Attractor reconstruction

Attractor reconstruction refers to methods for inference of geometrical and topological information about a dynamical attractor from observations.

[edit] Determinism and Phase Space

The modeling of a deterministic dynamical system relies on the concept of a phase space, the collection of possible system states. The system state at time t consists of all information needed to uniquely determine the future system states for times \(\geq \) t; e.g., in many cases, positions and velocities. For a system that can be modeled mathematically, the phase space is known from the equations of motion.

For experimental and naturally occurring chaotic dynamical systems, the phase space and a mathematical description of the system are often unknown. Attractor reconstruction methods have been developed as a means to reconstruct the phase space and develop new predictive models. One or more signals from the system must be observed as a function of time. The time series are then used to build a proxy of the observed states.

[edit] Whitney and Takens Embedding Theorems

The Whitney Embedding Theorem (Whitney 1936) holds that a generic map from an n-manifold to 2n+1 dimensional Euclidean space is an embedding: the image of the n-manifold is completely unfolded in the larger space. In particular, no two points in the n-dimensional manifold map to the same point in the (2n+1)-dimensional space. As 2n+1 independent signals measured from a system can be considered as a map from the set of states to 2n+1 dimensional space, Whitney's theorem implies that each state can be identified uniquely by a vector of 2n+1 measurements, thereby reconstructing the phase space.

The contribution of the Takens Embedding Theorem (Takens 1981) was to show that the same goal could be reached with a single measured quantity. Takens proved that instead of 2n+1 generic signals, the time-delayed versions \([y(t), y(t-\tau), y(t-2\tau), \ldots, y(t-2n\tau)]\) of one generic signal would suffice to embed the n-dimensional manifold. There are some technical assumptions that must be satisfied, restricting the number of low-period orbits with respect to the time-delay \(\tau\) and repeated eigenvalues of the periodic orbits. Similar theoretical results in (Aeyels 1981) from the mathematical control theory point of view, and a more empirical account (Packard et al. 1980) were published at about the same time.

The idea of using time delayed coordinates to represent a system state is reminiscent of the theory of ordinary differential equations, where existence theorems say that a unique solution exists for each \([y(t), \dot{y}(t), \ddot{y}(t), \ldots]\ .\) For example, in many-body dynamics under Newtonian gravitation, current knowledge of the position and momentum of each body suffices to uniquely determine the future dynamics. The time derivatives can be approximated by delay-coordinate terms as \([y(t), \frac{y(t)-y(t-\tau)}{\tau}, \frac{y(t)-2y(t-\tau)+y(t-2\tau)}{\tau^2}, \ldots]\ .\)

[edit] Fractal Attractors and Extensions

Emergence of chaos and fractal geometry in physical systems motivated a reassessment of the original theory, which applies to smooth manifold attractors. It was shown (Sauer et al. 1991) that a, possibly fractal, attractor of box-counting dimension d can always be reconstructed with m generic observations, or with m time-delayed versions of one generic observation, where m is any integer greater than 2d.

Figure 1: The Lorenz attractor

Figure 2: Times series formed by x coordinate

Figure 3: Reconstructed attractor

The figure shows a reconstruction of the fractal attractor for the well-known Lorenz system, whose fractal dimension is slightly larger than 2. The time series shown consists of the \(x\) coordinate of the system traced as a function of time. In the third image, triples of time series values \([x(t), x(t-\tau), x(t-2\tau)]\) are plotted. The topological structure of the Lorenz attractor is preserved by the reconstruction.

Embedding ideas were later extended beyond autonomous systems with continuously-measured time series. A version was designed for excitable media, where information may be transmitted by spiking events, extending usage to possible neuroscience applications (Sauer 1994). An embedding theorem for skew systems (Stark 1999) explores extensions of the methodology when one part of a system is driving another, and only the latter can be observed.

[edit] Attractor Reconstruction in Practice

Although the theory implies that an arbitrary time delay is sufficient to reconstruct the attractor, efficiency with a limited amount of data is enhanced by particular choices of the time delay \(\tau\ .\) Methods for choosing an appropriate time delay have centered on measures of autocorrelation and mutual information (Fraser & Swinney 1986). Further, in the absence of knowledge of the phase space dimension n, a choice of the number of embedding dimensions m must also be made. A number of ad hoc methods have been proposed that try to estimate whether the image has been fully unfolded by a given m-dimensional map. The approach of Kennel and Abarbanel (Kennel & Abarbanel 2002) is often used. This approach examines whether points that are near neighbors in one dimension are also near neighbors in the next higher embedding dimension. If not, then the image had not been fully unfolded. If all near neighbors remain so, then the unfolding is complete and the dimension is established. There are still unresolved issues as to what constitutes a near neighbor or a false near neighbor.

The success of embedding in practice depends heavily on the specifics of the application. In particular, the hypothesis of a generic observation function creating the time series is often problematic. A mathematically generic observation, by definition, monitors all degrees of freedom of the system. The extent to which this is true affects the faithfulness of the reconstruction. If there is only a weak connection from some degrees of freedom to the observation function, the data requirements for a satisfactory reconstruction may be prohibitive in practice. Other factors which limit success are difference in time scales between different parts of the system, as well as system and observational noise.

Applications of embedding time-series data (Ott et al. 1994, Kantz & Schreiber 1997) have been extensive since the Takens Embedding Theorem was published. Many techniques of system characterization and identification were made possible, including determination of unstable periodic orbits and symbolic dynamics, as well as approximation of attractor dimensions and Lyapunov exponents of chaotic dynamics. In addition, researchers have focused on methods of time series prediction and nonlinear filtering for noise reduction, the use of chaotic signals for communication, and for controlling chaos.

[edit] References

Abarbanel, H. (1996) Analysis of Observed Chaotic Data. New York: Springer-Verlag
Aeyels, D. (1981) Generic observability of differentiable systems. SIAM J. Control Optim. 19:595-603
Eckmann, J.-P., Ruelle, D. (1985) Ergodic theory of chaos and strange attractors. Reviews of Modern Physics 57:617-652
Fraser, A.M., Swinney, H.L. (1986) Independent coordinates for strange attractors from mutual information. Physical Review A 33:1134-1140
Kantz, H., Schreiber, T. (1997) Nonlinear Time Series Analysis. Cambridge:Cambridge University Press
Kennel, M and Abarbanel, H. (2002) Physical Review E 66: 026209
Ott, E., Sauer, T, Yorke, J.A. (1994) Coping with Chaos: Analysis of Chaotic Data and the Exploitation of Chaotic Systems. New York:Wiley Interscience
Packard, N., Crutchfield, J., Farmer, D., Shaw, R. (1980) Geometry from a time series. Physical Review Letters 45:712-715
Sauer, T., Yorke, J.A., Casdagli, M. (1991) Embedology. Journal of Statistical Physics 65:579-616
Sauer, T. (1994) Reconstruction of dynamical systems from interspike intervals. Phys. Rev. Lett. 72:3811-3814
Stark, J. (1999) Delay embeddings of forced systems I: Deterministic forcing. J. Nonlinear Sci. 9:255-332
Takens, F. (1981) Detecting strange attractors in turbulence. Lecture Notes in Mathematics 898, Berlin:Springer-Verlag
Whitney, H. (1936) Differentiable manifolds. Ann. Math. 37:645-680

Internal references

John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
Edward Ott (2006) Controlling chaos. Scholarpedia, 1(8):1699.
Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
James Murdock (2006) Unfoldings. Scholarpedia, 1(12):1904.