# Determining Lyapunov exponents from a time series

## Summary (4 min read)

### 1. Introduction

- This section includes table I which summarizes the model systems that are used in this paper.
- Section 3 is a review of the calculation of the complete spectrum of exponents for systems in which the defining differential equations are known.
- Which to their knowledge has not been published elsewhere.
- In section 5 the authors describe the algorithms for estimating the two largest exponents.
- The authors method is applied to model systems and experimental data in section 8, and the conclusions are given in section 9.

### t--,oo t pc(O)'

- Thus the Lyapunov exponents are related to the expanding or contracting nature of different directions in phase space.
- Since the orientation of the ellipsoid changes continuously as it evolves, the directions associated with a given exponent vary in a complicated way through the attractor.
- (See section 7. 1.) the sum of the first j exponents is defined by the long term exponential growth rate of a j-volume element.
- The magnitudes of the Lyapunov exponents quantify an attractor's dynamics in information theoretic terms.
- After this time the small initial uncertainty will essentially cover the entire attractor, reflecting 20 bits of new information that can be gained from an ad: ditional measurement of the system.

### 4. An approach to spectral estimation for experimental data

- In principle the authors might attempt the estimation of negative exponents by going to higher-dimensional volume elements, but information about contracting phase space directions is often impossible to resolve.
- In a system where fractal structure can be resolved, there is the difficulty that the volume elements involving negative exponent directions collapse exponentially fast, and are therefore numerically unstable for experimental data (see section 7.1).

### 5. Spectnd algorithm implementation

- The authors have implemented several versions of their algorithms including simple "fixed evolution time" programs for ~'1 and X1 ÷ hE, "variable evolution time" programs for XI+~:, and "interactive" programs that are used on a graphics machinet.
- This program is not sophisticated, but it is concise, easily understood, and useful for learning about their technique.
- The authors can also provide a highly efficient data base management algorithm that can be used in any of their programs to eliminate the expensive process of exhaustive search for nearest neighbors.
- The authors now discuss the fixed evolution time program for A t and the variable evolution time program for ~x + h2 in some detail.

### 6.1. Selection of embedding dimension and delay time

- In their data [2] for the Belousov-Zhabotinskii chemical reaction (see fig.
- In the first case the authors obtain a "pencil-like" region which obscures the folding region of the attractor.
- This structure opens up and grows larger relative to the total extent of the attractor for the larger values of ~', which is clearly desirable for their algorithms.
- A check of the stationarity of exponent estimates with t-is again recommended.

### 6.2. Evolution times between replacements

- In the Lorenz attractor, the separatrix between the two lobes of the attractor is not a good place to find a replacement dement.
- An element chosen here is likely to contain points that will almost immediately fly to opposite lobes, providing an enormous contribution to an exponent estimate.
- This effect is certainly related to the chaotic nature of the attractor, but is not directly related to the values of the Lyapunov exponents.
- This has the same effect as the catastrophes that can arise from too low a value of embedding dimension as discussed in section 6.1.
- While the authors are not aware of any foolproof approach to detecting troublesome regions of attractors it may be possible for an exponent program to avoid catastrophic replacements.

### 6.3. Shorter lengths versus orientation errors

- In this case their algorithm is closely related to obtaining the Lyapunov exponent of the map through a determination of its local slope profile [13] .
- The transverse vector component plays the role of the chord whose image under the map provides a slope estimate.
- This chord should obviously be no longer than the smallest resolvable structure in the 1-D map, a highly system-dependent quantity.
- Since the underlying maps of commonly studied model and physical systems have not had much detailed structure on small length scales (consider the logistic equation, cusp maps, and the Belousov-Zhabotinskii map [2] ) the authors have somewhat arbitrarily decided to consider 5-10% of the transverse attractor extent as the maximum acceptable magnitude of a vector's transverse component.

### 6.4. The accumulation of orientation errors

- If the authors can manage to evolve the vector for two orbits, the permissible initial orientation error is about 27 degrees, and so on.
- The authors see that a given orientation error at replacement time shrinks to a value negligible compared to the next orientation error, provided that propagation times are long enough.
- The ease of estimating the i th exponent depends on how small the quantity 2 (x'*~-x,)tr is.
- Problems arise when successive exponents are very close or identical.

### 7.1. Probing small length scales

- Even though infinitesimal length scales are not accessible, Lyapunov exponent estimation may still be quite feasible for many experimental systems.
- The same problem arises in calculations of the fractal dimension of strange attractors.
- In these calculations the authors hope that as the smallest accessible length scales are approached, scaling converges to the zero length scale limit.
- Similarly, provided that chaos production is nearly the same on infinitesimal and the smallest accessible length scales, their calculations on the small scales may provide accurate results.
- A successful calculation requires that one has enough data to approach the appropriate length scales, ignores anything on the length scale of the noise, and has an attractor with a macroscopic stretching/folding mechanism.

