bluuebunny/crossref_metadata_2025 huggingface dataset:
COMPOUND CHAOS
Authors: SINGH R. , MOHARIR P. S. , MARU V. M.
Published: October 2004

Compounding is a statistical notion. Essentially, it comprises of regarding the parameters in a particular statistical distribution as random variables with a prescribed distribution. The compound distribution then acquires the parameters of the compounding distribution as its own. As deterministic chaos, in spite of being deterministic, appears like a statistical phenomenon, the notion of compounding can be extended to chaotic systems. It is shown with illustrations that a chaotic system can be compounded by another chaotic system, giving rise to compound chaos which is, in general, “chaoticer”. The concept can also be used to make a periodic system chaotic, thus opening possibilities of “chaoticization”. Examples of compound chaos and chaoticization are given using Lorenz and Rössler systems, including their attractors and limit cycles as “compoundee” and/or “compounder” systems. The conclusions are based on quantitative studies of Lyapunov exponents and correlation dimensions.
ON ADAPTIVE SYNCHRONIZATION AND CONTROL OF NONLINEAR DYNAMICAL SYSTEMS
Authors: WU CHAI WAH , YANG TAO , CHUA LEON O.
Published: October 2004

In this paper, we study the synchronization of two coupled nonlinear, in particular chaotic, systems which are not identical. We show how adaptive controllers can be used to adjust the parameters of the systems such that the two systems will synchronize. We use a Lyapunov function approach to prove a global result which shows that our choice of controllers will synchronize the two systems. We show how it is related to Huberman-Lumer adaptive control and the LMS adaptive algorithm.

We illustrate the applicability of this method using Chua's oscillators as the chaotic systems. We choose parameters for the two systems which are orders of magnitude apart to illustrate the effectiveness of the adaptive controllers. Finally, we discuss the role of adaptive synchronization in the context of secure and spread spectrum communication systems. In particular, we show how several signals can be encoded onto a single scalar chaotic carrier signal.
LOOKING FOR CHAOS.
Authors: DANKOWICZ HARRY
Published: October 2004

This paper derives an alternative approach to the Melnikov method, which greatly reduces the amount of algebra involved in higher-order calculations. To illustrate this, a particular system is studied for which such a higher-order analysis is necessary, due to an identically vanishing first-order Melnikov function. The results of a second-order calculation imply the existence of transverse homoclinic orbits and, consequently, the existence of a horseshoe.
SPIRAL-PLUS-SADDLE ATTRACTORS AND ELEMENTARY MECHANISMS FOR CHAOS GENERATION
Authors: DENG BO
Published: October 2004

Rössler's spiral-plus-saddle (SPS) mechanism for chaotic attractors is systematically implemented. The method is based on the works of Rössler [1976, 1979] and Deng [1994]. Some more complex chaotic structures are also investigated experimentally when the SPS, branching, and relaxation-folding mechanisms for chaos generation are combined.
ON CHAOS DETECTION METHODS
Authors: VIBE KARIN , VESIN JEAN-MARC
Published: October 2004

Reliable chaos detection in real-world time series is attracting increasing attention in the scientific community. This work shows that it is possible to use chaos analysis methods such as attractor dimension estimation, Lyapunov exponents estimation and nonlinear prediction, under the condition that the limitations and drawbacks of the algorithms used are kept in mind. Three existing algorithms for chaos characterization are analyzed in terms of classification performances and robustness with respect to noise and data length. It is shown that all three help detect chaos and even classify different types of signals, but that their results are not devoid of ambiguity. An illustrative example is given, in which the algorithms presented are applied to heart rate variability signals, and directions of research are proposed for the design of a straightforward and simple chaos detection methodology.
AN ADAPTIVE APPROACH TO THE CONTROL AND SYNCHRONIZATION OF CONTINUOUS-TIME CHAOTIC SYSTEMS
Authors: DI BERNARDO MARIO
Published: October 2004

This paper is concerned with synchronization and control of chaotic nonlinear dynamical systems. First, a unified frame for both the synchronization and the control problem is described. Then by modifying a feedback plus feedforward controller, a discontinuous strategy is synthesized which exploits the boundedness of chaotic attractors and limit cycles. Finally, an adaptive approach is investigated and an adaptive estimation law is implemented. The result, which is both global and not reliant on complete knowledge of the systems involved, is rigorously proved by means of Lyapunov Theory. An application to the synchronization of two chaotic systems is presented.
EXACT SYNCHRONIZATION OF MISMATCHED CHAOTIC SYSTEMS
Authors: KOZLOV ALEXANDER K. , SHALFEEV VLADIMIR D. , CHUA LEON O.
Published: October 2004

In this letter we use adaptive parameter and state feedback control to synchronize two or more slightly mismatched chaotic systems. Chua's circuit with a smooth nonlinearity is used throughout to illustrate our approach. We specify the conditions under which the parameter of a slave system will automatically converge to the parameter of the master system. We also consider potential applications of the control system to problems of secure communications and synchronization of chaos in a chain of slightly different Chua's circuits.
BIFURCATIONS AND HOPF DEGENERACIES IN NONLINEAR FEEDBACK SYSTEMS WITH TIME DELAY
Authors: MOIOLA JORGE L. , CHIACCHIARINI HÉCTOR G. , DESAGES ALFREDO C.
Published: October 2004

