2.2 Dynamical systems - examples
Before we move on to a detailed description of the chaotic behaviour of systems possessing deterministic equations of motion, we should like to lead up to this subject by beginning with four typical examples.
Our first example is a mechanical system with a single degree of freedom and no loss of energy due to friction. Let a small mass be hung on a spring and allowed to oscillate vertically (fig. 2.2.1). For small deflections, the motion of this undampedand for the velocity x
Eliminating the time t from equs. (2.2.3) and (2.2.4), we find
spring pendulum can be expressed by the linear differential equation in the two variables displacement x and their second derivative with respect to time t
where wo is the frequency of the oscillation. Equation (2.2.1) has the general solution
where the integration constants A and B are determined by the initial conditions. Denoting the initial values of x and x at the instant t = 0 by x0 and xq, we obtain for the deflection x
The family of all ellipses in the i,i-plane designated by equ. (2.2.5) is called a phase portrait. On the right of fig. 2.2.1, the temporal course of the deflection x for a concrete initial condition is shown and on the left, the trajectories in the two- dimensional phase space for three different initial conditions. The state variables or coordinates x and x which span the phase space characterise the single-mass system uniquely.
Fig. 2.2.1 Single-degree-of-freedom oscillator without friction
In this idealised case where the energy is preserved, the mass returns periodically to its initial position. Also, the state of maximum deflection and zero velocity is
repeated with a period of
In natura, it is practically impossible to realise the assumption "no loss of energy due to friction". Loss of energy, for example due to air resistance, continuously decreases the deflection of the pendulum until at some stage, the mass remains in a state of equilibrium. During the transient phase, the aforementioned elliptical trajectories in the phase space then become spirals which all end in a point, the state of equilibrium (fig. 2.2.2). This point which captures all spirals is called an attractor, in this particular case a point attractor.
Fig. 2.2.2 Single-degree-of-freedom oscillator with friction
When the loss of energy of the pendulum due to friction is of a viscous nature, it can be reproduced by a damping term which depends to a first approximation linearly on the velocity. The linear differential equation (2.2.1) for a conservative system then takes the following standard form, again a linear system,
where the damping factor ( controls the fading of the periodic transient response. In the case of damping £ > 1, the system tends aperiodically towards the state of equilibrium. For ( < 1, the case of sub-critical damping, the amplitudes also decrease; but the movement retains qualitatively the appearance of an oscillatory process.
The loss of energy in a system can be compensated by means of the continuous supply of energy, e.g. a periodic external excitation (fig. 2.2.3). After a certain transient phase, the trajectories in the two-dimensional phase space approach
Fig. 2.2.3 Single-degree-of-freedom oscillator with friction, periodically excited; limit cycle
asymptotically a closed curve, the so-called limit cycle, which is run through periodically. Beside the point attractor, the limit cycle is the only possible attractor in the two-dimensional phase space.
For a harmonically excited, viscously damped oscillator, the equation of motion in linearised form becomes
Here, the periodic external excitation is reproduced by a sinus term on the right- hand side. Linear equations of this type are integrable and lead to a family of solutions, the individual curves of which are determined by the initial conditions. Since the dependence of a specific solution is relatively insensitive with regard to the initial conditions, small changes in the latter cause only small changes in the solution. Thus, our physical system is characterised by a "similar"connection between cause and effect.
The situation in the case of non-linear equations of motion is completely different. A simple dynamical system which leads to chaotic patterns of motion is the Duffing equation (see section 9.5). In its modified form, the restoring force is approximated by a polynomial of the third order, thus allowing, for example, the reproduction of larger deflections in an externally excited beam under transverse and normal force. In this special case, the differential ecmation takes the form
Phase portrait of the Duffing equation (chaotic)
(Ueda, 1980a,b; Moon and Holmes, 1979; Seydel, 1980). We stress that the non- linearity - in x, but not in time t - is a necessary condition for chaotic behaviour, though not sufficient in itself.
If the substitution X\ = x. .x'2 = x, .x'a = t is carried out, the non-autonomous differential equ. (2.2.8) with the second derivative with respect to time, the right- hand side of which depends explicitly on the time t, can be re-written as a system of non-linear, so-called autonomous first-order differential equations
This autonomous set of equations is equivalent to the non-autonomous equ. (2.2.8). As a generalisation, we can state that an autonomous differential equation may be put in the form
where the non-linear vector function F(x) does not depend explicitly on the time.
