Example Of Deterministic Chaos Research Articles

Singing of Neoconocephalus robustus as an example of deterministic chaos in insects

We use nonlinear time series analysis methods to analyse the dynamics of the sound-producing apparatus of the katydid Neoconocephalus robustus. We capture the dynamics by analysing a recording of the singing activity. First, we reconstruct the phase space from the sound recording and test it against determinism and stationarity. After confirming determinism and stationarity, we show that the maximal Lyapunov exponent of the series is positive, which is a strong indicator for the chaotic behaviour of the system. We discuss that methods of nonlinear time series analysis can yield instructive insights and foster the understanding of acoustic communication among insects.

Period Doubling and Chaos of Spontaneous Oscillatory Phenomena on Iron

A new self-oscillating system introducing spontaneous oscillatory phenomena on iron in a sulfuric acid solution of hydrogen peroxide is discussed. This phenomenon can be explained by a continuous oscillation between the active and passive state of iron. Unlike other published systems that only show one-peak oscillations, the potential of iron shows highly complex dynamics: depending on the concentration of hydrogen peroxide and sulfuric acid, a period doubling and non-periodic oscillations in a self-oscillating system were observed for the first time. The reactions were analyzed by methods based on chaos theory and could finally be identified as an example of deterministic chaos.

Mathematics. The Lorenz attractor exists

The Lorenz attractor is an example of deterministic chaos. Previously, the Lorenz attractor could only be generated by numerical approximations on a computer. Now we have a rigorous proof that confirms its existence.

Chaotic dynamics and subharmonic bifurcations in a non-linear system

Examples of deterministic chaos abound. A periodically forced system of phase-locked loops or a pendulum is considered. This system has shown to have various types of chaotic motion, in particular pendulum-type, Duffing-type and their combination-type under certain parameter conditions. There exist some parameter regions such that the combination-type of chaos motions and all possible types of subharmonic solutions occur simultaneously. In this case the system has enormous complexity of chaotic regimes which have been well understood theoretically.

Chaos and non-linear phenomena in renal vascular control

Renal autoregulation of blood flow depends on the functions of the tubuloglomerular feedback (TGF) system and the myogenic response of the afferent arteriole. Studies of the dynamic aspects of these control mechanisms at the level of both the single nephron and the whole kidney have revealed a variety of non-linear phenomena. In halothane-anesthetized, normotensive rats the TGF system oscillates regularly at 2-3 cycles/min because of the non-linearities and the time delays within the feedback system. Oscillations are present in single nephron blood flow, tubular pressure and flow, and in the tubular solute concentrations. Nephrons deriving their afferent arteriole from the same cortical radial artery are entrained, and consequently oscillate at the same frequency. Experimental studies have shown that the synchronization is due to an interaction of the TGF between nephrons. A necessary condition for the interaction is that the nephrons derive their blood supply from the same cortical radial artery. Development of hypertension is associated with a shift from periodic oscillations of tubular pressure to random-like fluctuations. Numerical analyses indicate that these fluctuations are an example of deterministic chaos. Experimental studies show that the development of hypertension is associated with an increase in strength of the interaction between nephrons. Mathematical models suggest that an increased nephron-nephron interaction could cause a bifurcation in the dynamics of TGF from periodic oscillations to deterministic chaos. In addition to the TGF mediated oscillation, experimental studies have also demonstrated the presence of a faster oscillation, this having a frequency of 120-160 mHz. This is caused by a mechanism intrinsic to the vascular wall, and presumably represents the well-known phenomenon of vasomotion. Using newly developed non-linear analytical methods non-linear interactions between vasomotion and the TGF mediated oscillation were detected both in single nephron and in whole kidney blood flow. The physiological significance of these non-linear phenomena in renal vascular control is discussed.

Chaotic behavior of the north-south asymmetry of sunspots?

North-south asymmetry in the distribution of sunspots was examined. Weak correlations between north-south asymmetry and sunspot number were found in several time lags. Higuchi's fractal dimension (1988) was calculated to evaluate irregularity in north-south asymmetry. The fractal dimension obtained is 1.90 ± 0.01 and this implies that the north-south asymmetry is highly irregular. The method of Sugihara and May (1990), based on the nonlinear prediction method, was used to distinguish between deterministic chaos and noise. The results do not confirm the idea that north-south asymmetry is an example of deterministic chaos.