An application of the well-developed frequency-domain approach to detect oscillations in nonlinear feedback systems with time delay is presented. The method depends on an early proof of the Hopf bifurcation theorem known as the Graphical Hopf Theorem (GHT). Several nondegeneracy conditions are included to apply the GHT in nonlinear systems with time delay. The singular conditions corresponding to degeneracies, which include static and dynamic bifurcations, as well as some special cases of degenerate Hopf bifurcations and multiple crossings, are also discussed. Two Single-Input Single-Output (SISO) feedback systems with odd nonlinearities are presented as examples to show that the proposed technique and a standard simulation method have very good agreement in the results, yet the GHT is much simpler in calculation. The first one shows an application of the GHT under classical Hopf conditions while the second emphasizes the presence of degenerate Hopf bifurcations and multiple crossings. For both examples, and others which have appeared recently in the literature, a considerable simplification of the formulas for recovering periodic solutions is also provided in this paper.
THE SLOW INVARIANT MANIFOLD OF A CONSERVATIVE PENDULUM-OSCILLATOR SYSTEM
Authors: GEORGIOU IOANNIS T. , SCHWARTZ IRA B.
Published: October 2004

We analyze the motions of a conservative pendulum-oscillator system in the context of invariant manifolds of motion. Using the singular perturbation methodology, we show that whenever the natural frequency of the oscillator is sufficiently larger than that of the pendulum, there exists a global invariant manifold passing through all static equilibrium states and tangent to the linear eigenspaces at these equilibrium states. The invariant manifold, called slow, carries a continuum of slow periodic motions, both oscillatory and rotational. Computations to various orders of approximation to the slow invariant manifold allow analysis of motions on the slow manifold, which are verified with numerical experiments. Motion on the slow invariant manifold is identified with a slow nonlinear normal mode.
EXPERIMENTAL TRACKING OF UNSTABLE ORBITS IN DRIVEN DIODE RESONATOR BY WEAK COUPLING
Authors: CHEOL SHIN JONG , KWUN SOOK-IL , KIM YOUNGTAE
Published: October 2004

The weak coupling method is demonstrated to stabilize and track the unstable orbits in the diode resonator system. Continuous tracking of the unstable orbits could be achieved over a wide range from the periodic down to the deep chaotic regime as the control parameter is varied continuously without changing the feedback control parameter. During the tracking of unstable orbits, hysteresis as well as switching of the attractors are observed and these are supposed to be the generic properties of continuous tracking.
ON THE GENERALITY OF THE UNFOLDED CHUA'S CIRCUIT
Authors: WU CHAI WAH , CHUA LEON O.
Published: October 2004

In this paper, we study the generality of Chua's oscillator by deriving a class of vector fields that Chua's oscillator is equivalent to. For the class of vector fields with a scalar nonlinearity, we prove that under certain conditions, two such vector fields are topologically conjugate if the Jacobian matrices at each point have the same eigenvalues and the equilibrium points are matched up. We show how these conditions are related to the complete state observability of a corresponding linear system. These results are used to show that the n-dimensional Chua's oscillator is topologically conjugate to almost every vector field in this class. We comment on the special case when the vector field is piecewise-linear and in particular when the vector field is 2-segment piecewise-linear. These results are illustrated by transforming several systems studied in the literature into equivalent Chua's oscillators.

We also extend some of these results to the case of several scalar nonlinearities. As a corollary we prove that almost all piecewise-linear vector fields with parallel boundary planes are topologically conjugate if the boundary planes and equilibrium points are the same and the eigenvalues in corresponding regions are the same. We also give a dual result of topological conjugacy.
MULTIPLE HOMOCLINIC BIFURCATIONS FROM ORBIT-FLIP I: SUCCESSIVE HOMOCLINIC DOUBLINGS
Authors: KOKUBU HIROSHI , KOMURO MOTOMASA , OKA HIROE
Published: October 2004

The purpose of this and forthcoming papers is to obtain a better understanding of complicated bifurcations for multiple homoclinic orbits. We shall take one particular type of codimension two homoclinic orbits called orbit-flip and study bifurcations to multiple homoclinic orbits appearing in a tubular neighborhood of the original orbit-flip. The main interest of the present paper lies in the occurrence of successive homoclinic doubling bifurcations under an appropriate condition, which is a part of the entire bifurcation for multiple homoclinic orbits. Since this is a totally global bifurcation, we need the aid of numerical experiments for which we must choose a concrete set of ordinary differential equations that exhibits the desired bifurcation. In this paper we employ a family of continuous piecewise-linear vector fields for such a model equation. In order to explain the cascade of homoclinic doubling bifurcations theoretically, we also derive a two-parameter family of unimodal maps as a singular limit of the Poincaré maps along homoclinic orbits. We locate bifurcation curves for this family of unimodal maps in the two-dimensional parameter space, which basically agree with those for the piecewise-linear vector fields. In particular, we show, using a standard technique from the theory of unimodal maps, that there exists an infinite sequence of doubling bifurcations which corresponds to the sequence of homoclinic doubling bifurcations for the piecewise-linear vector fields described above. Since our unimodal map has a singularity at a boundary point of its domain of definition, the doubling bifurcation is slightly different from that for standard quadratic unimodal maps, for instance the Feigenbaum constant associated with the accumulation of the doubling bifurcations is different from the standard value 4.6692.…
RING OSCILLATING SYSTEMS AND THEIR APPLICATION TO THE SYNTHESIS OF CHAOS GENERATORS
Authors: DMITRIEV A.S. , PANAS A.I. , STARKOV S.O.
Published: October 2004

