory
Arithmetic mean
Steady state (chemistry)
Phase transition
Waveform
Cycle (graph theory)
Coefficient
Spacetime
Extension (kinesiology)
11:08
Point (geometry)
Threedimensional space
Trajectory
Dynamical system
State of matter
Multiplication sign
Spiral
Parameter (computer programming)
Distance
Attractor
Dimensional analysis
Group representation
Frequency
Plane (geometry)
Physical system
Rotation
Area
Multiplication
Forcing (mathematics)
Model theory
Projective plane
Curve
Oscillation
Cartesian coordinate system
Oscillation
Permutation
Recurrence relation
Category of being
Arithmetic mean
Ring (mathematics)
Uniformer Raum
Order (biology)
Phase transition
Cycle (graph theory)
Resultant
Spacetime
15:14
Point (geometry)
State of matter
Multiplication sign
Coordinate system
Oscillation
Oscillation
Phase angle
Product (business)
Frequency
Plane (geometry)
Ring (mathematics)
Circle
Figurate number
Physical system
16:07
Point (geometry)
Trajectory
Dynamical system
Multiplication sign
Sheaf (mathematics)
Regular graph
Attractor
Phase angle
Alexander polynomial
Poincaré map
Plane (geometry)
Different (Kate Ryan album)
Position operator
Physical system
Sigmaalgebra
Mass flow rate
Surface
Consistency
Model theory
Oscillation
19 (number)
Numerical analysis
Orbit
Degree (graph theory)
Chaos theory
Poincaré, Henri
Arithmetic mean
Ring (mathematics)
Äquivariante Abbildung
Cycle (graph theory)
Figurate number
00:28
Deterministic nonlinear dynamic systems show many different kinds of behaviour ranging from periodic oscillations to irregular chaotic motions.
00:39
For deterministic systems the equation guarantees that the motions are identical for all times if the starting or initial conditions are precisely the same. We have analytic solutions for such systems only in particular welldefined situations where the equations are integrable. We are familiar with periodic behaviour of dynamic systems and with methods for analyzing it. But chaotic motions can also occur; in fact, they are often observed in driven nonlinear oscillators. This variety of behaviour makes the study of nonlinear systems so fascinating, but also difficult.
01:28
Classical techniques for characterizing the seemingly irregular behaviour usually fail. An equation of a type originally introduced by the German engineer Duffing will be used as mathematical model to explain modern techniques for analysis and characterization of nonlinear dynamic systems. This equation represents a nonautonomous periodically forced onedegreeoffreedom oscillator. In mechanical engineering, such an equation might model for example the motion of a sinusoidally forced structure undergoing large elastic deflections. The displacement is described by the coordinate x, the forcing frequency is omega, and the parameters m, d, c, and a represent mass, damping coefficient, spring constant, and forcing amplitude. All parameters are normalized and, hence, dimensionless. As a simple technical representation of the mathematical model a mechanical model will be considered. The model consists of a mass
02:52
which is connected to the ground by a combination of a nonlinear spring and a damper. The periodic forcing is
03:03
symbolically represented by an unbalanced rotor with frequency omega. For the simulations presented the following parameters are kept constant:  the mass m equal to 1  the spring constant c also = 1  the damping coefficient d = 0.1  and the forcing frequency omega = 2 pi. The only parameter which will be varied is the forcing amplitude a. Hence the forcing amplitude is considered as control parameter which is kept constant during the observation: it is varied in what is called a quasistatic way. Of primary interest is the longterm behaviour of the system. The system's steady state response to different values of the forcing amplitude will be demonstrated by digital simulations. 1. An oscillation of period 1 is obtained for the parameter a = 4.
04:18
The motion is already in the steady state. After each second, the motion repeats itself. 2. An oscillation of period
04:44
3 is observed for a = 9.
04:56
The process therefore repeats itself every 3 seconds. Again, we look only at the steady state behaviour after transients are damped out. 3. Chaotic motions are observed for a = 12. The oscillations
05:36
are no longer periodic but irregular, also called chaotic. Although certain patterns in the waveform repeat themselves at irregular intervals, we never observe exact repetition, the motion is truly nonperiodic. So far we have seen that the magnitude of forcing can significantly alter the dynamic behaviour of the oscillator. The equation used has an exceedingly complex response which is not even yet fully understood. 1. Bifurcation Responses of different periodicity exist, and even regimes of chaotic motions are by no means rare. The parameter dependence of the system's motion can be geometrically visualized by stability diagrams. Here is the damping coefficient d versus the forcing amplitude a. The a  d combinations, already used, are indicated by dots. Obviously there must be a transition from one behaviour to another. By means of careful analog and digital computer simulations the Japanese researcher Ueda could outline curves where the transitions take place. These transitions are called bifurcations. Blue regions indicate periodic, red regions chaotic oscillations. The diagram shows that although the mathematical model looks quite simple the system's behaviour is by no means trivial. 2. Acoustic demonstration Besides the demonstration of the motion itself it is also possible to hear what is going on within the system. Therefore, we have to increase the forcing frequency to an audible level. The following acoustic experiment has been produced by means of a fast analog simulation device with a forcing frequency increased to 400 Hertz.  If we repeat the simulations for the three parameter values, the period 1 oscillation produces this kind of of clear sound.  The sound is obviously different for the oscillation of period 3.  Chaotic oscillations produce a noisy signal. Many different frequencies add up to this kind of sound. So far we have made the sound experiments in a quasistatic manner. That means: We kept all parameters constant. Next we change a dynamically from 4 to 12. We hear a sequence of bifurcations. Periodic oscillations are followed by chaotic motions. Again periodic and chaotic motions can be recognized. Looking at graphs only, we cannot always observe essential features of the system's motion. To study the qualitative behaviour of dynamic systems we need a better geometrical concept. Such a concept, the socalled state space, was already introduced by Poincaré. A mechanical o scillator with a single generalized coordinate has a twodimensional state space associated with the displacement x and the velocity x dot. Changes in the actual state of the system are represented by a curve called trajectory or orbit. Each point of this curve carries (implicitly at least) a label recording the time of observation. The state space filled with trajectories is called phase portrait. Transients are damped out and eventually end up at the steady state. Next we produce the phase portraits for the three different values of the forcing amplitude a. For a = 4 the stationary solution is represented by this closed curve, a oneperiodic solution, closing itself after one forcing cycle. For the parameter value a = 9 a
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stable stationary periodthree solution is obtained. That is, the curve closes itself after three forcing cycles and, hence, the motion is periodic of order three.
