**4.1 Background**

Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system.[50]

Over the past decade, distinguishing deterministic chaos from noise has become an important problem in many diverse fields, e.g., physiology [51], economics [52]. This is due, in part, to the availability of numerical algorithms for quantifying chaos using experimental time series. In particular, methods exist for calculating correlation dimension (D2 ) [53], Kolmogorov entropy [54], and Lyapunov characteristic exponents. Dimension gives an estimate of the system complexity; entropy and characteristic exponents give an estimate of the level of chaos in the dynamical system.

The Grassberger-Procaccia algorithm (GPA) [53] appears to be the most popular method used to quantify chaos. This is probably due to the simplicity of the algorithm [55] and the fact that the same intermediate calculations are used to estimate both dimension and entropy.

However, the GPA is sensitive to variations in its parameters, e.g., number of data points [56], embedding dimension [56], reconstruction delay [57], and it is usually unreliable except for long, noise-free time series. Hence, the practical significance of the GPA is questionable, and the Lyapunov exponents may provide a more useful characterization of chaotic systems.

For time series produced by dynamical systems, the presence of a positive characteristic exponent indicates chaos. Furthermore, in many applications it is sufficient to calculate only the largest Lyapunov exponent ( λ1). However, the existing methods for estimating λ1 suffer from at least one of the following drawbacks: (1) unreliable for small data sets, (2) computationally intensive, (3) relatively difficult to implement. For this reason, we have developed a new method for calculating the largest Lyapunov exponent. The method is reliable for small data sets, fast, and easy to implement. "Easy to implement" is largely a subjective quality, although we believe it has had a notable positive effect on the popularity of dimension estimates.

For a dynamical system, sensitivity to initial conditions is quantified by the Lyapunov exponents. For example, consider two trajectories with nearby initial conditions on an attracting manifold. When the attractor is chaotic, the trajectories diverge, on average, at an exponential rate characterized by the largest Lyapunov exponent [58]. This concept is also generalized for the spectrum of Lyapunov exponents, λi (i=1, 2, ..., n), by considering a small

<sup>6</sup> Lyapunov exponent

Chaos is a phenomenon that occurs in many non-linear definable systems which show a high sensitivity to the primary conditions and semi random behavior. These systems will

Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of

Over the past decade, distinguishing deterministic chaos from noise has become an important problem in many diverse fields, e.g., physiology [51], economics [52]. This is due, in part, to the availability of numerical algorithms for quantifying chaos using experimental time series. In particular, methods exist for calculating correlation dimension (D2 ) [53], Kolmogorov entropy [54], and Lyapunov characteristic exponents. Dimension gives an estimate of the system complexity; entropy and characteristic exponents give an estimate of

The Grassberger-Procaccia algorithm (GPA) [53] appears to be the most popular method used to quantify chaos. This is probably due to the simplicity of the algorithm [55] and the fact that the same intermediate calculations are used to estimate both dimension and

However, the GPA is sensitive to variations in its parameters, e.g., number of data points [56], embedding dimension [56], reconstruction delay [57], and it is usually unreliable except for long, noise-free time series. Hence, the practical significance of the GPA is questionable, and the Lyapunov exponents may provide a more useful characterization of chaotic

For time series produced by dynamical systems, the presence of a positive characteristic exponent indicates chaos. Furthermore, in many applications it is sufficient to calculate only the largest Lyapunov exponent ( λ1). However, the existing methods for estimating λ1 suffer from at least one of the following drawbacks: (1) unreliable for small data sets, (2) computationally intensive, (3) relatively difficult to implement. For this reason, we have developed a new method for calculating the largest Lyapunov exponent. The method is reliable for small data sets, fast, and easy to implement. "Easy to implement" is largely a subjective quality, although we believe it has had a notable positive effect on the popularity

For a dynamical system, sensitivity to initial conditions is quantified by the Lyapunov exponents. For example, consider two trajectories with nearby initial conditions on an attracting manifold. When the attractor is chaotic, the trajectories diverge, on average, at an exponential rate characterized by the largest Lyapunov exponent [58]. This concept is also generalized for the spectrum of Lyapunov exponents, λi (i=1, 2, ..., n), by considering a small

**4. Reviewing the predictability of time series by the help of lyapunov** 

remain stable in the chaotic mode if they provide the Lyapunov exponent equations.

**exponent6**

**4.1 Background** 

chaos in a system.[50]

entropy.

systems.

 6

of dimension estimates.

Lyapunov exponent

the level of chaos in the dynamical system.

n-dimensional sphere of initial conditions, where n is the number of equations (or, equivalently, the number of state variables) used to describe the system. As time (t) progresses, the sphere evolves into an ellipsoid whose principal axes expand (or contract) at rates given by the Lyapunov exponents.

The presence of a positive exponent is sufficient for diagnosing chaos and represents local instability in a particular direction. Note that for the existence of an attractor, the overall dynamics must be dissipative, i.e., globally stable, and the total rate of contraction must outweigh the total rate of expansion. Thus, even when there are several positive Lyapunov exponents, the sum across the entire spectrum is negative.

Wolf et al. [59] explain the Lyapunov spectrum by providing the following geometrical interpretation. First, arrange the n principal axes of the ellipsoid in the order of most rapidly expanding to most rapidly contracting. It follows that the associated Lyapunov exponents will be arranged such that

$$
\lambda\_1 > \lambda\_2 > \dots > \lambda\_n
$$

where �� and �� correspond to the most rapidly expanding and contracting principal axes, respectively. Next, recognize that the length of the first principal axis is proportional to ����; the area determined by the first two principal axes is proportional to ���������; and the volume determined by the first k principal axes is proportional to ��������������. Thus, the Lyapunov spectrum can be defined such that the exponential growth of a k-volume element is given by the sum of the k largest Lyapunov exponents. Note that information created by the system is represented as a change in the volume defined by the expanding principal axes. The sum of the corresponding exponents, i.e., the positive exponents, equals the Kolmogorov entropy (K) or mean rate of information gain [58]:
