We report on structures in a scaledependent lyapunov exponent of an experimental chaotic map that arise due to discontinuities in the map. The regularity condition of lyapunov means, essentially, that the limit in. For small scale problems, numerical methods are well developed for long time dynamical processes, for example lyapunov exponents. Realizing that the difficulty in sea clutter modeling is due to the multiscale nature of sea clutter, we employ a new multiscale complexity measure, the scale dependent lyapunov exponent sdle, to better characterize the nonstationary and multiscale nature of sea clutter. Request pdf multiscale analysis of biological data by scaledependent lyapunov exponent physiological signals often are highly nonstationary i. Calculation lyapunov exponents for ode file exchange. A physical system with this exponent is conservative. Time series chaos detection and assessment via scale. Mar 18, 2004 lyapunov exponent calcullation for odesystem. A major area for future work is to extend these techniques to large scale. It has been further developed theoretically gao et al. While excellent scale dependent measures such as epsilon entropy and the finite size lyapunov exponent fsle have been proposed, simple algorithms have not been developed to reliably compute them.
Multiscale characterization of sea clutter by scale. In devising measures characterizing such complex time series, it is most desirable to incorporate explicitly the concept of scale in the measures. In mathematics the lyapunov exponent or lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. The scaledependent lyapunov exponent sdle is capable of such a fundamental task. Electronics and information engineering, huazhong university of science and technology hust, wuhan, p. Multiscale analysis of biological data by scaledependent. Research article multiscale characterization of sea. Here, we employ a new multiscale complexity measure, the scaledependent lyapunov. Cao3 1department of electrical and computer engineering, university of florida, gainesville, florida 32611, usa 2department of earth and atmospheric sciences, purdue university, west lafayette, indiana 47907, usa 3biosieve, 1026 spring.
This alternate definition will provide the basis of our spectral technique for experimental data. In particular, sdle can readily characterize all known types of signal data, including deterministic chaos, noisy chaos, random 1f. A simulation study to evaluate small sample performance is reported. Can economic time series be modeled by lowdimensional. Ames abstractimplementing statebased parameterized periodic trajectories on complex robotic systems, e. Lyapunov asserts that, under an additional regularity condition, stability remains valid for nonlinear perturbations w. A lyapunov exponent based method for online transient stability assessment p. Distinct gray scales correspond to observed periodic dynamics of different peri ods. The remainder of the paper is organized as follows. Vastano, determining lyapunov exponents from a time series, physica d, vol. Research article multiscale characterization of sea clutter. Their concept was introduced by lyapunov when studying the stability of nonstationary solutions of ordinary di. These equations are linear but time dependent and can be obtained via d.
Structured scaledependence in the lyapunov exponent of a. For systems whose equations of motions are explicitly known, there exist several methods for computing lyapunov exponents. Previous studies on heart rate variability hrv using chaos theory, fractal scaling analysis, and many other methods, while fruitful in many aspects, have produced much confusion in the literature. Time series from complex systems with interacting nonlinear and stochastic subsystems and hierarchical regulations are often multiscaled. Time series chaos detection and assessment via scale dependent lyapunov exponent many dynamical systems in a wide range of disciplines such as engineering, economy and biology exhibit complex behaviors generated by nonlinear components which might result in deterministic chaos. A practical test for noisy chaotic dynamics sciencedirect. In this paper, we illustrate the general use of two multiscale approaches that do not assume a stationary process, the adaptive fractal analysis afa gao et al.
The above equation can used to compute the lyapunov exponents limt. A lyapunov exponent of zero indicates that the system is in some sort of steady state mode. Observe a characteristic feature of the dependence intrinsic to the non. Utilising a pragmatic, physical approach, this selfcontained book provides a. On finitesize lyapunov exponents in multiscale systems. We introduce a statistical framework for testing the chaotic hypothesis based on the estimated lyapunov exponents and a consistent variance estimator. The alogrithm employed in this mfile for determining lyapunov exponents was proposed in a. The scale dependent lyapunov exponent sdle is capable of such a fundamental task. In section 2 we describe the sdle and study its properties. Therefore, it is necessary to bridge the gap between the concept of ftle and.
Universal scaling of lyapunov exponents in coupled chaotic. To calculate the lyapunov exponents numerically one usually solves the equations of motion for n perturbations and orthonormalizes them every k steps. Multiscale analysis of economic time series by scale. Iii, we show how the cumulantgenerating function can be. A technique for revealing scaledependent patterns in. Utilising a pragmatic, physical approach, this selfcontained book provides a comprehensive description of the concept. In particular, sdle can readily characterize all known types of signal data, including deterministic chaos, noisy chaos, random 1 f.
A numerical computation of the logarithm of the stretch. Cao3 1department of electrical and computer engineering, university of florida, gainesville, florida 32611, usa. Pullback attractor for the three dimensional nonautonomous. The present code estimates the largest lyapunov exponent lyapunov exponent henceforth in a noisy time series, and decides whether the data are chaotic or stochastic based on a confidence level the main advantage of this test is that it can be conducted directly on experimental data without the need to define the generating equations. The lyapunov characteristic exponents and their computation. We report on structures in a scale dependent lyapunov exponent of an experimental chaotic map that arise due to discontinuities in the map. The predictability time is found to scale with the integral time scale in both the reacting and inert jet. Transitions from desynchronization to synchronization of trajectories occur when the conditional lyapunov exponent changes from positive to negative refs. The exponent is positive, so numerics lends credence to the hypothesis that the rossler.
