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Deterministic nonlinear dynamic systems show many different kinds of behaviour ranging from periodic oscillations to irregular chaotic motions.

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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 well-defined 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.

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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 non-autonomous periodically forced one-degree-of-freedom 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

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which is connected to the ground by a combination of a nonlinear spring and a damper. The periodic forcing is

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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 long-term 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.

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The motion is already in the steady state. After each second, the motion repeats itself. 2. An oscillation of period

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3 is observed for a = 9.

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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

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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 non-periodic. 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 so-called state space, was already introduced by Poincaré. A mechanical o scillator with a single generalized coordinate has a two-dimensional 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 one-periodic solution, closing itself after one forcing cycle. For the parameter value a = 9 a

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stable stationary period-three solution is obtained. That is, the curve closes itself after three forcing cycles and, hence, the motion is periodic of order three.

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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 non-crossing trajectories, the non-autonomous system has to be rewritten as an autonomous system by means of an additional dimension. By introducing the time we end up with a three-dimensional 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 three-dimensional 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 t-axis is 4 pi over omega. Starting at a point B means shifting the trajectory by an increment of 2 pi over omega. The complete three-dimensional representation gives a better impression of the behaviour than the two-dimensional 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 start-up transient, after which the motion settles down towards an attractor. In the three-dimensional 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 two-dimensional projection of the three-dimensional phase portrait. This projection is of course in agreement with the former two-dimensional portrait. A and B represent the intersections of the trajectory with the x, x-dot plane at times t equal to integer multiples of the forcing period. A three-dimensional 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 x-dot determine a plane R2 and the periodic

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forcing is represented by a circle S1. The combination of

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both the plane and the circle represents a cylindrical ring. This is an example of a Cartesian product. Every point

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in the resulting three-dimensional scheme represents a state of the system. A slice is removed for better visibility. Compared with our original three-dimensional 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

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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 period-one solution is represented by a trajectory, which closes itself after once going around the ring. The orbit of the period-three solution closes itself after three forcing cycles. A non-periodic 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 so-called 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

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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 one-periodic orbit is represented by a single point in the sigma-plane; the period-three-solution 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 point-symmetric to the figure obtained for phi = 0. This point-symmetry holds for all values of phi. Continuing our journey along the ring, we eventually return to our starting position.