Autonomous Chaos Research Articles

The Existence of Autonomous Chaos in EDM Process

The dynamical evolution of electrical discharge machining (EDM) has drawn immense research interest. Previous research on mechanism analysis has discussed the deterministic nonlinearity of gap states at pulse-on discharging duration, while describing the pulse-off deionization process separately as a stochastic evolutionary process. In this case, the precise model describing a complete machining process, as well as the optimum performance parameters of EDM, can hardly be determined. The main purpose of this paper is to clarify whether the EDM system can maintain consistency in dynamic characteristics within a discharge interval. A nonlinear self-maintained equivalent model is first established, and two threshold conditions are obtained by the Shilnikov theory. The theoretical results prove that the EDM system could lead to chaos without external excitation. The time series of the deionization process recorded in the EDM experiments are then analyzed to further validate this theoretical conclusion. Qualitative chaotic analyses verify that the autonomous EDM process has chaotic characteristics. Quantitative methods are used to estimate the chaotic feature of the autonomous EDM process. By comparing the quantitative results of the autonomous EDM process with the non-autonomous EDM process, a deduction is further made that the EDM system will evolve towards steady chaos under an autonomous state.

A design principle for posttranslational chaotic oscillators.

SummaryChaos behavior has been observed in various cellular and molecular processes. Here, we modeled reversible phosphorylation dynamics to elucidate a design principle for autonomous chaos generation that may arise from generic enzymatic reactions. A comprehensive parameter search demonstrated that the reaction system composed of a set of kinases and phosphatases and two substrates with two modification sites exhibits chaos behavior. All reactions are described according to the Michaelis-Menten reaction scheme without exotic functions being applied to enzymes and substrates. Clustering analysis of parameter sets that can generate chaos behavior revealed the existence of motif structures. These chaos motifs allow the two-substrate species to interact via enzyme availability and constrain the two substrates' dynamic changes in phosphorylation status so that they occur at different timescales. This chaos motif structure is found in several enzymatic reactions, suggesting that chaos behavior may underlie cellular autonomy in a variety of biochemical systems.

New Results to a Three-Dimensional Chaotic System With Two Different Kinds of Nonisolated Equilibria

In the paper by Liu et al. (2009, “A Novel Three-Dimensional Autonomous Chaos System,” Chaos Solitons Fractals, 39(4), pp. 1950–1958), the three-dimensional (3D) chaotic system x·=-ax-ey2,y·=by-kxz,z·=-cz+mxy is investigated, and some of its dynamics according to theoretical and numerical analyses only for the parameters (a, e, b, k, c, m) = (1, 1, 2.5, 4, 5, 4) are discussed. In 2013, the same chaotic system x·1=-ax1- fx2x3,x·2=cx2-dx1x3,x·3=-bx3+ex22 by Li et al. (2013, “Analysis of a Novel Three-Dimensional Chaotic System,” Optik, 124(13), pp. 1516–1522) was mainly discussed by numerical simulation. In this article, by some deeper investigations, combining some numerical simulations, we formulate some new results of the system. First, after some problems in the first paper are pointed out, we display that its parameters e, k, and m may be kicked out by some homothetic transformations. Second, some of its rich nonlinear dynamics hiding and not found previously, such as the stability and Hopf bifurcation of its isolated equilibria, the behavior of its nonisolated equilibria, the existence of singular orbits (including singularly degenerate heteroclinic cycle, homoclinic and heteroclinic orbits, etc.), and its dynamics at infinity, etc., are clearly formulated. What's more interesting, we find, this system has two different kinds of nonisolated equilibria Ex and Ez, and new chaotic attractors can be bifurcated out with the disappearance of Ex, but this system has no such properties at Ez. In the meantime, several problems about the existence of singular orbits deserving further investigations are presented. Our results better complement and improve the known ones.