Thus, the introduction of additional variables transforms the non-autonomous equ. (2.2.8) with second derivatives with respect to time into a system of three first- order differential equations. Such a set of equations which is expressed directly in the terms of displacement and velocity considerably simplifies a qualitative and quantitative discussion of the course of the trajectories in the phase space (the concept of the phase space is elucidated in section 2.3).
The phase portrait of equ. (2.2.9) (note variables xi,±i) shows intersecting trajectories corresponding to specific choices of control parameters and initial conditions (fig. 2.2.4). Trajectories of the autonomous type of equ. (2.2.9) which do not intersect in the extended phase space with the coordinates Xi,Xi and t do so in the projection. It is easy to imagine that in the range of a non-periodic solution (i.e. an irregular respectively chaotic motion) for t —> oo, the phase portrait will be blackened out in a relatively short time. For this reason, this representation must be ruled out as an indicator of chaos since it would be futile to try to identify the corresponding trajectories which remain adjacent over a long time but diverge exponentially from one another at some stage.
The most widespread form of a display of the equations of motion consists of plotting the deflection, for example, over the time axis (fig. 2.2.5). However, to deduce a chaotic behaviour by means of this method is not realistic and is doomed to failure because the observed period of time is necessarily finite. Although the bizarre, non-periodic course of the curve in fig. 2.2.5 strongly suggests an erratic course of motion, it remains uncertain whether a periodic motion might not be established after all, were one to observe a longer time interval.
Fig. 2.2.5 Transient response of the motion
It may require skill and effort to distinguish between regular and irregular behaviour on the basis of the transient response of the variables of state of nonlinear dynamic models; it is, however, even more difficult to differentiate in an experimentally established transient response between background noise and deterministic chaos. In order to assess dynamic behaviour, it is also possible to apply, apart from the representation in the phase space and the temporal change of individual variables of state, the power spectrum method, well known to the engineer (see section 5.3). The recorded histograms, which are evaluated in this method by means of a Fourier analysis, yield clearly defined peaks in the power spectrum diagram for periodic or quasi-periodic motions; in contrast, continuous curves or curves with a high noise level indicate stochastic behaviour. This means that in the erratic case, the measured variables can no longer be represented as a discrete superimposition of oscillations: from a critical value of this variable of state in the Fourier-transformed signal onwards, a very high noise component in
Fig. 2.2.6 Stream lines, transient response and power spectrum in dependence on the Reynolds number Re; cf. (Feynmann et al., 1977)
the spectrum emerges. Since these power spectra can easily be ascertained experimentally, however, they offer themselves as a means of characterising the irregular behaviour of real systems. At the moment, however, we know of no method which enables us to eliminate the random influences involved in the experiment in order to make a strict distinction in the power spectrum between chaotic behaviour due to stochastic causes and that due to deterministic contributions. We can illustrate this with an example taken from fluid dynamics (fig. 2.2.6).As a consequence, simple spectral analysis does not suffice as a single tool for describing erratic behaviour in chaotic physical systems. Other methods of measuring must be found in order to describe the wide spectrum of regular and irregular motions qualitatively.
In the literature, a multitude of possible methods for characterising the occurrence of chaotic motions is presented. Apart from the power spectrum and autocorrelation, the two classic tools, we propose to concentrate on the following criteria: Lyapunov exponents, dimensions and Kolmogorov-Sinai entropy. An extensive discussion of these concepts can be found in the context of dissipative systems and attractors in Chapter 5.
The examples of mechanical systems mentioned here make it clear that the concept of the attractor plays a central role in the description of the behaviour of damped systems subject to deterministic equations of motion. We basically differentiate between two types: regular attractors and strange attractors. On the one hand, there are three classic types of motion: equilibrium, periodic motion and quasi- periodic motion. All three states are associated with regular attractors since, in the case of damping, the system tends towards one of these three states after the transient phase. The point attractor corresponds to the state of equilibrium, the limit cycle to periodic motion and the torus to quasi-periodic motion. On the other hand, there is a class of deterministic, but erratic, i.e. chaotic, motions which are not predictable if the initial conditions are subject to small fluctuations. Such long-term behaviour (t —> oo) is associated with the concept of the strange attractor.