11:33
The parameter value a = 12 results in a stationary chaotic oscillation. That is, the behaviour observed is not transient, although the curve never closes itself and we may eventually come arbitrarily close to our starting point. This kind of behaviour is called recurrent. A particular state of a dynamic system is recurrent, if after a sufficiently long time the system returns arbitrarily close to that state. This kind of motion is called stationary with respect to its longterm behaviour because it stays within a certain area and therefore should not be considered to be transient. To preserve the property of noncrossing trajectories, the nonautonomous system has to be rewritten as an autonomous system by means of an additional dimension. By introducing the time we end up with a threedimensional state space. If we take into account the forcing period T = 2 pi over omega and look at an oscillation of period 2, we arrive at this kind of threedimensional phase portrait. After two forcing cycles we match the same coordinates x and x dot. We moved from a point A via B to a point A. The distance between these points along the taxis is 4 pi over omega. Starting at a point B means shifting the trajectory by an increment of 2 pi over omega. The complete threedimensional representation gives a better impression of the behaviour than the twodimensional portrait. All previous solutions have been asymptotically stable. That is, neighbouring trajectories, shown in blue, will converge to these solutions and eventually, as time increases, end up there. For this reason these solutions are called attractors. This fundamental idea can be generalized to higher dimensions. Dissipative dynamic systems typically exhibit a startup transient, after which the motion settles down towards an attractor. In the threedimensional state space the trajectories of a forced oscillator tend to spiral around the time axis. In the phase projection this gives rise to a rotation about the origin. If we look down the time axis from the end we see the twodimensional projection of the threedimensional phase portrait. This projection is of course in agreement with the former twodimensional portrait. A and B represent the intersections of the trajectory with the x, xdot plane at times t equal to integer multiples of the forcing period. A threedimensional state space model does not visualize the periodic forcing in a geometric way. A better representation is obtained by means of a ring model as follows: x and xdot determine a plane R2 and the periodic
15:15
forcing is represented by a circle S1. The combination of
15:21
both the plane and the circle represents a cylindrical ring. This is an example of a Cartesian product. Every point
15:30
in the resulting threedimensional scheme represents a state of the system. A slice is removed for better visibility. Compared with our original threedimensional figure, the phase angle omega t of the driving oscillation has replaced the time coordinate, where the period is still capital T = 2 pi over omega. All solutions of the driven oscillator are embedded in this ring. Periodic solutions are represented
16:09
by closed trajectories, our solution of "period two" by this picture. Again, blue trajectories determine transient behaviour. They spiral around, getting closer and closer to the periodic solution with each forcing cycle. The periodone solution is represented by a trajectory, which closes itself after once going around the ring. The orbit of the periodthree solution closes itself after three forcing cycles. A nonperiodic or chaotic trajectory never closes itself. Embedded in the ring such a trajectory after a considerable time duration will become an impenetrable tangled skein. For a better understanding of its very complicated shape, we replaced it by this body. The ring model gives a much better impression of what is going on in a driven nonlinear oscillator, but it may still be difficult to analyze the chaotic dynamic behaviour. Now we inspect a surface of section, the socalled Poincaré map. This surface, orignially introduced by Henri Poincaré at the end of the last [19th]century, can be of arbitrary shape, as long as it cuts the trajectories transversally. But in our case a plane is most appropriate, that is the plane sigma at phi equal 0 degree. Here it coincides with our original plane R2 . By means of the surface, the continuous
18:27
system is discretized. The flow of the system, represented by trajectories, is replaced by points in the plane sigma. This type of discrete mapping is called point mapping or Poincaré mapping. By means of this procedure we gain a much better insight into the system's behaviour. Periodic solutions are represented by a consistent number of points, e. g. the oneperiodic orbit is represented by a single point in the sigmaplane; the periodthreesolution is represented by three points; the chaotic motion is visualized by infinitely many points. Although the attractor is generated by one single trajectory, the points wander over the surface in an erratic seemingly random fashion. After a sufficient number of points is plotted, they form a geometric figure called "strange attractor". The intertwining of the continuous trajectory did not reveal much about the dynamics  except that it was complex. The Poincaré sections however, show a figure with a certain regularity even for a chaotic trajectory. By placing the surface of section at different phase angles, we get different images of the same solution. For the chaotic solution we will travel along the ring by continuously changing the phase angle phi. The figure at 180 degrees is pointsymmetric to the figure obtained for phi = 0. This pointsymmetry holds for all values of phi. Continuing our journey along the ring, we eventually return to our starting position.