We consider in this paper a problem of the design of deterministic chaos generators with given spectral properties. It is shown that implementation of such generators is possible in the class of ring self-oscillating systems. Examples of the synthesis of systems with different types of spectrum are exhibited. The question of practical implementation of such generators is discussed.
A NUMERICAL TOOLBOX FOR HOMOCLINIC BIFURCATION ANALYSIS
Authors: CHAMPNEYS A.R. , KUZNETSOV YU. A. , SANDSTEDE B.
Published: October 2004

This paper presents extensions and improvements of recently developed algorithms for the numerical analysis of orbits homoclinic to equilibria in ODEs and describes the implementation of these algorithms within the standard continuation package AUTO86. This leads to a kind of toolbox, called HOMCONT, for analysing homoclinic bifurcations either as an aid to producing new theoretical results, or to understand dynamics arising from applications. This toolbox allows the continuation of codimension-one homoclinic orbits to hyperbolic or non-hyperbolic equilibria as well as detection and continuation of higher-order homoclinic singularities in more parameters. All known codimension-two cases that involve a unique homoclinic orbit are supported. Two specific example systems from ecology and chemical kinetics are analysed in some detail, allowing the reader to understand how to use the the toolbox for themselves. In the process, new results are also derived on these two particular models.
COMPUTER-GENERATED PROOFS OF PHASE PORTRAITS FOR PLANAR SYSTEMS
Authors: GUCKENHEIMER JOHN , MALO SALVADOR
Published: October 2004

This paper introduces new algorithms for automatically constructing proofs for the correctness of phase portraits obtained from numerical integration of structurally stable planar polynomial vector fields. These algorithms are based upon the concept of rotated vector fields and the use of interval arithmetic to obtain rigorous bounds on the accuracy of floating point arithmetic calculations. Phase portrait features such as the existence or non-existence of limit cycles in particular regions are proved while avoiding the awkwardness associated with error estimates for the accuracy of approximate trajectories obtained from numerical integration. In some cases this method extends local analysis to obtain a complete, rigorous description of the flow structure of a system.
ABOUT TWO-DIMENSIONAL PIECEWISE CONTINUOUS NONINVERTIBLE MAPS
Authors: CHRISTIAN MIRA
Published: October 2004

This paper is devoted to some features of a class of two-dimensional piecewise continuous noninvertible maps, with only one line of discontinuity. It describes properties of basin boundaries, chaotic areas, and their bifurcations. Such properties are not encountered with continuous noninvertible maps.
FAMILIES OF TRANSVERSE POINCARÉ HOMOCLINIC ORBITS IN 2N-DIMENSIONAL HAMILTONIAN SYSTEMS CLOSE TO THE SYSTEM WITH A LOOP TO A SADDLE-CENTER
Authors: KOLTSOVA O. YU. , LERMAN L. M.
Published: October 2004

We prove the theorem: if an n-degrees-of-freedom Hamiltonian system has an equilibrium of the saddle-center type (there is a pair of simple eigenvalues ±iω; the rest of the spectrum consists of eigenvalues with nonzero real parts) with a homoclinic orbit to it then this system, and all those close to it, have transversal Poincaré homoclinic orbits to Lyapunov periodic orbits if some genericity conditions are satisfied. These conditions are pointed out explicitly. Thus a new criterion of nonintegrability has been obtained.
CONVERGENCE OF INVARIANT MEASURES FOR DEGENERATING SEQUENCES OF HYPERBOLIC MAPPINGS WITH SINGULARITIES
Authors: SATAEV E. A.
Published: October 2004

This paper is devoted to presenting and giving a sketch of the proof of the theorem which states that, if the sequence of hyperbolic mappings with singularities converges to degenerating piecewise expanding mapping, then the corresponding sequence of measures of a Sinai-Bowen-Ruelle type converges to an absolutely continuous invariant measure.
AN EXPERIMENTAL STUDY OF A POPULATION OF RELAXATION OSCILLATORS WITH A PHASE-REPELLING MEAN-FIELD COUPLING
Authors: BRAILOVE ADAM A. , LINSAY PAUL S.
Published: October 2004

The dynamical behavior of a system of coupled relaxation oscillators is studied experimentally. A population of up to 15 coupled electronic op-amp relaxation oscillators is used as a prototype for the real collections of limit-cycle oscillators frequently found in many physical, biological, and technological systems. The oscillators interact via an all-to-all or mean-field coupling. The rarely studied case of antiferromagnetic interactions, in which oscillators tend to repel each other in phase, is considered. The behavior of the system is significantly different from the predictions of the limited theory that is currently available. The novel behavior observed includes the existence of numerous distinguishable phase-locked states and an exponential distribution of the duration of transients. The critical coupling strength necessary for the oscillator system to completely phase-lock is measured.

A simple geometrical model of the dynamics of the system during the transient is presented as a means of understanding the exponential distribution of transient lengths. In addition, a phase-response oscillator model is shown to exhibit similar transient behavior.
ON THE DESTRUCTION OF THREE-DIMENSIONAL TORI
Authors: FEUDEL U. , SAFONOVA M. A. , KURTHS J. , ANISHCHENKO V. S.
Published: October 2004

We have studied the direct transition from three-frequency oscillations on T3 to chaos. The onset of chaos is analyzed where the torus loses its smoothness in the neighbourhood of a resonance. We have proposed two methods to reveal the destruction of the torus T3. The first method is based on the iteration of parts of the attractor to demonstrate stretching and folding. The second one is related to the evolution of the critical surfaces of noninvertible maps, which are generalizations of the critical points in one-dimensional maps. With both techniques we can show that the torus is indeed broken at the onset of chaos. An important advantage of these methods is that they can be easily applied even in cases where the expansion is much smaller than the contraction. Note that the estimation of the box dimension of such attractors yields integer values, because the broken structure of the attractor is too tiny to be detected by estimating the capacity.
NOISE TOLERANCE OF THE AGOY CONTROL METHOD FOR STABILIZING CHAOTIC SYSTEMS
Authors: GLASS KATHRYN , ADACHI MASAHARU , MEES ALISTAIR
Published: October 2004

Presented by D. Auerbach, C. Grebogi, E. Ott, and J.A. Yorke in Auerbach et al. [1992], the AGOY control method allows us to stabilize periodic points of a chaotic map without knowledge of the underlying equations governing the system, and using only a one-dimensional time series. We are interested to discover the effect of observational noise on the success of the algorithm. In this paper, we will investigate the tolerance of this algorithm under observational noise and compare this tolerance to that of other stabilization algorithms.
SUPPRESSION OF CHAOS THROUGH CHANGES IN THE SYSTEM VARIABLES THROUGH POINCARÉ AND LORENZ RETURN MAPS
Authors: GUTIÉRREZ J.M. , IGLESIAS A. , GÜÉMEZ J. , MATÍAS M. A.
Published: October 2004

The present paper introduces a new chaos suppression method that is able to stabilize N- dimensional flows by perturbing the state variables in a discrete way when the flow crosses a suitably chosen Poincaré section. Two versions of the method are presented, depending on whether the perturbations are introduced on a proportional or an additive fashion. The perturbations are applied to the corresponding N−1-dimensional Poincaré map, or, in the case of strongly dissipative systems, to a suitable one-dimensional Lorenz map. The method is applied to two different three-variable flows: Rössler spiral chaos model and an isothermal three-variable autocatalator model introduced by Peng et al.
CHARACTERIZATION OF CHAOTIC ATTRACTORS IN EXTENDED SYSTEMS
Authors: CHEN CHIEN-CHONG , HWANG CHYI , WOLF EDUARDO E. , CHANG HSUEH-CHIA
Published: October 2004

It is shown that the traditional delayed embedding method, which is often applied to measurements at a single spatial point, could only account for partial chaotic dynamics of an extended system. To catch the whole dynamics of an extended system, we apply in this letter the Karhunen-Loeve (K-L) procedure to construct a series of tangent maps from spatially distributed measurements. The tangent maps obtained are then used for computing certain ergodic invariants. The procedure is illustrated with the experimental data obtained from a catalytic wafer, which is a spatially extended system exhibiting complex dynamics. The computed results show clearly that the K-L procedure is more suitable than the delayed embedding method for deciphering the full chaotic behavior of an extended system.
STRANGE NONCHAOTIC ATTRACTORS FROM QUASIPERIODICALLY FORCED UEDA’S CIRCUIT
Authors: LIU ZHONG , ZHU ZHIWEN
Published: October 2004

In this paper, we discuss the existence of strange nonchaotic attractors for the Ueda’s circuit with two-frequency quasiperiodic excitation. We have calculated the Lyapunov exponents, Poincaré map, winding number and frequency spectrum of characterizing the attractors. The results show that the Ueda’s circuit does indeed exhibit strange nonchaotic attractors. Because of the simplicity, it is expected that the circuit would be useful for the theoretical study and experimental observation of the strange nonchaotic attractors.
VISUALIZING RELATIVE EQUILIBRIA AND BIFURCATIONS BY PAINTING HAMILTONIANS ON PERSONAL COMPUTERS
Authors: PASCUA P. , RUBIO J.L. , VIARTOLA A. , FERRER S.
Published: October 2004

In this paper, we present a graphical tool running on a PC for studies of integrable systems of one degree of freedom with several parameters, in particular for perturbed Hamiltonian systems with symmetries whose orbit space is the two-dimensional sphere S2 after they are reduced by normalization. The pioneering work was performed by Coffey et al. [1990], who developed a graphical method on a Connection Machine in order to paint the Hamiltonian function ℋ* over phase space, instead of numerical integrations of the corresponding differential equations. We have pursued their idea and the procedure proposed here renders pictures at the rate of one picture per minute.

For research or teaching activities, the above mentioned method may be used in two directions: (a) in the stability analysis of equilibria and their bifurcations in parameter space; (b) for the specification of initial conditions and the corresponding qualitative long term global dynamics, in order to make proper choices for numerical studies of the full problem.

As examples of how to interact with the program, and of the different role played by physical and dynamical parameters, we consider the dynamics of the Laplace–Runge–Lenz vector in two perturbed Keplerian systems of recent interest: the generalized Van der Waals effect in the hydrogen atom, and Harrington’s model for triple stellar systems.
BIFURCATION STRUCTURE IN A TRANSMISSION SYSTEM MODELLED BY A TWO-DIMENSIONAL ENDOMORPHISM
Authors: GICQUEL NATHALIE
Published: October 2004

This paper concerns the bifurcation properties of a D.P.C.M. transmission system with an order-2 transversal predictor. These properties are related to the organization of some bifurcation curves in a parameter plane.
CRISIS-INDUCED INTERMITTENCY IN A SPATIALLY EXTENDED NONLINEAR OPTICAL SYSTEM
Authors: SAUER M. , KAISER F.
Published: October 2004

In this contribution we investigate the bifurcations of a spatially extended nonlinear optical system displaying two-band chaos. Low-dimensional systems with a two-band chaotic attractor undergo a crisis in which the two bands merge, thus representing an example of crisis-induced intermittency. We generalize the concept of crisis-induced intermittency to spatially extended systems and show that the bifurcations of the model system can be well understood within this framework. In particular, the new phenomenon of a spatially localized attractor crisis is discussed. This new type of crisis is due to the space dependence of the bifurcation parameter. It is further demonstrated by calculation of correlation functions that the crises generate spatiotemporal chaos. The results are discussed with relevance to experimental observation.
BIFURCATION STRUCTURE AT 1/3-SUBHARMONIC RESONANCE IN AN ASYMMETRIC NONLINEAR ELASTIC OSCILLATOR
Authors: REGA G. , SALVATORI A.
Published: October 2004

The attractor-basin bifurcation structure in an asymmetric nonlinear oscillator representative of the planar finite forced dynamics of elastic structural systems with initial curvature is studied at the 1/3-subharmonic resonance regime. Local and global analyses are made by means of different computational tools to obtain frequency-response curves of coexisting regular solutions, bifurcation diagrams ensuing from different sets of initial conditions, manifolds structure of direct and inverse saddles corresponding to unstable periodic solutions, basins of attraction at different values of the control parameter. Deep insight into the global dynamics of the system and its evolution is achieved through the analysis of synthetic attractor-basin-manifold phase portraits. The topological mechanisms which entail onset and disappearance of various attractors, and the main and secondary evolutions to chaos, are identified. Special attention is devoted to the analysis of sudden bifurcational events characterizing the system global dynamics, associated with the topological behavior of the invariant manifolds of several direct and inverse saddles. Features of basin metamorphosis, attractor-basin accessibility, and window occurrence are examined. The approach followed, consisting in combined bifurcation analysis of the attractor-basin structure and of the manifold structure, is thought to be useful for a variety of dynamical systems.
DIMENSIONAL ANALYSIS OF THE ELECTRICAL ACTIVITY IN FIBRILLATING ISOLATED HEARTS
Authors: CASALEGGIO A. , CORANA A. , RANJAN R. , THAKOR N.V.
Published: October 2004

The dynamical behavior of the electrophysiological response at the epicardium of isolated Langendorff-perfused rabbit heart during different induced arrhythmic conditions is studied with the use of the correlation dimension D2 and Lyapunov exponents. Different conditions are investigated: normal sinus rhythm (NSR), normal sinus rhythm after ischemia (NSRI), and ventricular fibrillation with and without perfusion (VFWP and VF, respectively). Both single lead and simultaneous recordings from four leads are analyzed. The correlation dimensions during VF and VFWP are significantly higher than during NSR or NSRI, although low-dimensional attractors could not be clearly observed in VF and VFWP.

Our conclusions are (a) that isolated hearts behave periodically during NSR, probably because the heart does not have to cope with time-varying demands either from the nervous system or from the metabolism; (b) during NSRI, spatially different regions of the heart show periodicities or chaotic behavior with low-dimensions, possibly due to electrophysiological inhomogeneity of the heart tissue; (c) during VF and VFWP, an intrinsic change in the dynamics seems to occur, and the heart behaves as a high-dimensional system, wherein VFWP signals show a greater spatial homogeneity than do VF signals.

We propose a possible interpretation of the results in terms of the synchronization of cells by a pacemaker and desynchronization as a result of malignant arrhythmias.
PROBABILISTIC CONTROL OF CHAOS: THE β-ADIC RENYI MAP UNDER CONTROL
Authors: ANTONIOU I. , BASIOS V. , BOSCO F.
Published: October 2004

We propose a new, probabilistic, approach for the control of chaotic systems. This approach is illustrated by a specific method, for the control of any periodic orbit of the simplest piecewise linear chaotic map, namely the β-adic Renyi map. As these chaotic maps are structurally stable, they cannot be controlled using conventional control methods without significant change of the dynamics. Our method consists in the probabilistic coupling of the original system with a controlling system. The chosen periodic orbit of the original system is a global attractor for the probability densities. The generalized spectral decomposition of the associated Frobenius–Perron operator provides a spectral condition of controllability for chaotic dynamical systems.
FEIGENBAUM’S UNIVERSALITY IN A LOW-DIMENSIONAL FLUID MODEL
Authors: MESEGUER A. , MARQUÈS F. , SÁNCHEZ J.
Published: October 2004

We present a low-dimensional truncated model for a viscous fluid contained in a two-dimensional square box, obtained by truncating a dynamical system of amplitudes for the velocity field. This low-dimensional model exhibits a route to chaos via a period doubling cascade (Feigenbaum’s Scenario). In order to compute with high accuracy the period doubling, a numerical method based on the first order variational equations and a Poincaré map has been developed. This methodology can also be applied to the analysis of bifurcations of periodic orbits in low-dimensional ordinary differential equations. This method allows to detect not only the presence of bifurcations but also the computation of stable and unstable periodic orbits. On the other hand, the chaotic dynamics of the system is analyzed in detail by the computation of the Liapunov exponents for long-time integrations. For this purpose, a numerical scheme based on renormalization techniques has been constructed.
DYNAMICS OF LATTICE DIFFERENTIAL EQUATIONS
Authors: CHOW SHUI-NEE , MALLET-PARET JOHN , VAN VLECK ERIK S.
Published: October 2004

In this paper recent work on the dynamics of lattice differential equations is surveyed. In particular, results on propagation failure and lattice induced anisotropy for traveling wave or plane wave solutions in higher space dimensions spatially discrete bistable reaction–diffusion systems are considered. In addition, analysis of and spatial chaos in the equilibrium states of spatially discrete reaction–diffusion systems are discussed.
THE STOCHASTIC NATURE OF CARDIAC PROPAGATION DUE TO THE DISCRETE CELLULAR STRUCTURE OF THE MYOCARDIUM
Authors: SPACH MADISON S.
Published: October 2004

The object of this paper is to describe cardiac conduction phenomena caused by the discrete nature of cardiac cellular structure. Recent results show that the myocardial architecture creates inhomogeneities of electrical load at the microscopic level that cause cardiac propagation to be stochastic in nature. That is, the excitatory events during propagation are constantly changing and disorderly in the sense of varying intracellular events and delays between cells. A unique feature of the stochastic nature of cardiac propagation is that electrical boundaries produced by cellular myocardial architecture create inhomogeneities of electrical load that affect conduction inside individual cells and influence conduction delays across gap junctions, as well as at muscle bundle junctions. This process produces discontinuous propagation as a primary reflection of the nonuniformities of electrical load due to the irregular arrangement of the cellular borders and the associated nonuniform distribution of their electrical interconnections. A fundamental consequence of the stochastic nature of normal propagation at a microscopic level is that it provides a major protective effect against arrhythmias by re-establishing the general trend of wave front movement after small variations in excitation events occur. When the diversity at a very small size scale decreases throughout the tissue, such as occurs when there are regularly repeating relatively isolated groups of cells, larger fluctuations of load can develop and be distributed over more cells than occurs normally. The myocardial architecture may then fail to re-establish a smoothed wave front and re-entry can develop. These relatively new discontinuous conduction phenomena provide important theoretical and experimental challenges to synthesize a complete theory linking continuous and discontinuous media as applied to cardiac conduction. The results show that it will be important to distinguish differences in wave front movement and conduction block caused by mechanisms of continuous media versus wave front movement and block imposed by directional or localized changes in cellular connectivity; i.e., the topology of the electrical connections between cells (gap junctions).
EFFECTS OF LOCAL ISCHEMIA AND TRANSIENT CONDUCTION BLOCKS ON THE INDUCTION OF CARDIAC REENTRIES
Authors: BARDOU ALAIN , ACHOUR SOUMEYA , AUGER PIERRE , CHASSÉ JEAN-LUC
Published: October 2004

The present work is aimed at investigating the effects of ischemia and transient blocks on induction of ventricular fibrillation. Electrophysiologic effects of local ischemia induced by minor myocardial infarction (variation of the dispersion of refractory periods as well as conduction velocity) on initiation of reentry mechanisms were studied by means of computer simulations based on a cellular automata model of propagation of activation wave through a ventricular surface element. A local ischemic area was simulated, showing that altered refractoriness can facilitate initiation of reentry phenomena. The probability of reentries is a function of the size of ischemic zone and of the spatial rate of dispersion of refractory periods. Our results show that the dispersion of refractoriness is an important parameter in triggering cardiac reentries. Above a threshold value of dispersion, even very small ischemic zones can induce spiral waves.

We also study the effects of transient conduction blocks on reentries. The role of transient blocks in the triggering of reentries is investigated by use of a two-dimensional set of discretely coupled cells described by the Van Capelle and Durrer model. We study the effect of the size and of the time duration of the transient block on the induction of spiral waves. For each block size, we determine a time window in which spiral waves are initiated.
INTERACTION OF RE-ENTRIES IN COUPLED PARALLEL FIBERS
Authors: DE CASTRO M. , PÉREZ-MUÑUZURI V. , MARIÑO I.P. , GÓMEZ-GESTEIRA M. , CHUA L.O. , PÉREZ-VILLAR V.
Published: October 2004

The interaction between two re-entries and between a single re-entry and a defect was studied both numerically and experimentally in two coupled parallel excitable fibers. This study was carried out as a function of the chirality of the re-entries and the initial difference of phase between them in two homogeneous fibers. Unstable bound states were found to drift even in homogeneous systems due to the spatial symmetry breaking generated by the presence of interacting re-entries in the medium.
CONTINUOUS CONDUCTIVE VOLUME AFFECTS THE PROPAGATION OF SIGNALS IN DISCRETE SYSTEMS
Authors: MUÑUZURI A.P. , DE CASTRO M. , HOFER E. , PÉREZ-MUÑUZURI V. , GÓMEZ-GESTEIRA M. , PÉREZ-VILLAR V.
Published: October 2004

The effect of a continuous conductive volume on the propagation of signals in discretely coupled nonlinear electronic circuits is presented. Two different coupling mechanisms between the circuits are considered at the same time (through linear resistors and through a continuous conductive volume) trying to mimic the behavior observed in some experiments from cardiac electrophysiology. The effects of the geometry of the conductive medium and the distance from the measuring point to the electronic fiber are studied experimentally.
AVALANCHE-COLLAPSE MECHANISM: A MODEL FOR EXCITABILITY
Authors: PLAZA F. , VELARDE M.G.
Published: October 2004

We propose a new type of excitability mechanism, based on an avalanche-collapse process. We show the differences and similarities between this and the already known excitability concepts for biological and mechanical systems. An experiment is also proposed with a laser containing a saturable absorber inside the cavity.
DISCRETE COUPLING AND PROPAGATING SIGNALS
Authors: SCHÜTZ PETER , BODE MATHIAS
Published: October 2004

Biological systems quite often consist of a large number of discrete cells, which are diffusively coupled in an inhomogeneous way and use to be inhomogeneous on their own, too. To study pattern formation in such a system, it is necessary to model the dynamics of quantities like voltages or chemical concentrations. If the coupling is sufficiently strong, a description by continuous reaction–diffusion models is possible. Thereby inhomogeneous coupling leads to a space-dependent diffusion constant. As examples for dynamical behavior are likely to appear in such systems, we study front propagation on a chain of inhomogeneously spaced cells as well as in a two-dimensional cluster of cells, which has the shape of a bottle-necked channel. Using multiple scale methods to reduce the dynamics to a few degrees of freedom, we find phenomena like pinning, transition to oscillations of the front position, oscillatory pinning, reflection and transmission.
SPIRAL TIP MOVEMENT: THE ROLE OF THE ACTION POTENTIAL WAVELENGTH IN POLYMORPHIC CARDIAC ARRHYTHMIAS
Authors: STARMER C. FRANK , STAROBIN JOSEF
Published: October 2004

Sudden cardiac death is often preceded by re-entrant (spiral wave) tachyarrhythmias associated with a prolongation of the action potential wavelength. Potassium channel blockade, hypokalemia and genetic alterations in membrane ion channels (e.g. long QT syndrome) can all lead to extension of the action potential wavelength. In experiments with cardiac cells, we have found that potassium channel blockade can also promote transient oscillation of the membrane potential which can then initiate transient or sustained rhythm disturbances. Using a simple FitzHugh–Nagumo model of an excitable cell, we observed that extending the time required for an excited cell to return to the rest state (equivalent to potassium channel blockade or hypokalemia) amplified the region of spiral tip movement (meandering). When the region of meandering was large relative to the total myocardial surface, the QRS amplitude and configuration varied from one re-entrant cycle to the next and resembled the patterns seen in the cardiac arrhythmia, torsade de pointes. By changing media parameters in order to slow conduction, the spatial wavelength of the action potential was reduced which diminished the relative size of the meandering region and increased the degree of uniformity in QRS shape, i.e. the peak amplitude and QRS configuration were relatively constant from cycle to cycle. Injecting a small, continuous depolarizing current near the spiral tip reduced the local threshold of excitability and reduced the size of the meandering region. These studies support a unified model for initiation and evolution of polymorphic re-entrant arrhythmias and suggest new strategies for arrhythmia management based on control of spiral tip movement.
INTERACTIONS BETWEEN VORTICES AND LESIONS AND ECTOPIC FOCI IN A CML MODEL OF CARDIAC TISSUE
Authors: ZHANG H.
Published: October 2004

Interactions between a localised lesion, and ectopic focus and vortices are studied in CML models of myocardium. Decrease in the size of a localised lesion may change the motion of a trapped spiral wave from a periodic to a quasiperiodic rotation in 2D models. In a 3D model, a localised lesion gives rise to distortion and then breakdown of a trapped scroll wave, generating spatio-temporal irregularity. The interaction between a spiral wave and an ectopic focus is dependent on the differences in their frequencies and phases.
FRACTAL EVOLUTION IN DETERMINISTIC AND RANDOM MODELS
Authors: FUSSY SIEGFRIED , GRÖSSING GERHARD , SCHWABL HERBERT
Published: October 2004

One-dimensional coupled map lattices or quantum cellular automata with any additionally implemented temporal feedback operations (involving some memory of the system’s states) and a normalization procedure after each time step exhibit a universal dynamic property called fractal evolution [Fussy & Grössing, 1994]. It is characterized by a power-law behavior of a system’s order parameter with regard to a resolution-like parameter which controls the deviation from an undisturbed (i.e., feedback-less) system’s evolution and provides a dynamically invariant measure for the emerging spatiotemporal patterns.

By comparison with another, simpler model without memory, where the patterns are generated randomly, the underlying principles of fractal evolution are studied. It is shown that our system evolving entirely deterministically, exhibits properties occurring usually only in random models, where the global measures, up to a certain degree, are calculable. Other properties like the fractal evolution exponent remain in general computationally irreducible due to the self-referential feedback dynamics. A specific model with an approximate estimation of the fractal evolution exponent is discussed.

The stability of fractal evolution with respect to the dependence of pattern formation on the systems variables is also analyzed.
ON DYNAMICAL SYSTEMS GENERATED BY TWO ALTERNATING VECTOR FIELDS
Authors: KLÍČ A. , POKORNÝ P.
Published: October 2004

Dynamical systems with time evolution determined by two alternating vector fields are investigated both analytically and numerically. When the two vector fields are related by an involutory diffeomorphism G then the fixed points of G (either isolated or non-isolated) are shown to give rise to branches of periodic solutions of the resulting non-autonomous system. The method of averaging is used for small switching periods. Detailed numerical study of both conservative (“blinking vortex”) and dissipative (“blinking nodes”, “blinking cycles” and “blinking Lorenz”) systems shows that the technique of blinking can be used to initiating and controlling of chaos.
REAL LIFE
Authors: MAR GARY , DENIS PAUL ST.
Published: October 2004

