bluuebunny/crossref_metadata_2025 huggingface dataset:
Zero-Dispersion Nonlinear Resonance
Authors: Soskin S. M. , Luchinsky D. G. , Mannella R. , Neiman A. B. , McClintock P. V. E.
Published: September 2003

Underdamped oscillators that possess a maximum or minimum in their dependence of eigenfrequency on energy have recently been shown to exhibit a range of unusual phenomena. Because they are associated with the presence of an extremum in whose vicinity the eigenfrequency is almost energy-independent, they have been named zero-dispersion phenomena. They manifest themselves both in the deterministic dynamics and in the presence of noise. When the oscillator is driven by a weak periodic force at a frequency close to that of the extremum, a novel type of nonlinear resonance, zero-dispersion nonlinear resonance (ZDNR) can occur. A giant response then arises even in the absence of resonance between the drive frequency and any eigenoscillation of the system. The properties of ZDNR, the nature of the transition from ZDNR to conventional nonlinear resonance as relevant parameters are varied, the occurrence of dynamical chaos associated with ZDNR, and the influence of noise, are analysed and discussed for both Hamiltonian and dissipative systems.
Control of Chaotic Impacts
Authors: Casas Fernando , Grebogi Celso
Published: September 2003

We apply controlling chaos techniques to select the desired sequence of impacts in a map that captures universal properties of impact oscillators near grazing. For instance, we can choose the period and then stabilize an unstable periodic orbit with, say, one impact per period involved in the grazing bifurcations that take place in the system.
Short-Wave Instability in Extended Systems with Additional Symmetry
Authors: Tribelsky Michael I.
Published: September 2003

Stability of steady spatially periodic patterns in systems with an additional continuous group of symmetry is discussed. It is shown that different systems with the same dimensionality of the continuous group of symmetry display remarkable similarity in all qualitative features of the pattern stability problem. Attention is called to the fact that, beside an extra band of slowly varying modes, the additional symmetry may yield a mixture of different scales in the final dispersion equation for pattern's perturbations, so that the stability conditions become unusually sensitive to very fine details of the problem. A one-dimensional partial differential equation governing seismic waves in viscoelastic media is considered as a particular example. The equation exhibits short-wave instability and additional invariance under the transformation u → u + const. , where the order parameter u(x, t) is associated with the displacement velocity. The analytical study of the equation is supplemented by computer simulations. The simulations show that the system undergoes a bifurcation from the trivial state with u ≡ 0 to well-developed chaos directly and the transition to the chaos is smooth, without any discontinuity. The chaos is characterized by excitation of a big number (in a boundless system — continuum) of coupled modes localized generally in two narrow subbands, centered around the critical wavenumber for the short-wave instability and the wavenumber equals zero, respectively.
Rule Dynamics of Electronic Turbulence in Semiconductors Caused by Impact Ionization Avalanche
Authors: Aoki Kazunori
Published: September 2003

Complex dissipative structures and turbulence of filamentary currents have been studied numerically in a model of one-level impact ionization avalanche in semiconductors. The model includes the important parts of (1) transverse diffusion term of impact-ionized carriers in the equation of the generation-recombination process and (2) drift/diffusion currents in the dielectric relaxation of the transverse field. Under a periodically-driven regime using a dc + ac field E0 + Eac sin (2πf0t) the spatiotemporal patterns of the electron density (fast variable) and subsequently the current density exhibit period doubling bifurcation route to chaos as a function of E0, which is ruled by the pattern of the transverse field (slow variable).
Chaotic Evolution of a Parametric Instability in a Piezoelectric Semiconductor Plasma
Authors: Shah Ijaz , Anwar Ghazala , Shah H. A. , Abdullah T. , Anis Alam M.
Published: September 2003

We consider the chaotic evolution of three parametrically interacting waves in a piezoelectric semiconductor plasma. The evolution equation has been derived using the coupled mode theory. The waves are an extraordinary wave, ordinary wave and upper hybrid acoustic wave. The extraordinary wave is a growing wave and the other two are damped modes. The equations are transformed and put in a standard form and made dimensionless by suitable substitution of new variables. Finally a set of three equations is obtained. The real and imaginary parts of the equations are then separated. The three imaginary equations can be reduced to one resulting in a system of four equations. Analytical investigation are made of the stationary points. The evolution of equations for various parametric values is done numerically. Time series analysis and phase space plots of the chaotic evolution of the amplitude are also investigated numerically. Lyapunov exponents are computed and the results are presented graphically.
Replicating Spots in Reaction-Diffusion Systems
Authors: Lee Kyoung J. , Swinney Harry L.
Published: September 2003

We review the phenomenon of replicating spots in reaction-diffusion systems and discuss the mechanism of replication. This phenomenon was discovered in recent experiments on a ferrocyanide-iodate-sulfite reaction-diffusion system. Patterns form in a thin gel layer that is in contact with a continuously fed stirred reservoir. Patterns of spots are observed to undergo a continuous process of growth and multiplication through cell division and death through overcrowding. A similar phenomenon is also found in numerical simulations in one dimension on a four-species model of the ferrocyanide-iodate-sulfite reaction and in simulations in two dimensions of simpler two-species reaction-diffusion models: Gray–Scott model by J. Pearson and FitzHugh–Nagumo model by A. Hagberg and E. Meron.
Group Invariances, Unfoldings and the Bifurcation of Capillary-Gravity Waves
Authors: Jones M. C. W.
Published: September 2003

An investigation is made into the waves which may arise from an interaction between the Mth and Nth harmonics of a fundamental mode. The problem is studied by transforming it into a nonlinear differential equation for the angle between the free surface and the horizontal. Solutions of this equation lie in an appropriate infinite dimensional Banach space but by using classical methods of analysis the equation may be reduced to a finite set of algebraic equations. By making use of certain group invariance properties of the original operator, information may be obtained about the structure of these equations and their solutions. The results are interpreted in the context of the original problem.
Lyapunov Spectral Analysis on Random Data
Authors: Ikeguchi Tohru , Aihara Kazuyuki
Published: September 2003

In this paper, an algorithm for estimating all Lyapunov exponents, or Lyapunov spectra of deterministic dynamical systems, is applied to random time series, that is, the time intervals of gamma ray emissions of cobalt. The algorithm used in this paper is that proposed by Sano and Sawada, which estimates Jacobian matrices of an assumed dynamical system from the data points on a possible attractor reconstructed. As a result, the largest Lyapunov exponent of the cobalt data is estimated to be positive, the same as the case of deterministic chaos. Although the sum of all Lyapunov exponents is positive in lower reconstructed dimensions, the values are estimated to be negative when the reconstructed dimension becomes higher. This result is also the same as deterministic chaos in dissipative dynamical systems. It is an indication that naive application of estimating algorithm for Lyapunov spectra to real data does not necessarily lead us to correct results.

In order to check the appearance of positive Lyapunov exponents, firstly an analysis by the local versus global (LVG) plots is adopted. The results by LVG plots on the cobalt data are clearly different from those on typical nonlinear dynamical systems in lower dimensional reconstructed state space. Secondly, a direct approach for estimating largest Lyapunov exponents is applied. The results show that estimated Lyapunov exponents on cobalt data do not necessarily indicate existence of positive values. Lastly, a test of statistical hypothesis with surrogate data is also applied to the cobalt data under the null hypotheses that the cobalt data is produced from a linear stochastic system. The results show that statistical significances between the original and surrogate data are so small that some null hypotheses on the cobalt data cannot be rejected; the results imply that an analysis with surrogate data sets could avoid spurious interpretation that random time series, such as the cobalt data, has low-dimensional nonlinear deterministic dynamics with positive Lyapunov exponents.
Correction of the Correlation Dimension for Noisy Time Series
Authors: Kugiumtzis D.
Published: September 2003

In the computation of the correlation dimension of chaotic time series corrupted with observational noise, the scaling region is often masked resulting in deteriorated estimates. Here a simple method is proposed to correct the corrupted correlation integral based on the statistical properties of the Euclidean norm used to compute the noisy point interdistances. When the noise level is known, the corrected slope from the noisy data is very close to the slope for the noise-free data. Thus if the scaling property of the noise-free attractor holds for distances around the noise amplitude then the correct dimension can be inferred. The problem of estimating the correct noise level is discussed and a simple approach is proposed. Simulations with synthetic noisy chaotic time series demonstrate the efficiency of the correction scheme. Furthermore, the correction scheme is used to enhance correlation dimension estimates for the Taylor–Couette chaotic data and for EEG data from epileptic seizures.
Bifurcations of Numerically Simulated Marangoni Flows in Floating Zones
Authors: Menke C.
Published: September 2003

Marangoni or thermocapillary convection in a two-dimensional cylindrical half zone configuration under microgravity is studied numerically. The time-dependent simulations take into account convection and conduction in the melt, heat transfer between the melt and the ambient, and deformations of the free melt/gas surface of the half zone. A modified Marker and Cell (MAC) method is used to compute the flow and the temperature fields. The algorithm is applied especially to silicon melts. Above a critical temperature difference in the melt, the steady state becomes unstable and oscillatory thermocapillary convection occurs. The relevant control parameter for the onset of oscillations is the Marangoni number. As the Marangoni number increases, the phenomenon of period doubling is observed in the simulations. After a sequence of period doubling bifurcations, the flow becomes turbulent.
Nonlinear H∞ Synchronization of Chaotic Lur'e Systems
Authors: Suykens J. A. K. , Vandewalle J. , Chua L. O.
Published: September 2003

In this paper a master-slave synchronization scheme for Lur'e systems is investigated, which consists of vector field modulation by means of the message signal. Using a full state error feedback mechanism and formulating the synchronization problem within the standard plant framework of modern control theory, it is shown how the original message can be recovered for a class of binary-valued continuous time reference inputs or message signals. Input/output properties of the synchronization scheme are analyzed using the system-theoretical approach of dissipativity with a quadratic storage function and a supply rate of finite L2-gain. The feedback matrix is designed such that the influence of the exogenous input on a tracking error is minimized, corresponding to a so-called nonlinear H∞ synchronization scheme. Channel noise is taken into account in the design. The method is illustrated on Chua's circuit and an n-double scroll circuit.
Transient Chaos-Hyperchaos Intermittency in Coupled Nonautonomous Oscillating Circuits
Authors: Zhu Zhiwen , Liu Zhong
Published: September 2003

In this paper, we show by computer simulations that a novel transient process exhibits a characteristic chaos-hyperchaos intermittency in a system of two coupled chaotic circuits. The transient Lyapunov exponents are used to illustrate the transient process. The exponents of power laws of average transient lifetime of chaos-hyperchaos intermittency and average length of time intervals during which the system is in chaotic state are found to be approximately -3.0 and -1.66, respectively.
Experimental Demonstration of Chaotic Synchronization in the Modified Chua's Oscillators
Authors: Yin Yuan-Zhao
Published: September 2003

In this paper we demonstrate experimentally the properties of bifurcation, chaos, synchronization and the secure communication idea by making use of the modified Chua's oscillators.
Nonlinear Identification and Cluster Analysis of Chaotic Attractors from a Real Implementation of Chua's Circuit
Authors: Aguirre Luis A. , Rodrigues Giovani G. , Mendes Eduardo M. A. M.
Published: September 2003

This paper discusses the identification of nonlinear polynomial models from real data sets measured from an implementation of Chua's circuit. Data filtering issues are also discussed. The identified models are analyzed and validated and suggest that the major characteristics of the real system have been incorporated in the models. Lyapunov exponents, correlation and Lyapunov dimensions are estimated for the identified models and some practical difficulties concerning the estimation of such invariants are discussed. Cluster analysis of the identified models has been performed and it is in accord with a number of results reported in the literature. Five of the identified models are presented in the paper and because of their simple structure, it is believed that such models will be useful in the study of nonlinear dynamics of Chua's circuit.
Modeling Qualitative Changes in Bimanual Movements
Authors: Scheffczyk C. , Zaikin A. , Rosenblum M. , Engbert R. , Krampe R. , Kurths J.
Published: September 2003

In the cognitive sciences the study of complex rhythmic movements is a challenging problem which is a subject of extensive research. Experiments on bimanual movements are paradigms for studying the ability of humans on timing and coordination. Such experiments give insights into the control mechanisms of the central nervous system and also reflect the functional state and level of training of the person tested. In a recent study on bimanual polyrhythm production the existence of phase transitions in dependence on the speed of performance has been shown. In this paper we present an iterated map model to explain main findings of these experiments. The model consists of two iterated maps describing the dynamics of the finger movements. The essential properties of the model are a nonlinear correction function and a coupling mechanism between the two maps. Numerical simulations show that the model is in qualitative agreement with the experimentally observed phenomena.
Complex Order from Disorder and from Simple Order in Coarse-Graining Invariant Orbits of Certain Two-Dimensional Linear Cellular Automata
Authors: Barbé André
Published: September 2003

This paper considers three-dimensional coarse-graining invariant orbits for two-dimensional linear cellular automata over a finite field, as a nontrivial extension of the two-dimensional coarse-graining invariant orbits for one-dimensional CA that were studied in an earlier paper. These orbits can be found by solving a particular kind of recursive equations (renormalizing equations with rescaling term). The solution starts from some seed that has to be determined first. In contrast with the one-dimensional case, the seed has infinite support in most cases. The way for solving these equations is discussed by means of some examples. Three categories of problems (and solutions) can be distinguished (as opposed to only one in the one-dimensional case). Finally, the morphology of a few coarse-graining invariant orbits is discussed: Complex order (of quasiperiodic type) seems to emerge from random seeds as well as from seeds of simple order (for example, constant or periodic seeds).
Controlling Chaos Using Input–Output Linearization Approach
Authors: Yu Xinghuo
Published: September 2003

In this article the input–output linearization approach is used for controlling chaos. It is shown that by using only partial states, the entire chaotic system is stabilizable, provided the zero dynamics is stable. Generally speaking, trajectories of chaotic systems do not grow exponentially and are usually bounded. In particular, for dissipative chaotic systems the stable zero dynamics can always be found. Hence the stabilization as well as tracking periodic signals are possible. The Lorenz system is used to inform the discussion. Simulation results are presented to show the effectiveness of the approach.
Fractal-Like Basin Boundaries and Indeterminate Transition in Nonlinear Resonance of a Toda Oscillator
Authors: Iobe Atsushi , Abe Yutaka
Published: September 2003

We have numerically investigated the unpredictable jump in a nonlinear damped Toda oscillator. It was shown that three stable steady-states exist in a certain frequency region of the external force in this system, and these multiple steady-states are realized due to the overlap of the primary resonance branch with the secondary resonance branch. Even though no homoclinic tangles exist in the present system, we found that the unpredictable jump can occur via a saddle-node (SN) bifurcation, which occurs in the course of heteroclinic events. It is pointed out that the origin of the unpredictable jump (or unpredictable transition) is due to the complex basin structure which remains after the SN bifurcation.
Examining the Robustness of Coupled Map Lattices against Discretization
Authors: Goto Kentaro , Yamaguchi Akihiro , Tsuda Ichiro
Published: September 2003

Cellular automata deduced by discretization of coupled map lattices are investigated. The following results were obtained by the numerical experiments: 29 states on each cell is sufficient for equivalent dynamics, and at least 26 states are necessary for qualitative dynamics. We also develop a theory to examine the conditions for the discretization.
Interactive Computation, Parameter Continuation, and Visualization
Authors: Maddocks John H. , Manning Robert S. , Paffenroth Randy C. , Rogers Kathleen A. , Warner Jeremy A.
Published: September 2003

Nonlinear problems arising in modeling applications are frequently parameter dependent, so that families of solutions are of interest. Such problems naturally lend themselves to interactive computation that exploits parameter continuation methods combined with visualization techniques. Visualization provides both understanding of the solution set and feedback for computational steering. We describe various issues that have arisen in our investigations of problems of this general type.
New Nonlinear Adaptive Filters with Applications to Chaos
Authors: Rauf Fawad , Ahmed Hassan M.
Published: September 2003

We present a new approach to nonlinear adaptive filtering based on Successive Linearization. Our approach provides a simple, modular and unified implementation for a broad class of polynomial filters. We refer to this implementation as the layered structure and note that it offers substantial computational efficiency over previous methods.

A new class of Polynomial Autoregressive filters is introduced which can model limit cycle and chaotic dynamics. Existing geometric methods for modeling and characterizing chaotic processes suffer from several drawbacks. They require a huge number of data points to reconstruct the attractor geometry and performance is severely limited by noisy experimental measurements.

We present a new method for processing chaotic signals using nonlinear adaptive filters. We demonstrate the modeling, prediction and filtering of these signals. We also show how the prediction error growth rate can be used to estimate the effective Lyapunov exponent of the chaotic map. Our approach requires orders of magnitude fewer data points and is robust to noise in the experimental data. Although reconstruction of the attractor geometry is unnecessary, the adaptive filter contains most of the geometric information.
Detection of Teleseismic Events in Seismic Sensor Data Using Nonlinear Dynamic Forecasting
Authors: Short Kevin M.
Published: September 2003

In this paper we consider the use of nonlinear dynamic (NLD) forecasting as a signal processing tool for seismic applications. The specific problem considered here arises in monitoring nuclear tests and nuclear treaty compliance, where the presence of ubiquitous background noise obscures the seismic signals associated with the tests. The problem is that the signal from a distant teleseismic event can be attenuated so that it is lost in the background noise, and since the noise overlaps the frequency band occupied by the teleseisms, frequency-based techniques provide only marginal improvements in detection capabilities. For the work in this paper, we studied a test set of actual seismic sensor data prepared by the Air Force Technical Applications Center (AFTAC). The data set was composed of background seismic noise which contained or had added to it a number of hidden teleseismic signals. This data was analyzed to determine if techniques of NLD forecasting could be used to detect the hidden signals. For this test case, it was possible to predict the behavior of the seismic background sufficiently well that when the predicted background behavior was removed, the hidden signals became evident. However, some of the weaker signals were very close to the residual noise level, so the ability to detect these events is compromised.
Chaos and Topological Evolution in Cosmology
Authors: Tomaschitz Roman
Published: September 2003

An elementary account on the chaoticity of galactic world-lines in an open universe is given. A new type of cosmic evolution by global metrical deformations, unpredicted by Einstein's equations, is pointed out. Physical effects of this evolution are backscattering of electromagnetic fields and particle creation in quantum fields. We review in an untechnical way how global metrical deformations of the open and multiply connected spacelike slices induce angular fluctuations in the temperature of the cosmic microwave background radiation.
Nonlinear Science and the Laws of Nature
Authors: Prigogine Ilya
Published: September 2003

For well defined classes of dynamical systems (chaos, nonintegrable systems in the thermodynamic limit) it is now possible to formulate the basic laws of the probabilistic level. This leads to an extension of classical and quantum theory to include time symmetry breaking. This leads to a unification of the time reversible description of dynamics and the evolutionary view of thermodynamics.
Nonlinear Science — The Impact of Biology
Authors: Holden A. V.
Published: September 2003

Nonlinear science has primarily developed from applications of mathematics to physics. The biological sciences are emerging as the dominant growth points of science and technology, and biological systems are characterized by being information dense, spatially extended, organized in interacting hierarchies, and rich in diversity. These characteristics, linked with an increase in available computing power and accessible memory, may lead to a nonlinear science of complicated interacting systems that will link different types of mathematical objects within a framework of algebraic models of computing systems. Examples, drawn from current work on intracellular, cellular, tissue, organ, and integrative physiology of an individual, are outlined within the theory of synchronous concurrent algorithms. Possible directions in population dynamics and applications to ecosystem management are outlined.
Quantum-Dot Devices and Quantum-Dot Cellular Automata
Authors: Porod Wolfgang
Published: September 2003