The lces are asymptotic measures characterizing the average rate of growth or shrinking of small perturbations to the solutions of a dynamical system. Lyapunov exponents an overview sciencedirect topics. However, a subset of trajectories may still be asymptotically periodic or quasiperiodic in a chaotic system. Srivastava, senior member, ieee abb ab, global research lab. Exploring the lyapunov instability properties of highdimensional.
Localized finitetime lyapunov exponent for unsteady flow analysis jens kasten1, christoph petz1, ingrid hotz1, bernd r. Here, we employ a new multiscale complexity measure, the scaledependent lyapunov exponent sdle, to characterize hrv. Conditional lyapunov exponent criteria in terms of ergodic. Box 2077, west lafayette, indiana 47996, usa 2affymetrix, inc. Jun 30, 2009 previous studies on heart rate variability hrv using chaos theory, fractal scaling analysis, and many other methods, while fruitful in many aspects, have produced much confusion in the literature. When the perturbation is still small compared to largescale. The lyapunov exponent is the average of the logarithm of the stretching factor of each perturbation. Distinguishing chaos from noise by scaledependent lyapunov exponent j. The lyapunov exponent can also be found using the formula. The orbit is a neutral fixed point or an eventually fixed point. Positivity of the lyapunov exponent is an operational definition of chaos. The conditional lyapunov exponent is defined for investigating chaotic synchronization refs. As the lyapunov exponent measures the rate of divergence or convergence of two nearby initial points of a dynamical system.
Localized finitetime lyapunov exponent for unsteady flow. Time dependent control lyapunov functions and hybrid zero dynamics for stable robotic locomotion shishir kolathaya, ayonga hereid and aaron d. Here, we employ a new multiscale complexity measure, the scale dependent lyapunov exponent sdle. While excellent scale dependent measures such as epsilon entropy and the finite size lyapunov exponent fsle. System is deterministic, the irregular behavior is due to. Take the case of two identical simple harmonic oscillators with different amplitudes.
Time dependent control lyapunov functions and hybrid zero. Lyapunov exponents and strange attractors in discrete and. Special class for gifted youth, electronics and information engineering, huazhong university of. The longtime dynamics of solutions for the threedimensional primitive equations of large scale atmosphere was considered, and the. Multiscale characterization of sea clutter by scaledependent. Characterizing heart rate variability by scaledependent lyapunov exponent jing hu,1,2 jianbo gao,1,a and wenwen tung3 1pmb intelligence llc, p. The lyapunov exponent is a number that measures stability.
Scale dependent lyapunov exponent sdle analysis sdle is a multiscale complexity measure first introduced in 2006 gao et al. Lyapunov exponents lie at the heart of chaos theory, and are widely used in studies of complex dynamics. Universal scaling of lyapunov exponents in coupled chaotic oscillators. Distinguishing chaos from noise by scaledependent lyapunov.
Lyapunov exponents and the information tranfer index. Distinguishing chaos from noise by scale dependent lyapunov exponent j. A lyapunov exponent based method for online transient. Lyapunov exponents for continuoustime dynamical systems. Scaledependent lyapunov exponent sdle analysis sdle is a multiscale complexity measure first introduced in 2006 gao et al. While excellent scaledependent measures such as entropy and the finite size lyapunov exponent fsle have been proposed, simple algorithms. Characterizing heart rate variability by scaledependent.
Characterizing heart rate variability by scaledependent lyapunov. This may be done through the eigenvalues of the jacobian matrix j 0 x 0. Wolf et al determining lyapunov exponents from a time series 287 the sum of the first j exponents is defined by the long term exponential growth rate of a jvolume element. Multiscale analysis of heart rate variability in non. Lyapunov exponents, which provide a qualitative and quantitative characterization of. In this page, the lyapunov exponent is applied to an equation that jumps between stability and instability, between chaos and order the logistic equation. Lyapunov exponent as a metric for assessing the dynamic. Multiscale signal processing by scaledependent lyapunov exponent adviser. Whereas the global lyapunov exponent gives a measure for the total predictability of a system, it is sometimes of interest to estimate the local predictability around a point x 0 in phase space. While excellent scaledependent measures such as epsilon entropy and the finite size lyapunov exponent fsle have been proposed, simple algorithms have not been developed to reliably compute them. Structured scaledependence in the lyapunov exponent. To understand why multiscale analysis of sea clutter is essential, it is important to highlight a few empirical observations that point to the multiscale. Realizing that the difficulty in sea clutter modeling is due to the multiscale nature of sea clutter, we employ a new multiscale complexity measure, the scaledependent lyapunov exponent sdle, to better characterize the nonstationary and multiscale nature of sea clutter.
Keywords fractal dimension, lyapunov exponent, primitive equations msc classification. Given a deterministic system which remains in a bounded region of state space, chaos is defined by a statistic called the lyapunov exponent. Characterizing heart rate variability by scale dependent lyapunov exponent jing hu,1,2 jianbo gao,1,a and wenwen tung3 1pmb intelligence llc, p. In a onedimensional system, the lyapunov exponent reflects the logarithm of the geometric average growth of infinitesimal uncertainties. Pdf distinguishing chaos from noise by scaledependent. As we shall show, the sdle can not only resolve the above dilemma, but also conveniently deal with outliers. Fourth, an analysis of the local lyapunov exponent is performed to demonstrate that this metric can also determine. Quantitatively, two trajectories in phase space with initial separation vector diverge provided that the divergence can be treated within the linearized approximation at a rate given by.