Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map : Mechanisms and their characterizations

A simple quasiperiodically forced one-dimensional cubic map is shown to exhibit very many types of routes to chaos via strange nonchaotic attractors (SNAs) with reference to a two-parameter $(A-f)$ space. The routes include transitions to chaos via SNAs from both one frequency torus and period doubled torus. In the former case, we identify the fractalization and type I intermittency routes. In the latter case, we point out that atleast four distinct routes through which the truncation of torus doubling bifurcation and the birth of SNAs take place in this model. In particular, the formation of SNAs through Heagy-Hammel, fractalization and type--III intermittent mechanisms are described. In addition, it has been found that in this system there are some regions in the parameter space where a novel dynamics involving a sudden expansion of the attractor which tames the growth of period-doubling bifurcation takes place, giving birth to SNA. The SNAs created through different mechanisms are characterized by the behaviour of the Lyapunov exponents and their variance, by the estimation of phase sensitivity exponent as well as through the distribution of finite-time Lyapunov exponents.Comment: 27 pages, RevTeX 4, 16 EPS figures. Phys. Rev. E (2001) to appea

Strange nonchaotic attractors in driven excitable systems

Through quasiperiodic forcing, an excitable system can be driven into a regime of spiking behavior that is both aperiodic and stable. This is a consequence of strange nonchaotic dynamics: the motion of the system is on a fractal attractor and the largest Lyapunov exponent is negative

Testing For Nonlinearity Using Redundancies: Quantitative and Qualitative Aspects

A method for testing nonlinearity in time series is described based on information-theoretic functionals -- redundancies, linear and nonlinear forms of which allow either qualitative, or, after incorporating the surrogate data technique, quantitative evaluation of dynamical properties of scrutinized data. An interplay of quantitative and qualitative testing on both the linear and nonlinear levels is analyzed and robustness of this combined approach against spurious nonlinearity detection is demonstrated. Evaluation of redundancies and redundancy-based statistics as functions of time lag and embedding dimension can further enhance insight into dynamics of a system under study.Comment: 32 pages + 1 table in separate postscript files, 12 figures in 12 encapsulated postscript files, all in uuencoded, compressed tar file. Also available by anon. ftp to santafe.edu, in directory pub/Users/mp/qq. To be published in Physica D., [email protected]

A simple method for detecting chaos in nature

Chaos, or exponential sensitivity to small perturbations, appears everywhere in nature. Moreover, chaos is predicted to play diverse functional roles in living systems. A method for detecting chaos from empirical measurements should therefore be a key component of the biologist's toolkit. But, classic chaos-detection tools are highly sensitive to measurement noise and break down for common edge cases, making it difficult to detect chaos in domains, like biology, where measurements are noisy. However, newer tools promise to overcome these limitations. Here, we combine several such tools into an automated processing pipeline, and show that our pipeline can detect the presence (or absence) of chaos in noisy recordings, even for difficult edge cases. As a first-pass application of our pipeline, we show that heart rate variability is not chaotic as some have proposed, and instead reflects a stochastic process in both health and disease. Our tool is easy-to-use and freely available

Chaos at Fifty

In 1963 Edward Lorenz revealed deterministic predictability to be an illusion and gave birth to a field that still thrives. This Feature Article discusses Lorenz's discovery and developments that followed from it.Comment: For an animated visualization of the Lorenz attractor, click here http://www.youtube.com/watch?v=iu4RdmBVdp

Mixing Bandt-Pompe and Lempel-Ziv approaches: another way to analyze the complexity of continuous-states sequences

In this paper, we propose to mix the approach underlying Bandt-Pompe permutation entropy with Lempel-Ziv complexity, to design what we call Lempel-Ziv permutation complexity. The principle consists of two steps: (i) transformation of a continuous-state series that is intrinsically multivariate or arises from embedding into a sequence of permutation vectors, where the components are the positions of the components of the initial vector when re-arranged; (ii) performing the Lempel-Ziv complexity for this series of `symbols', as part of a discrete finite-size alphabet. On the one hand, the permutation entropy of Bandt-Pompe aims at the study of the entropy of such a sequence; i.e., the entropy of patterns in a sequence (e.g., local increases or decreases). On the other hand, the Lempel-Ziv complexity of a discrete-state sequence aims at the study of the temporal organization of the symbols (i.e., the rate of compressibility of the sequence). Thus, the Lempel-Ziv permutation complexity aims to take advantage of both of these methods. The potential from such a combined approach - of a permutation procedure and a complexity analysis - is evaluated through the illustration of some simulated data and some real data. In both cases, we compare the individual approaches and the combined approach.Comment: 30 pages, 4 figure

On the chaotic evolution of baroclinic instability of wave-wave interactions with topography

De Szoeke (1986) developed an asymptotic solution for the nonlinear evolution of a type of baroclinic instability of a two-layer quasi-geostrophic model over topography. He found that under certain conditions pairs of hybrid modes interacting with topography could become unstable in the linearized model. He also found that the addition of the nonlinearity stabilized the flow which he analyzed using an expansion in small ε, a measure of the topographic height. This work is extended and modified by first considering a slightly varying mean flow (time dependent parametric forcing) which produces chaotic behavior, and then considering friction which allows chaos on a strange attractor. This chaos is examined in various ways including using Melnikov\u27s method. The original unforced system without friction can be called critically nonchaotic in that only a very small amount of forcing produces significant chaotic behavior. We then investigate whether the original system can be chaotic without any variation in the mean flow. An additional term is included in the asymptotic system to form a six variable system which can become chaotic. We also look at a more general, nonasymptotic, initial value problem consisting of sixteen variables, assuming small amplitudes, which also can become chaotic. Finally, we consider an asymptotic expansion in small α, the aspect ratio of the top to the bottom layer, assuming ε = O (1), which is often more realistic in the ocean. It is found that at the conditions of greatest initial growth of the amplitudes the system can be chaotic with the largest amplitude in the top layer