We discuss novel nanoelectronic architecture paradigms based on cells composed of coupled quantum-dots. Boolean logic functions may be implemented in specific arrays of cells representing binary information, the so-called Quantum-Dot Cellular Automata (QCA). Cells may also be viewed as carrying analog information and we outline a network-theoretic description of such Quantum-Dot Nonlinear Networks (Q-CNN). In addition, we discuss possible realizations of these structures in a variety of semiconductor systems (including GaAs/AlGaAs, Si/SiGe, and Si/SiO 2), rings of metallic tunnel junctions, and candidates for molecular implementations.
Computation, Continuation and Bifurcation Analysis of Periodic Solutions of Delay Differential Equations
Authors: Luzyanina Tatyana , Engelborghs Koen , Lust Kurt , Roose Dirk
Published: September 2003

We present a new numerical method for the efficient computation of periodic solutions of nonlinear systems of Delay Differential Equations (DDEs) with several discrete delays. This method exploits the typical spectral properties of the monodromy matrix of a DDE and allows effective computation of the dominant Floquet multipliers to determine the stability of a periodic solution. We show that the method is particularly suited to trace a branch of periodic solutions using continuation and can be used to locate bifurcation points with good accuracy.
Checkerboard Spiral Waves in a 2D Coupled Map Lattice
Authors: Sbitnev Valery I.
Published: September 2003

An intermittency of spatiotemporal chaos and checkerboard spiral waves is observed in a 2D coupled sigmoid map lattice. A rotating arm of such a spiral consists of sites oscillating in opposite phases with respect to nearest neighbors. A region of the spiral waves exhibited in the control parameter space is presented. The spatiotemporal intermittency stems from a crisis-induced intermittency of chaotic wanderings and zigzag burstings that is shown in this work.
Lossless Digital Filter Overflow Oscillations; Approximation of Invariant Fractals
Authors: Ashwin Peter , Chambers W. , Petkov G.
Published: September 2003

We investigate second order lossless digital filters with two's complement overflow. We numerically approximate the fractal set D of points that iterate arbitrarily close to the discontinuity. For the case of eigenvalues of the associated linear map of the form eiθ with θ/π ∉ Q we present evidence that D has positive two dimensional Lebesgue measure. For θ/π ∈ Q we confirm that D has Lebesgue measure zero. As a by-product we get estimates of the exterior dimension of D. These results imply that if such filters are realized using finite-precision arithmetic then they will have a sizeable fraction of orbits that are periodic with high period overflows.
Cyclic Dynamics in Romantic Relationships
Authors: Gragnani Alessandra , Rinaldi Sergio , Feichtinger Gustav
Published: September 2003

Minimal models composed of two ordinary differential equations are considered in this paper to mimic the dynamics of the feelings between two persons. In accordance with attachment theory, individuals are divided into secure and non-secure individuals, and synergic and non-synergic individuals, for a total of four different classes. Then, it is shown that couples composed of secure individuals, as well as those composed of non-synergic individuals can only have stationary modes of behavior. By contrast, couples composed of a secure and synergic individual and a non-secure and non-synergic individual can experience cyclic dynamics. In other words, the coexistence of insecurity and synergism in the couple is the minimum ingredient for cyclic love dynamics. The result is obtained through a detailed local and global bifurcation analysis of the model. Supercritical Hopf, fold and homoclinic bifurcation curves are numerically detected around a Bogdanov–Takens codimension-2 bifurcation point. The existence of a codimension-2 homoclinic bifurcation is also ascertained. The bifurcation structure allows one to identify the role played by individual synergism and reactiveness to partners love and appeal. It also explains why ageing has a stabilizing effect on the dynamics of the feelings. All results are in agreement with common wisdom on the argument. Possible extensions are briefly discussed at the end of the paper.
A Model of the Dynamics of The Newton–Leipnik Attractor
Authors: Lofaro Thomas
Published: September 2003

The dynamics and bifurcations of the Newton–Leipnik equations are presented. Numerical computations and local stability calculations suggest that the dynamics of the Newton–Leipnik equations are related to the dynamics and bifurcations of a family of odd, symmetric, bimodal maps. The numerically computed dynamics and bifurcations of the Newton–Leipnik equations are compared with the dynamics and bifurcations of a family of odd, symmetric, bimodal maps to motivate the connection between the two systems.
Low Dimensional Traveling Interfaces in Coupled Map Lattices
Authors: Carretero-González R.
Published: September 2003

We study the dynamics of the traveling interface arising from a bistable piecewise linear one-way coupled map lattice. We show how the dynamics of the interfacial sites, separating the two superstable phases of the local map, is finite dimensional and equivalent to a toral map. The velocity of the traveling interface corresponds to the rotation vector of the toral map. As a consequence, a rational velocity of the traveling interface is subject to mode-locking with respect to the system parameters. We analytically compute the Arnold's tongues where particular spatiotemporal periodic orbits exist. The boundaries of the mode-locked regions correspond to border-collision bifurcations of the toral map. By varying the system parameters it is possible to increase the number of interfacial sites corresponding to a border-collision bifurcation of the interfacial attracting cycle. We finally give some generalizations towards smooth coupled map lattices whose interface dynamics is typically infinite dimensional.
Secondary Bifurcations and High Periodic Orbits in Voltage Controlled Buck Converter
Authors: Di Bernardo M. , Fossas E. , Olivar G. , Vasca F.
Published: September 2003

Period doubling route to chaos has been shown to occur in the voltage controlled DC/DC buck converter, both experimentally and numerically. A chaotic attractor was found at the end of the sequence, suddenly followed by an increase of its size. In this paper new secondary bifurcations and high periodic phenomena, coexisting with the main sequence are detected and analyzed over the same range of parameters. A(synchronous)-switching and stroboscopic maps, unstable orbits, bifurcation diagrams, invariant manifolds and basins of attraction are outlined. These tools are put together to reveal the dynamical richness of this nonsmooth system.
Chaos and Predictability of Climate
Authors: Hameed Sultan , Zhang Minghua
Published: September 2003

While day-to-day atmospheric fluctuations are not predictable after two weeks or so because of the chaotic nature of the atmospheric dynamics, there is evidence that certain aspects of the atmospheric circulation are predictable on time scales much longer than two weeks in the presence of a time dependent climate forcing. This paper illustrates distinctions between weather predictability and climate predictability by using a truncated atmospheric model in an explicitly chaotic parameter regime. An external forcing with time scale longer than the weather tends to modulate the statistical characteristics of the chaotic atmospheric variations to enhance the predictability of time-averaged climate states. Moreover, there exist climate variables which are more predictable than others.
Spiral Waves in Spatially Discrete λ-ω Systems
Authors: Paullet Joseph E. , Ermentrout G. Bard
Published: September 2003

The existence of stable spiral wave solutions in a spatially discrete λ-ω system is proven. The waves are shown to exist only if the coupling parameter d is sufficiently small. There is a critical value dc such that if d>dc, the spiral wave solution ceases to exists. This is in agreement with the behavior of such waves in spatially continuous λ-ω systems on finite domains.
Edge of Chaos and Local Activity Domain of FitzHugh-Nagumo Equation
Authors: Dogaru Radu , Chua Leon O.
Published: September 2003

The local activity theory [Chua, 97] offers a constructive analytical tool for predicting whether a nonlinear system composed of coupled cells, such as reaction-diffusion and lattice dynamical systems, can exhibit complexity. The fundamental result of the local activity theory asserts that a system cannot exhibit emergence and complexity unless its cells are locally active. This paper gives the first in-depth application of this new theory to a specific Cellular Nonlinear Network (CNN) with cells described by the FitzHugh–Nagumo Equation. Explicit inequalities which define uniquely the local activity parameter domain for the FitzHugh–Nagumo Equation are presented. It is shown that when the cell parameters are chosen within a subset of the local activity parameter domain, where at least one of the equilibrium state of the decoupled cells is stable, the probability of the emergence of complex nonhomogenous static as well as dynamic patterns is greatly enhanced regardless of the coupling parameters. This precisely-defined parameter domain is called the "edge of chaos", a terminology previously used loosely in the literature to define a related but much more ambiguous concept. Numerical simulations of the CNN dynamics corresponding to a large variety of cell parameters chosen on, or nearby, the "edge of chaos" confirmed the existence of a wide spectrum of complex behaviors, many of them with computational potentials in image processing and other applications. Several examples are presented to demonstrate the potential of the local activity theory as a novel tool in nonlinear dynamics not only from the perspective of understanding the genesis and emergence of complexity, but also as an efficient tool for choosing cell parameters in such a way that the resulting CNN is endowed with a brain-like information processing capability.
Multifractals Defined by Nonlinear Cellular Automata
Authors: Voorhees Burton
Published: September 2003

It is known that the space-time output patterns of additive cellular automata may be scaled to yield fractals. In this paper, nonadditive cellular automata rules are grouped into equivalence classes, each class defined in terms of a characteristic nonadditive rule. This rule defines a multifractal uniquely associated to each class. It is shown that these multifractals can be generated by a form of matrix substitution system, called concatenation substitution. This allows easy computation of fractal dimensions. It is likely that the multifractal spectra for each class does not possess the usual inverted U shape. A conserved quantity is found to be associated with the concatenation substitution system, and is shown to play a role similar to Langton's λ-parameter.
Biologically Modeled Noise Stabilizing Neurodynamics for Pattern Recognition
Authors: Chang Hung-Jen , Freeman Walter J. , Burke Brian C.
Published: September 2003

We present a distributed KIII model for the olfactory neural system. Low-level Gaussian noise is introduced to the receptors and anterior olfactory nucleus, which biologically models the peripheral and central sources of noise. The additive noise numerically makes the model stable and robust in respect to repeated input-induced state transitions, while improving the simulations of EEG potentials and multiunit activity from the olfactory system. This hybrid dynamics generates a 1/f aperiodic state, which provides an unpatterned basal state for every module to stay in while there is no significant stimulus. Any external input may guide the system to a certain patterned state. The mechanism is fast, fully parallel, under modulatory control, and flexible in absorbing new patterns from unpredictable environments.
On the Structure of the Bursting Region for a Neuronal Model
Authors: Dick O. E. , Akoev G. N. , Braun C. , Kniffki K.-D. , Flockerzi D. , Bedrov Y. A.
Published: September 2003

We examine the question of whether the bursting region of a neuronal model is simply or multiply connected in the parameter space. Taking into account that the presence of slow oscillations of intracellular calcium concentration is a necessary condition for a bursting mode, we analyze the subysystem responsible for the slow component of bursting. Explicit parametric form of the Hopf bifurcation is given. It enables us to reduce the problem to determining the number of the bifurcation curves by finding the number of discontinuities of the expressions giving the bifurcation parameter values. The results obtained suggest that the bursting region is simply connected over the physiologically possible parameter range.
Minimum Requirements for Stochastic Resonance in Threshold Systems
Authors: Hess S. M. , Albano A. M.
Published: September 2003

We present results of numerical simulations which demonstrate that sufficient conditions for a system to exhibit stochastic resonance are the presence of input noise and a system threshold in a system subject to a subthreshold periodic signal.
Chaotic Vibrations of the One-Dimensional Wave Equation Due to a Self-Excitation Boundary Condition.
Authors: Chen Goong , Hsu Sze-Bi , Zhou Jianxin
Published: September 2003

Consider the initial-boundary value problem of the linear wave equation wtt-wxx=0 on an interval. The boundary condition at the left endpoint is linear homogeneous, injecting energy into the system, while the boundary condition at the right endpoint has cubic nonlinearity of a van der Pol type. We show that the interactions of these linear and nonlinear boundary conditions can cause chaos to the Riemann invariants (u,v) of the wave equation when the parameters enter a certain regime. Period-doubling routes to chaos and homoclinic orbits are established. We further show that when the initial data are smooth satisfying certain compatibility conditions at the boundary points, the space-time trajectory or the state of the wave equation, which satisfies another type of the van der Pol boundary condition, can be chaotic. Numerical simulations are also illustrated.
Geometry of Mixed-Mode Oscillations in the 3-D Autocatalator
Authors: Milik Alexandra , Szmolyan Peter , Löffelmann Helwig , Gröller Eduard
Published: September 2003

We present a geometric explanation of a basic mechanism generating mixed-mode oscillations in a prototypical simple model of a chemical oscillator. Our approach is based on geometric singular perturbation theory and canard solutions. We explain how the small oscillations are generated near a special point, which is classified as a folded saddle-node for the reduced problem. The canard solution passing through this point separates small oscillations from large relaxation type oscillations. This allows to define a one-dimensional return map in a natural way. This bimodal map is capable of explaining the observed bifurcation sequence convincingly.
Basins of Attraction in Experimental Nonlinear Oscillators
Authors: Virgin L. N. , Todd M. D. , Begley C. J. , Trickey S. T. , Dowell E. H.
Published: September 2003

This paper addresses two relatively simple but fundamental questions in nonlinear oscillations: Given an arbitrary initial condition where will the trajectory go, and how long will it take to get there? These related questions are addressed from an experimental perspective where generating global transient behavior has received relatively little attention, despite the fact that a global view of transient behavior provides a much more complete description of the dynamics of a system than a traditional concentration on steady-state behavior. Three different physical systems are studied, each of which exhibits a specific behavior heavily influenced by transient global effects.
Estimation of the Dominant Singular Values for SVD Based Noise Reduction Methods
Authors: Pilgram B. , Schappacher W.
Published: September 2003

In this paper we present an algorithm for the estimation of the dominant singular values or respectively the noise floor for Singular Value Decomposition (SVD) based noise reduction methods. The algorithm presented here uses the SVD of the trajectory matrix, is based on matrix perturbation analysis, and is applied for a noise filtering method based on time-varying state-space modeling. The efficiency of the algorithm is then studied by the estimation of the correlation integral and, respectively, the correlation dimension of noisy and filtered time series. For testing of the filtering method with the noise floor estimation algorithm integrated, the Lorenz equations, which have a chaotic attractor, were used. To determine the effects of the noise and the noise reduction method, Gaussian random noise was added to the simulated system as a measurement error. The proposed noise floor estimation algorithm turns out to be well suited for SVD based noise filtering methods where the number of dominant singular values is not known a priori.
Spin Rotor Stabilization of a Dual-Spin Spacecraft with Time Dependent Moments of Inertia
Authors: Lanchares V. , Iñarrea M. , Salas J. P.
Published: September 2003

We consider a dual-spin deformable spacecraft, in the sense that one of the moments of inertia is a periodic function of time such that the center of mass is not altered. In the absence of external torques and spin rotors, by means of the Melnikov's method we prove that the body motion is chaotic. Stabilization is obtained by means of a spinning rotor about one of the principal axes of inertia.
Coexistence of Periodic Roll Motion and Chaotic One in a Forced Flooded Ship
Authors: Murashige Sunao , Aihara Kazuyuki
Published: September 2003

This letter describes the coexistence of periodic and chaotic roll motion of a flooded ship in waves. We found experimentally, both with a flooded ferry model and with a simplified box-shaped model, that the two types of roll motion can coexist under the same wave condition. A trajectory reconstructed in a delay-coordinate state space from the time series data of the measured roll angle looks like a low-dimensional strange attractor. Moreover, a mathematical model for the simplified box-shaped ship shows the coexistence of a periodic solution and a chaotic one with a positive maximum Liapunov exponent.
A Three-Dimensional Autowave Turbulence
Authors: Biktashev V. N.
Published: September 2003

Autowave vortices are topological defects in autowave fields in nonlinear active media of various natures and serve as centers of self-organization in the medium. In three-dimensional media, the topological defects are lines, called vortex filaments. Evolution of three-dimensional vortices, in certain conditions, can be described in terms of evolution of their filaments, analogously to that of hydrodynamical vortices in LIA approximation. In the motion equation for the filament, a coefficient called filament tension, plays a principal role, and determines qualitative long-time behavior. While vortices with positive tension tend to shrink and so either collapse or stabilize to a straight shape, depending on boundary conditions, vortices with negative tension show internal instability of shape. This is an essentially three-dimensional effect, as two-dimensional media with the same parameters do not possess any peculiar properties. In large volumes, the instability of filaments can lead to propagating, nondecremental activity composed of curved vortex filaments that multiply and annihilate in an apparently chaotic manner. This may be related to a mechanism of cardiac fibrillation.
On Bifurcations Leading to Chaos in Chua's Circuit
Authors: Bykov V. V.
Published: September 2003

Bifurcations and the structure of limit sets are studied for a three-dimensional Chua's circuit system with a cubic nonlinearity. On the base of both computer simulations and theoretical results a model map is proposed which allows one to follow the evolution in the phase space from a simple (Morse-Smale) structure to chaos. It is established that the appearance of a complex, multistructural set of double-scroll type is stimulated by the presence of a heteroclinic orbit of intersection of the unstable manifold of a saddle periodic orbit and unstable manifold of an equilibrium state of saddle-focus type.
Synchronization and Clustering in a Network of Three Globally Coupled Neural Oscillators
Authors: Kim Seunghwan , Kook Hyungtae , Lee Sang Gui , Park Myung-Han
Published: September 2003

Collective dynamics of three globally coupled Hodgkin–Huxley neurons with a symmetric synaptic coupling has been studied as a paradigm of one of the simplest but nontrivial example of physiology-based coupled oscillator networks capable of displaying synchrony and clustering. Rich phase dynamics have been observed and the phase diagram for various phase states, in particular the synchronized state and clustered states, have been constructed by direct numerical simulations of the full system. We find that the computed phase diagram in the synaptic parameter space of the reversal potential and the signal propagation time delay reveals a rich bifurcation structure and agrees with one from bifurcation analysis of the reduced phase model. Implication of our findings in connection with two-coupled neurons and larger neural networks is discussed.
Can Neurons Distinguish Chaos from Noise?
Authors: Lindner John F. , Meadows Brian K. , Marsh Tracey L. , Ditto William L. , Bulsara Adi R.
Published: September 2003

Recent studies suggesting evidence for determinism in the stochastic activity of the heart and brain have sparked an important scientific debate: Do biological systems exploit chaos or are they merely noisy? Here, we analyze the spike interval statistics of a simple integrate-and-fire model neuron to investigate how a real neuron might process noise and chaos, and possibly differentiate between the two. In some cases, our model neuron readily distinguishes noise from chaos, even discriminating among chaos characterized by different Lyapunov exponents. However, in other cases, the model neuron does not decisively differentiate noise from chaos. In these cases, the spectral content of the input signal may be more significant than its phase space structure, and higher-order spectral characterizations may be necessary to distinguish its response to chaotic or noisy inputs.
Probability Distribution of the Residence Times in Periodically Fluctuating Metastable Systems
Authors: Mantegna Rosario N. , Spagnolo Bernardo
Published: September 2003

We investigate experimentally and numerically the probability distribution of the residence times in periodically fluctuating metastable systems. The experiments are performed in a physical metastable system which is the series of a biasing resistor with a tunnel diode in parallel to a capacitor. The numerical simulations are performed in an overdamped model system with a time-dependent potential. We investigate both the cases where the system is deterministically overall-stable and overall-unstable. In the overall-unstable regime, the experimental and the numerically investigated systems show noise enhanced stability in the presence of a finite amount of noise. The determined P(T) is multi-peaked with an exponentially decaying envelop. We note that the shape of the nth peak in the P(T) is roughly fitted by a Gaussian function with standard deviation independent of n.