Simple Chaotic System Research Articles

A piece-wise affine contracting map with positive entropy

We construct the simplest chaotic system with a two-point attractor. 1 If f : X → X is an isometry of the metric space (X, d), then the topological entropy vanishes: htop(f) = 0 (for definitions and notations consult e.g. [4]). This follows from the fact, that the iterated distance dn = max 0≤i<n (f i)∗(d) equals d. If f is distance non-increasing, the same equality holds and again htop(f) = 0. Whenever f can have discontinuities of some tame nature, so that f is piece-wise continuous, even the isometry result becomes difficult. In dimension 2 for invertible maps it was proven by Gutkin and Haydn [3]. In arbitrary dimension Buzzi proved that piece-wise affine isometries have zero topological entropy [2]. In the same paper after the theorem (remark 4) it is claimed that the result holds for arbitrary piece-wise (non-strictly) contracting maps. This latter is however wrong and the goal of this note is to present a counter-example. Example: Let X be a rhombus ADBC with vertices (±1, 0), (0,±1), see the figure below. Let O be its center and P,Q, R, S be on the sides as is shown.

On multiscale entropy analysis for physiological data

We perform an analysis of cardiac data using multiscale entropy as proposed in Costa et al. [Multiscale entropy analysis of complex physiological time series, Phys. Rev. Lett. 89 (2002) 068102]. We reproduce the signatures of the multiscale entropy for the three cases of young healthy hearts, atrial fibrillation and congestive heart failure. We show that one has to be cautious how to interpret these signatures in terms of the underlying dynamics. In particular, we show that different dynamical systems can exhibit the same signatures depending on the sampling time, and that similar systems may have different signatures depending on the time scales involved. Besides the total amount of data we identify the sampling time, the correlation time and the period of possible nonlinear oscillations as important time scales which have to be involved in a detailed analysis of the signatures of multiscale entropies. We illustrate our ideas with the Lorenz equation as a simple deterministic chaotic system.

The lifetime of CFC substitutes studied by a network trained with chaotic mapping modified genetic algorithm and DFT calculations

The hydrohaloalkanes have attracted much attention as potential substitutes of chlorofluorocarbons (CFCs) that deplete the ozone layer and lead to great high global warming. Having a short atmospheric lifetime is very important for the potential substitutes that may also induce ozone depletion and yield high global warming gases to be put in use. Quantitative structure–activity relationship (QSAR) studies were presented for their lifetimes aided by the quantum chemistry parameters including net charges, Mulliken overlaps, E HOMO and E LUMO based on the density functional theory (DFT) at B3PW91 level, and the C-H bond dissociation energy based on AM1 calculations. Outstanding features of the logistic mapping, a simple chaotic system, especially the inherent ability to search the space of interest exhaustively have been utilized. The chaotic mapping aided genetic algorithm artificial neural network training scheme (CGANN) showed better performance than the conventional genetic algorithm ANN training when the structure of the data set was not favorable. The lifetimes of HFCs and HCs appeared to be greatly dependent on their energies of the highest occupied molecular orbitals. The perference of the RMSRE comparing to RMSE as objective function of ANN training was better for the samples of interest with relatively short lifetimes. C2H6 and C3H8 as potential green substitutes of CFCs present relatively short lifetimes.

A SIMPLE SMOOTH CHAOTIC SYSTEM WITH A 3-LAYER ATTRACTOR

This Letter reports a simple smooth quadratic autonomous chaotic system with four unstable equilibrium points and three homoclinic orbits, and can display a 3-layer attractor among some others. It is the first example of a simple smooth quadratic system that we know, which can exhibit Šhilnikov chaos without numerical evidence of a route to chaos through period-doubling bifurcations.

A novel pattern classification scheme using the Baker's map

We demonstrate a novel application of nonlinear systems in the design of pattern classification systems. We show that pattern classification systems can be designed based upon training algorithms designed to control the qualitative behaviour of a nonlinear system. Our paradigm is illustrated by means of a simple chaotic system—the Baker's map. Algorithms for training the system are presented and examples are given to illustrate the operation and learning of the system for pattern classification tasks.

Constructing a new chaotic system based on the S̆ilnikov criterion

Based on the S̆ilnikov criterion, a simple quadratic chaotic system is constructed, which has a single equilibrium point. The formation mechanism shows that this chaotic system has Smale horseshoes (homoclinic chaos), and numerical simulation demonstrates that there is a route to chaos through period-doubling bifurcations. In particular, the method of finding chaotic systems can be used to construct rather arbitrary chaotic attractors of even number of scrolls and arbitrary odd number of scrolls.

Breaking time reversal in a simple smooth chaotic system

Within random matrix theory, the statistics of the eigensolutions depend fundamentally on the presence (or absence) of time reversal symmetry. Accepting the Bohigas-Giannoni-Schmit conjecture, this statement extends to quantum systems with chaotic classical analogs. For practical reasons, much of the supporting numerical studies of symmetry breaking have been done with billiards or maps, and little with simple, smooth systems. There are two main difficulties in attempting to break time reversal invariance in a continuous time system with a smooth potential. The first is avoiding false time reversal breaking. The second is locating a parameter regime in which the symmetry breaking is strong enough to transform the fluctuation properties fully to the broken symmetry case, and yet remain weak enough so as not to regularize the dynamics sufficiently that the system is no longer chaotic. We give an example of a system of two coupled quartic oscillators whose energy level statistics closely match with those of the Gaussian unitary ensemble, and which possesses only a minor proportion of regular motion in its phase space.

A NEW CHAOTIC SYSTEM AND ITS GENERATION

This Letter introduces a relatively simple three-dimensional continuous autonomous chaotic system, which can display complex 2- and 4-scroll attractors in simulations. Its generation and basic dynamical behaviors are briefly described.

Stochastic Methods for Sequential Data Assimilation in Strongly Nonlinear Systems

This paper considers several filtering methods of stochastic nature, based on Monte Carlo drawing, for the sequential data assimilation in nonlinear models. They include some known methods such as the particle filter and the ensemble Kalman filter and some others introduced by the author: the second-order ensemble Kalman filter and the singular extended interpolated filter. The aim is to study their behavior in the simple nonlinear chaotic Lorenz system, in the hope of getting some insight into more complex models. It is seen that these filters perform satisfactory, but the new filters introduced have the advantage of being less costly. This is achieved through the concept of second-order-exact drawing and the selective error correction, parallel to the tangent space of the attractor of the system (which is of low dimension). Also introduced is the use of a forgetting factor, which could enhance significantly the filter stability in this nonlinear context.

Dripping faucet with ants

The formation of droplets of ants is observed under certain experimental conditions. An aggregate forms at the end of a rod, the size of this aggregate fluctuates, and a droplet containing up to 40 ants eventually falls down. When the flux of incoming ants is sufficient, this process can continue for several hours, leading to the formation and fall of tens of droplets. This phenomenon is reminiscent of a leaky faucet, a well-known example of a simple chaotic system. It is found that the similarity is more than apparent: the time series of drop-to-drop intervals appears to result from a nonlinear low-dimensional dynamics, and the interdrop increments exhibit long-range anticorrelations.

Resonances of dynamical systems and Fredholm-Riesz operators on rigged Hilbert spaces

Resonances of dynamical systems are defined as the singularities of the analytically continued resolvent of the restriction of the Frobenius-Perron operator to suitable test-function spaces. A sufficient condition for resonances to arise from a meromorphic continuation to the entire plane is that the Frobenius-Perron operator is a Fredholm-Riesz operator on a rigged Hilbert space. After a discussion of spectral theory in locally convex topological vector spaces, we illustrate the approach for a simple chaotic system, namely the Rényi map.

Long-Range Atmospheric Predictability Using Space–Time Principal Components

The long-term predictability of 70-kPa geopotential heights is examined by a series of hindcast experiments over a validation period of 40 years using empirical models. Only the North Atlantic sector is considered. Significant skill is found up to lead times of one to two months for forecasts of time averages and of weather regime occurrence frequencies. The empirical schemes produce forecasts of the conditional probability of occurrence of a predictand within its natural terciles. These probabilistic forecasts are compared for two sets of predictors. The (spatial) principal components of the Atlantic large-scale flow (S-PCs) and its space–time principal components (ST-PCs) obtained from multichannel singular spectrum analysis (MSSA). These latter predictors achieve a good compromise between explained variance and predictability. In particular, the skill of a one-step model, where predictand's conditional probabilities are obtained directly from an analog method, is compared with a two-step model, which first forecasts the ST-PCs and then specifies the predictand's conditional probabilities. The one- step model is systematically beaten by the ST-PC scheme for lead times beyond 10 days. An attempt is made to explain why ST-PCs perform better than S-PCs in the long run by applying the forecast schemes to a simple low-order chaotic dynamical system. The key factor seems to be that for a dynamical system displaying low-frequency behavior and nonlinear spells of oscillations, the MSSA expansion gathers these phenomena into a few leading ST-PCs. These ST-PCs are therefore good candidates to quantify the concept of atmospheric “predictable” components.

Fick's law and fractality of nonequilibrium stationary states in a reversible multibaker map

Nonequilibrium stationary states are studied for a multibaker map, a simple reversible chaotic dynamical system. The probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function. The equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of deRham-type functional equations. To select the physical states, the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type. This system corresponds to a diffusive flow experiment through a slab of material. For long times, any initial distribution approaches the stationary one obeying Fick's law. At stationarity, the intracell distribution is singular in the stable direction and expressed by the Takagi function, which is continuous but has no finite derivatives. The result suggests that singular measures play an important role in the dynamical description of non-equilibrium states.

Accuracy of the semiclassical approximation for chaotic scattering.

The semiclassical approximation for scattering probabilities is tested for a simple chaotic system consisting of a particle in one dimension scattering from a localized potential that varies periodically in time. Good agreement between semiclassical and exact quantum mechanical results is found even for relatively large de Broglie wavelengths.

Neuromodels of analytic dynamic systems

In contrast to recent work aimed at using neural networks for relatively ‘long term’ prediction of time series, this paper examines how neural networks designed for short term prediction can form very good approximation models, valid over a large region of the phase space, after having been trained on as few as 500 pointsfrom a single trajectory of the underlying dynamic system. This is illustrated using four dynamic systems of increasing complexity, including a simple chaotic system and a more realistic system describing rigid body rotation using orthogonal torques under a chaotic control regime.

Transport as a dynamical property of a simple map

We study a simple chaotic system given by a map which displays diffusion. The decaying eigenstates of the Frobenius-Perron operator of the map are constructed. These eigenstates are associated with the Ruelle resonances. The eigenstate associated with diffusion is considered in particular. The result implies a deep connection between the thermodynamical properties of a system and its underlying exact dynamics.

INDIRECT ADAPTIVE CONTROL OF A CHAOTIC SYSTEM

Abstract The adaptive control of a simple chaotic system to a steady reference is investigated. An indirect adaptive control scheme with least-mean-squares (LMS) parameter estimation yields a fractal basin boundary between stable and unstable control. This boundary appears self-similar, indicative of a fractal structure, and has a fractal dimension of 2.61. Adaptive control using a least-squares (LS) estimator appears globally stable with the rate of adaptation a complex function of the distance from the desired set point. The effect of control parameters, noise and a step-change perturbation of the system are also reported.

Heat transfer between two degrees of freedom.

A particularly simple chaotic nonequilibrium open system with two Cartesian degrees of freedom, characterized by two distinct temperatures T(x) and T(y), is introduced. The two temperatures are maintained by Nose-Hoover canonical-ensemble thermostats. Both the equilibrium (no net heat transfer) and nonequilibrium (dissipative) Lyapunov spectra are characterized for this simple system.

Switched-capacitor broadband noise generator for CMOS VLSI

A switched-capacitor circuit is reported for the generation of broadband white noise in MOS VLSI. It is based on the implementation of a very simple chaotic discrete-time system. The concept is demonstrated via a 3 μm CMOS double-metal double-poly monolithic prototype yielding a 4 V peak-to-peak signal with a flat power density spectrum from DC to about half the clock frequency.

Macroscopic behavior in a simple chaotic Hamiltonian system : Otto E. Rössler, Institute for Physical and Theoritical Chemistry, University of Tübingen, 7400, WEST GERMANY.

The BSCCO type intrinsic Josephson junction has been modeled as a stack of inductively coupled long Josephson junctions, which were described by a system of coupled sine-Gordon equations. In a system of 10 long Josephson junctions coupled to a linear cavity, we numerically investigate how the cavity perturbs fluxon motion in the junctions. Fluxons in neighboring junctions are repulsive leading to anti-phase motion. The cavity provides a force which perturbs this anti-phase motion, and may even force the fluxons to build a square lattice, i.e. to perform in-phase motion. For different values of the inductive coupling strength, we investigate the cavity current, fluxon phase difference, and current–voltage characteristic. The stack-cavity system with in-phase fluxon motion may be utilized as a THz oscillator.