Figure 5.16. The reconstructed state space of the blood oxygen concentration signal with t = 9 sec. projected in two dimensions (a) before filtering, and (b) after filtering.

Figure 5.16. The reconstructed state space of the blood oxygen concentration signal with t = 9 sec. projected in two dimensions (a) before filtering, and (b) after filtering.

With a filtered signal and the proper state space, we can investigate certain properties of the process that generated the signal. One critical question we need to address is that of classifying or identifying the source of the observations. A prerequisite for devising a suitable approximation scheme is to make predictions on the system behavior, and to solve the mathematical model if available.

In the case of linear systems, we again have the powerful tool of Fourier analysis. The locations of the sharp peaks in the frequency spectrum are characteristic of the process under investigation. If we drive the linear system with more power, the peaks will get higher, and if we observe the system starting from a different origin in time, there will be a phase shift associated with the measurements, yet in all cases, the locations of the peaks are the same. Quantities such as the locations of peaks in frequency domain are invariants of a linear system dynamics, and can be used to classify the system. A powerful example of classifying systems using frequency contents is voice recognition, where the frequency spectrum of a speech signal reveals the identity of the speaker with almost no doubt.

We have argued that frequency domain techniques are not very useful for nonlinear systems, especially when they are operated in dynamically rich regimes. Still, there are other invariants that are specific in classifying and identifying the signal source. These invariants are quantities that remain unchanged under various operations on the dynamics or the orbit. Most importantly, they remain unchanged under small perturbations in initial conditions, other than on countable specific points. Some of the invariants remain unchanged throughout the operation of the system. This guarantees that they are insensitive to initial conditions, which is apparently not true for the individual orbits. Some of them are guage invariants and stay unchanged under smooth coordinate transformations, and others are topological invariants, which are purely geometric properties of the vector field describing the dynamics. Among these invariants are the local and global Lyapunov exponents, and various fractal dimensions. Chaotic systems are notorious for the unpredictability of their orbits, and the limited predictability of chaotic systems is quantified by the local and global Lyapunov exponents of the system. Fractal dimensions associated with the source, on the other hand, reveal the topology of its attractor.

One of the hallmarks of chaotic behavior is the sensitivity of any orbit to small changes in initial condition, which is quantified by a positive Lyapunov exponent. Because of this sensitivity, it is inappropriate to compare two orbits generated by a nonlinear process directly. Generically, they will be totally uncorrelated. However, the invariants of the system will enable us to identify the source of an observation, since they are unchanged properties of the attractor of the system that has a peculiar geometry. These invariants are as useful in identifying nonlinear systems, as the Fourier spectrum is for linear systems. Therefore, system identification in nonlinear systems means establishing a set of invariants for each system of interest, and then comparing the invariants of the observation to the database of invariants.

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