In Conway’s Game of Life every cell is either fully alive (has the value of 1) or completely dead (has the value 0). In Real Life this restriction to bivalence is lifted to countenance “real-valued” degrees of life and death. Real Life contains Conway’s Game of Life as a special case; however, Real Life, in contrast to Conway’s Game of Life, exhibits sensitive dependence on initial conditions which is characteristic of chaotic systems.
A SIMPLE PIECEWISE-LINEAR NON-AUTONOMOUS CIRCUIT WITH CHAOTIC BEHAVIOR
Authors: LACY J.G.
Published: October 2004

A non-autonomous circuit having a simple nonlinear element designed to be accurately piecewise-linear modeled, is described. Chaotic behavior is experimentally shown and confirmed by two simulation methods.
CONTROLLING PATTERN FORMATION IN A CNN OF CHUA’S CIRCUITS
Authors: SHALFEEV VLADIMIR D. , KUZNETSOV ALEXEY S.
Published: October 2004

In this letter we consider the nonlinear dynamics of a 2-dimensional CNN (cellular neural networks) made of a two-dimensional array of Chua’s oscillators, interconnected via nonlinear coupling. We focus our attention on the possibility of intelligently controlling the pattern formation process by applying an external signal from independent current sources to the cells of the CNN, or by an intelligent choice of initial conditions.
CHUA’S EQUATION WITH CUBIC NONLINEARITY
Authors: HUANG ANSHAN , PIVKA LADISLAV , WU CHAI WAH , FRANZ MARTIN
Published: October 2004

In this tutorial paper we present one of the simplest autonomous differential equations capable of generating chaotic behavior. Some of the fundamental routes to chaos and bifurcation phenomena are demonstrated with examples. A brief discussion of equilibrium points and their stability is given. For the convenience of the reader, a short computer program written in QuickBASIC is included to give the reader a possibility of quick hands-on experience with the generation of chaotic phenomena without using sophisticated numerical simulators. All the necessary parameter values and initial conditions are provided in a tabular form. Eigenvalue diagrams showing regions with particular eigenvalue patterns are given.
SPREAD SPECTRUM COMMUNICATIONS BASED ON CHAOTIC SYSTEMS
Authors: LIPTON J. M. , DABKE K. P.
Published: October 2004

Spread spectrum communication systems based on chaotic carrier sequences are examined for their security and robustness in the presence of noise and other interferences. Both discrete and continuous time chaotic carriers are examined and methods for determining the effectiveness of such carriers are discussed. An experimental implementation of a chaotic communication system was built and tested, and then compared with theory and the results from simulations.
A STATISTICAL METHOD FOR DETECTING CYCLES IN DISCRETE DYNAMICAL SYSTEMS
Authors: LOHMANN MARKUS , WENZELBURGER JAN
Published: October 2004

This paper introduces a statistical method for detecting cycles in discrete time dynamical systems. The continuous state space is replaced by a discrete one consisting of cells. Hashing is used to represent the cells in the computer’s memory. An algorithm for a two-parameter bifurcation analysis is presented which uses the statistical method to detect cycles in the discrete state space. The output of this analysis is a colored cartogram where parameter regions are marked according to the long-term behavior of the system. Moreover, the algorithm allows the computation of basins of attraction of cycles.
ON THE QUESTION OF DISSIPATION AND PARAMETRIC EXCITATION IN CHAOTIC NONAUTONOMOUS RESONATORS
Authors: SARAFIAN G. , KAPLAN B. Z.
Published: October 2004

Three categories of second order resonators are treated in this work: A. Nonautonomous driven resonators whose reactors are nonlinear and resistances are positive. (The resonators are similar to the RL Diode circuit). B. Nonautonomous driven resonators whose reactors are linear (and maintained constant in time) and whose dissipation is due to a nonlinear resistor which consists of a negative resistance part. C. Autonomous resonators whose reactors are parametrically excited. (The reactances of the resonators of category C are made directly periodically time varied). The introduction of category C is mainly for auxiliary purposes, in order to bridge difficulties that occur in comparing resonators of categories A and B.

Resonators of both categories A and B demonstrate similar dynamic behaviors, since both are capable in certain circumstances of exhibiting bifurcation phenomena and chaos. This demonstrated similarity is puzzling. This is due to the fact that the resonators of category A are fully dissipative, while the resonators of category B should exhibit negative dissipation for part of their movement. This negative dissipation in resonators of category B is essential for enabling them to exhibit chaotic behavior. A first step in resolving the puzzle is made by recalling the fact that parametrically pumping reactances in resonators of category C may induce an effective negative resistance in their circuit, and as a result they are somewhat similar to resonators of category B. A complete resolution of the puzzle should, however, include yet another step, where one shows that members of categories A and C possess affinity. A great part of the work is, therefore, devoted to demonstrating that resonators of categories A and C are related in exhibiting similarity in their dynamic features. (This happens although the reactances of resonators of category A are maintained constant in time, while the resonators of category C are periodically time varied). One of the objectives is to convince the reader that some of the dynamic interactions occurring in resonators of category A are “parametric"-like in nature. Hence, an association between categories A and C is demonstrated.
COMMUTING POLYNOMIALS AND POLYNOMIALS WITH SAME JULIA SET
Authors: ATELA PAU , HU JUN
Published: October 2004

It has been known since Julia that polynomials commuting under composition have the same Julia set. More recently in the works of Baker and Eremenko, Fernández, and Beardon, results were given on the converse question: When do two polynomials have the same Julia set? We give a complete answer to this question and show the exact relation between the two problems of polynomials with the same Julia set and commuting pairs.