A NEW AUTONOMOUS CHAOS GENERATOR FROM STATE CONTROLLED-CELLULAR NEURAL NETWORKS

In this study, a new autonomous chaos generator is presented via State Controlled-Cellular Neural Networks (SC-CNNs). From the numerical and circuit simulations, it has been shown that the proposed SC-CNN consisting of three cells can be easily used in chaotic applications as a chaos generator. After presenting the double scroll generation, the presented SC-CNN system is used either in multi-scroll chaotic attractor generation by adding a trigonometric function generator and or to construct a SC-CNN based chaotic masking system with feedback algorithm.

Chaos in fractional ordered Liu system

Present paper deals with fractional version of a dynamical system introduced by C. Liu, L. Liu and T. Liu [C. Liu, L. Liu, T. Liu, A novel three-dimensional autonomous chaos system, Chaos Solitons Fractals 39 (4) (2009) 1950–1958]. Numerical investigations on the dynamics of this system have been carried out. Properties of the system have been analyzed by means of Lyapunov exponents. Furthermore the minimum effective dimensions have been identified for chaos to exist in commensurate and incommensurate orders. It is noteworthy that the results obtained are consistent with the analytical conditions given in the literature.

A novel three-dimensional autonomous chaos system

This letter reports a novel three-dimensional autonomous chaotic system. Some basic dynamical properties, such as Lyapunov exponents, Poincare mapping, fractal dimension, continuous spectrum and chaotic behaviors of this new transverse butterfly attractor are studied. Furthermore, the forming mechanism of its compound structure obtained by merging together two simple attractors after performing one mirror operation has been investigated by detailed numerical as well as theoretical analysis.

Delayed Feedback Control of Chaos in an Electronic Double-Scroll Oscillator

Abstract We present experimental results on stabilizing unstable periodic orbits of an autonomous chaos oscillator based on a simple electronic circuit. Control is achieved by applying the difference between the actual and a delayed output signal of the oscillator. The quality of chaos control can be measured via the strength of perturbation. The dependence on the delay time shows a characteristic resonance-type behavior.

Antiphase dynamics and chaos in self-pulsing erbium-doped fiber lasers

We present experimental evidence of antiphase dynamics in self-pulsing erbium-doped fiber lasers operating simultaneously at 1.536- and 1.55-microm wavelengths. Autonomous chaos is also observed for several ranges of pumping rates. The fractal dimension of the chaotic attractor, calculated from experimental data by use of the Procaccia method, is found to depend on the pumping ratio but always remains smaller than 4.1, regardless of the chaotic regime considered.

Autonomous chaotic behaviour of the slime mould Dictyostelium discoideum predicted by a model for cyclic AMP signalling.

How sustained oscillations lose their periodicity and thus give rise to chaos was first analysed in mathematical models, then observed in chemical systems such as the Belousov-Zhabotinsky reaction where chaos is autonomous because it originates from endogenous kinetic mechanisms. In contrast, chaos can also be obtained by periodically forcing an oscillatory system, as shown, for example, in cardiac cells and yeast glycolysis. Biochemical evidence for autonomous chaos has been obtained both in vitro for the peroxidase reaction and in enzymatic models not based directly on experimental systems. We report here the occurrence of autonomous chaos in a realistic model for the cyclic AMP signalling system of the slime mould Dictyostelium discoideum, based on receptor modification. This model is also capable of bursting, a phenomenon characteristic of some pacemaker neurones such as R15 in Aplysia. Whereas bursting has not been observed in D. discoideum, our model suggests that 'aperiodic signalling' in the mutant Fr17 provides the first example of autonomous chaos occurring spontaneously at the cellular level.

Hysteresis chaos generator

AbstractThis paper discusses the hysteresis generator which is drawing attention as an autonomous chaos generator. However, this generator has not been discussed theoretically for its chaos generation mechanism. A fundamental hysteresis chaos generator is proposed deriving analytically the condition that chaos is generated in the circuit. The analysis of the circuit is performed by mapping procedure, and the Poincare' mapping is derived analytically as a one‐dimensional mapping from the fact that the circuit is a second‐order piecewise‐linear differential equation. By calculating based on the mapping, the generation condition for the chaos is represented explicitly for the parameters In this paper, the definition for the chaos is that the derived mapping has a unique absolutely continuous invariant measure and is ergodic. The generation of the chaos is verified by experiment.