Economic Complexity: Chaos, Sunspots, Bubbles, and Nonlinearity

The contents of this volume comprise the proceedings of the International Symposia in Economic Theory and Econometrics conference held in 1987 at the IC^T2 (Innovation, Creativity, and Capital) Institute at the University of Texas at Austin. The essays present fundamental new research on the analysis of complicated outcomes in relatively simple macroeconomic models. The book covers econometric modelling and time series analysis techniques in five parts. Part I focuses on sunspot equilibria, the study of uncertainty generated by nonstochastic economic models. Part II examines the more traditional examples of deterministic chaos: bubbles, instability, and hyperinflation. Part III contains the most current literature dealing with empirical tests for chaos and strange attractors. Part IV deals with chaos and informational complexity. Part V, Nonlinear Econometric Modelling, includes tests for and applications of nonlinearity.

Economic Complexity: Chaos, Sunspots, Bubbles and Nonlinearity.

The contents of this volume comprise the proceedings of the International Symposia in Economic Theory and Econometrics conference held in 1987 at the IC^T2 (Innovation, Creativity, and Capital) Institute at the University of Texas at Austin. The essays present fundamental new research on the analysis of complicated outcomes in relatively simple macroeconomic models. The book covers econometric modelling and time series analysis techniques in five parts. Part I focuses on sunspot equilibria, the study of uncertainty generated by nonstochastic economic models. Part II examines the more traditional examples of deterministic chaos: bubbles, instability, and hyperinflation. Part III contains the most current literature dealing with empirical tests for chaos and strange attractors. Part IV deals with chaos and informational complexity. Part V, Nonlinear Econometric Modelling, includes tests for and applications of nonlinearity.

Chaotic seismic faulting with a mass-spring model and velocity-weakening friction

We present a systematic analysis of the dynamical behavior introduced by fault zone heterogeneities, using a simple mass-spring model with velocity-weakening friction. The model consists of two sliding blocks coupled to each other and to a constant velocity driver by elastic springs. The state of this system can be characterized by the positions of the two blocks relative to the driver. Symmetry stabilizes the system and generates only cyclic behavior. For an asymmetric system where the frictional forces for the two blocks are not equal, the solutions exhibit chaotic behavior. The transition from stable cyclic behavior to chaos is characterized by the period-doubling route to chaos. Lyapunov exponents are computed to quantify the deterministic chaos and to locate the onset of the chaotic evolution in parameter space. In many examples of deterministic chaos, chaotic behavior of a low-order system implies chaos in similar higher order systems. Thus, our results provide substantial evidence that crustal deformation is an example of deterministic chaos.

Is the Applicability of Fractal Statistics to Sedimentary Structures the Result of Scale-Invariant Stochastic Processes or Deterministic Chaos?

Fractal statistics are the only statistics that are scale invariant. Examples in tectonics include distributions of faults, displacements on faults, distributions and permeabilities of fractures, and distributions of folds. Many aspects of sedimentology are also fractal including distributions of sedimentary sequences, variations in permeability, and shapes of boundaries. Since the underlying processes are likely to be scale invariant, it is reasonable to conclude that the number-size statistics of oil fields will be fractal. Log-normal statistics are often applied; they are not scale invariant. Two explanations for fractal statistics can be given. They may be the result of scale-invariant stochastic processes. Random walk (Brownian noise) is one example. Topography generally resembles Brownian noise, a power-law spectrum with fractal dimension D = 1.5. Alternatively fractal statistics can be the result of deterministic chaos. Turbulent flows are examples of deterministic chaos, the governing equations are deterministic but the resulting flows are statistical. Tectonic displacements can be shown to be the result of deterministic chaos; it is likely that erosion is another example.

Are earthquakes an example of deterministic chaos?

We use a simple mass‐spring model to systematically examine the dynamical behavior introduced by fault zone heterogeneities. The model consists of two sliding blocks coupled to each other and to a constant velocity driver by elastic springs. The state of this system can be characterized by the positions of the two blocks relative to the driver. A simple static/dynamic friction law is used. When the system is symmetric, we observe cyclic behavior. For an asymmetric system where the fractional forces for the two blocks are not equal, the solutions exhibit deterministic chaos. Chaotic windows occur repeatedly between regions of limit cycles on bifurcation diagrams. Our model behavior is similar to that of the one‐dimensional logistic map. In many examples of deterministic chaos, chaotic behavior of a low order system implies chaos in similar higher order systems. Thus, our results provide substantial evidence that earthquakes are an example of deterministic chaos.

Explanation of random vibrations in cutting on grounds of deterministic chaos

This article presents a model of orthogonal cutting on an elastic system. The dynamics of the system is described by the differential equations in which the forces are expressed by empirical nonlinear relations. The oscillations of the system determined by numerical solutions of nonlinear differential equations exhibit chaotic characters in certain domains of typical parameters and thus represent a new example of deterministic chaos. This indicates that, besides the irregular properties of cut material, nonlinear dynamics must also be taken into account in order to explain the random character of vibrations observed in cutting systems.