In this section the notions of nonlinear dynamics are briefly reviewed  to set the stage for recent achievements ,
The state of a dynamical system is most intuitively visualized geometrically as a point in a space, called state space or phase space, spanned by all the dependent variables of the system. Variables may be voltages, currents, temperatures, pressures, etc. In Fig. 1 these variables are called xp x2, xp .. ,*w,The initial state, at time t0 of the hypothetical system is marked as x0. The dynamic evolution, i. e. the change of the system with time, then is given by a curve in the state space of the system, called trajectory or orbit. Time is a parameter to be affixed to the points of the curve, as shown by the two times tj and t? At these times the system is characterized by the values of the variables given by the respective points they are affixed, to. Alternatively, the time may be added to the list of dependent variables as xm + j to yield an extended state space.
It is a simple but intriguing question what happens to a whole volume of initial conditions as time proceeds. The gross answer is given by the theorem of Liouville. The volume stays constant in conservative systems although it may deform in a most peculiar way, and it shrinks in dissipative systems as they are usually encountered in physics. It is clear that the rate of shrinking depends on the physical system (or its model) as does the final volume (the limit set) and its form. The limit set for t —» °° is composed of one or more attractors. Each attractor has its own set of initial conditions leading to it in the course of time, the set being called its basin. There exist different types of attractors. The most simple one is a fixed point, i. e., a single permanent state— an equilibrium or state of rest, where there is no alteration anymore.
The next, more involved attractor, is called a limit cycle. It is a closed curve in state space that is traversed again and again by the system. Any self-excited oscillator, for instance the heart beat, belongs to this class. Radio and TV (still) rely on limit cycles (simple sine waves in this case) as carriers of information that is modulated onto them. A fixed point, in whatever high-dimensional state space, is zero-dimensional. A limit cycle is one-dimensional.
The next simple attractor fills a two-dimensional manifold, a torus. A trajectory on the torus is a quasiperiodic motion. The quasiperiodic motion on a torus comes about through two incommensurable frequencies being present simultaneously.
The most intriguing and variable type of attractor is the strange or chaotic attractor. It needs an at least three-dimensional state space to occur in the case of continuous dynamics and has a fractal dimension, not an integer one (0, 1,2 in the previous cases), except for very special, accidental cases. Figure 2 shows graphical representations of the four types of attractors mentioned. The abundance of strange attractors in all disciplines of science now is a well established fact (see, e.g., [1-6]). Their characterization is mainly done via their fractal dimension (static property) and their Lyapunov spectrum (dynamic property). These notions are discussed below after having revealed how strange attractors may be obtained from measured data.
2.3 Poincar6 sections
As fractal attractors of continuous dynamical systems must be more than two-dimensional, they can be visualized in printing (plane of the paper) only by projection. This often yields unwieldy pictures except for the beautiful standard examples of the Lorenz or the Rossler attractor (Fig. 2d). Often a better view of the structure of the attractor can be obtained by a method invented by Poincare, whereby one dimension can be gained. It develops its full power for attractors in three-dimensional state spaces. In this method for visualizing an attractor a plane S, the Poincare section plane (a hyperplane of dimension m -1 in general), is chosen in the state space that is intersected transversally by the trajectories. To display an attractor only the section points of the attractor in the plane S are plotted. The points hop around in the plane. For a quasiperiodic motion they group themselves to a closed curve (dimension one). They form definite structures with (presumably) selfsimilar properties in the case of a strange attractor and show no signs of structure in the case of white (or also coloured) noise. Figure 3 shows an example of a strange attractor of the driven, double-well Duffing oscillator x + dx - x + ofi - a sin (Ot for d = 0.2, a - 0.3, and (0 = 1.24. The Poincare section plane is chosen as 0 = Wt mod In in the extended state space. The folded structure of the attractor, not at all consisting of homogeneously scattered points, can be noticed. As the times at which the system is sampled are successive integer multiples of the period of the driving, the method in this case just delivers a stroboscopic phase portrait of the system as well known to physicists and engineers.
A system normally depends on some parameters, e. g., temperature, and may show a totally different behaviour - expressing itself in a totally different attractor - at a different parameter value. For instance, a fixed point may change into a limit cycle signalling that the system now has acquired the ability to oscillate. Such a change in the qualitative behaviour of a system is called a bifurcation. A bifurcation occurs at some parameter value called a bifurcation point. It is a point in parameter space that may be high-dimensional when there are many parameters. A system or, more precisely, a family of systems then is best characterized by the set of parameter values at which a bifurcation occurs, the bifurcation set. When the bifurcation set is plotted in the parameter space, it generates regions of topological similar behaviour and thereby a parameter space diagram, also called phase diagram.
There are four basic types of local bifurcations, the period-doubling, the saddle-node, the transcritical, and the Hopf bifurcation . The period-doubling bifurcation has become widely known through its scaling behavior . This bifurcation likes to appear in a cascade when a parameter is altered (Fig. 4), and often there appears a complete, i. e., infinite cascade of period doublings leading to aperiodic motion via a period that becomes infinitely long.
A saddle-node bifurcation appears, e.g., at those points, where the resonance curve of a nonlinear oscillator turns over. At these points, the oscillation gets unstable, and the system has to settle down somewhere else.
Figure 4: Bifurcation diagram of the periodically pumped idealfour-level laser: q =- q + nq + sn, n = pjl +pm sin at) - bn - nq for s = 10~7,p0 = 6 • 10-4,pm = /, and b=0 (from Ref. ). The transcritical bifurcation appears in systems, where coexistent stable and unstable fixed points exist and where one of them does not depend on a parameter. Then, upon alteration of this parameter, a stable and an unstable fixed point may collide and exchange their stability at a bifurcation point via a transcritical bifurcation. The simple laser rate equations (see, e.g., ) without spontaneous emission term show a transcritical bifurcation for the photon number at the laser threshold.
At a Hopf bifurcation a limit cycle is generated. Self-excited oscillations derive their limit cycle from a Hopf bifurcation. The standard example is the van der Pol oscillator (, see also Fig. 2).
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