### 7.2. Noise

- It is ironic that measurement noise is not a problem unless large amounts of data are available to define the attractor.
- Noise is only detectable when the point density is high enough to provide replacements near the noise length scale.
- This suggests a simple approach to the noise problem, simply avoiding principal axis vectors whose magnitude is smaller than some threshold value the authors select.
- Avoiding noisy length scales is not a trivial matter, as noise may not be of constant amplitude throughout an attractor and the noise length scale may be difficult to determine.
- 6d the authors confirm that a straightforward small distancg cutoff works in the case of the Rossler attractor by showing stationarity of the estimated exponent over a broad range of cutoff values.

### 7.3. Low pass filtering

- Filtering dramatically altered the shape of the reconstructed attractor, but the estimated values of hi differed by at most a few percent for reasonable cutoff frequencies.
- A similar calculation for the Rossler attractor indicated that the low-frequency cutoff could be moved all the way down to the attractor's sole large spectral feature before the exponent estimate showed any noticeable effect.
- In a simple study of multiperiodic data with added white noise [3] the estimated exponent returned (very nearly) to zero for a sufficient amount of filtering.
- Unfiltered experimental data for the Belousov-Zhabotinskii reaction [2] ; b) the same data, low-pass filtered with a filter cutoff of 0.046 Replacement frequencies in the region of stationarity for these results were approximately 0.005 Hz. c) the data are over-filtered at 0.023 H_z. h 1 differs by only 20% from the exponent estimate for unfiltered data.

### N i~=l~ki od_j

- The authors have ignored several prefactors that might modify these estimates by at most an order of magnitude.
- From these values eq. ( 24) predicts to first order that between = 10 d and -30d points are necessary to fill out a d-dimensional attractor, independent of how many nonnegative exponents the authors are calculating.
- Note that lengthening the time series not only allows more time for convergence but also provides more replacement candidates at each replacement step.
- This would provide the same sampling resolution for the slope profile of the map (see ref. 13 ) in each dimension.
- The last and simplest point the authors consider is the required number of points per orbit, P.

### 8. Results

- The authors emphasize that no explicit use was made of the differential equations defining the model systems, except to produce a dynamical observable (the x-coordinate time series) which was then treated as experimental data.
- For the equations that define each system see table I.
- The quoted uncertainty values for each system were calculated either from the known values of the exponents or from the variation of their results with changes in parameters.

### 9. Conclusions

- The structure of this program is very similar to the program for hi: locate the two nearest neighbors of the first delay coordinate point, determine the initial area defined by this triple, enter the main program loop for repeated evolution and replacement.
- Triples are evolved EVOLV steps through the attractor and replacement is performed.
- The authors approach to triple replacement is a two step process; first the authors find a small set of points which, together with the fiducial trajectory, define small separation vectors and lie close to the required two-dimensional subspace.
- The authors then determine which of all of the possible pairs of these points will make the best replacement triple.
- The relative importance of replacement lengths, skewness, orientation changes, etcetera, are weighted by the user chosen factors.

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##### References

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### "Determining Lyapunov exponents from..." refers background in this paper

...1 illustrates the expanding, "slower than exponential," and contracting character of the flow for a three,dimensional system, the Lorenz model [23]....

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### "Determining Lyapunov exponents from..." refers background or methods in this paper

...We should obtain an embedding if m is chosen to be greater than twice the dimension of the underlying attractor [34]....

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...Given the time series x(t), an m-dimensional phase portrait is reconstructed with delay coordinates [2, 33, 34], i....

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...In principle, when using delay coordinates to reconstruct an attractor, an embedding [34] of the original attractor is obtained for any sufficiently large m and almost any choice of time delay ~-, but in practice accurate exponent estimation requires some care in choosing these two parameters....

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...The wellknown technique of phase space reconstruction with delay coordinates [2, 33, 34] makes it possible to obtain from such a time series an attractor whose Lyapunov spectrum is identical to that of the original attractor....

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4,323 citations

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### Additional excerpts

...Mackey-Glass: [27]...

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3,628 citations

### "Determining Lyapunov exponents from..." refers background or methods in this paper

...Given the time series x(t), an m-dimensional phase portrait is reconstructed with delay coordinates [2, 33, 34], i....

[...]

...The wellknown technique of phase space reconstruction with delay coordinates [2, 33, 34] makes it possible to obtain from such a time series an attractor whose Lyapunov spectrum is identical to that of the original attractor....

[...]