bluuebunny/crossref_metadata_2025 huggingface dataset:
BOUNDARY CRISES ARISING FROM SADDLE-NODE BIFURCATIONS IN PERIODICALLY FORCED MODELS
Authors: FOSTER E.A.D.
Published: October 2004

An identical bifurcation sequence is identified for models of predator-prey interaction and glycolytic reaction under strong periodic forcing. It is determined that a symmetric saddle-node bifurcation gives rise to local chaotic attractors. These attractors are annihilated by a boundary crisis, but a stable periodic orbit persists. When this remaining periodic orbit subsequently loses stability, an intermittency transition to chaos is observed. Through the construction of return maps, the essential flow behavior is reduced to that of one-dimensional noninvertible maps. The equivalent scenario in these maps is determined to be a tangent bifurcation, followed by a boundary crisis and a subcritical period halving. A remarkable feature of this previously unrecognized bifurcation sequence is the existence of a chaotic set that changes from attracting to nonattracting and back to attracting.
CONTROL STRATEGIES FOR CHAOTIC NEUROMODULES
Authors: STOLLENWERK NICO , PASEMANN FRANK
Published: October 2004

Different strategies for control of chaotic systems are discussed: The well known Ott-Grebogi-Yorke algorithm and two alternative algorithms based on least-squares minimisation of the one step future deviation. To compare their effectiveness in the neural network context they are applied to a minimal two neuron module with discrete chaotic dynamics. The best method with respect to calculation effort, to neural implementation, and to controlling properties is the nonlinear least squares method. Furthermore, it is observed in simulations that one can stabilise a whole periodic orbit by applying the control signals only to one of its periodic points, which lies in a distinguished region of phase space.
FEEDBACK CONTROL OF A QUADRATIC MAP MODEL OF CARDIAC CHAOS
Authors: BRANDT MICHAEL E. , CHEN GUANRONG
Published: October 2004

This paper describes methods based on engineering feedback principles for controlling cardiac chaos modeled by a quadratic map. We first discuss briefly the linear approach which is effective at controlling the map to fixed point or near fixed point target trajectories. We then describe and demonstrate self-tuning methods that can control the chaotic orbits of the map to arbitrary period one, two, or three target trajectories. We conclude with a discussion comparing the merits of this approach to the OGY method used by Garfinkel et al. [1992] for control of complex rhythms in rabbit septal preparations, and to the flat-topped quadratic mapping method of Glass and Zeng [1994].
STABILIZATION OF CHAOTIC DYNAMICS OF ONE-DIMENSIONAL MAPS BY A CYCLIC PARAMETRIC TRANSFORMATION
Authors: LOSKUTOV ALEXANDER Yu. , TERESHKO VALERY M. , VASILIEV KONSTANTIN A.
Published: October 2004

We consider one-dimensional maps, the logistic map and an exponential map, in those sets of parameter values which correspond to their chaotic dynamics. It is proven that such dynamics may be stabilized by a certain cyclic parametric transformation operating strictly within the chaotic set. The stabilization is a result of the creation of stable periodic orbits in the initially chaotic maps. The period of these stable orbits is a multiple of the period of the cyclic transformation. It is shown that stabilized behavior cannot be destroyed by a weak noise smearing of the required parameter values. The regions where the behavior stabilization takes place are numerically estimated. Periods of the created stabile periodic orbits are calculated.
THE PALIS AND TAKENS’ PROBLEM ON THE FIRST HOMOCLINIC TANGENCY INSIDE THE HORSESHOE
Authors: KIRIKI SHIN
Published: October 2004

We introduce a general definition of first homoclinic tangency, and construct a one-parameter family of planar diffeomorphisms from Smale’s n-fold horseshoe diffeomorphism, which has the first homoclinic tangency that is a first bifurcation introduced by Palis and Takens. Moreover, the limit capacity of its maximal invariant set associated with this first bifurcation can be as small as is required.
BIFURCATIONS AND CHAOS IN A PENDULUM WITH CIRCULAR SUPPORT MOTION
Authors: DOOREN R. VAN
Published: October 2004

We study the effect of circular support motion for a pendulum system in comparison with the usual case of forced motion by an external periodic torque with fixed support. The characteristics of the bifurcation diagram and the chaotic attractors are analysed and compared.
SUB-CRITICAL FLIP BIFURCATIONS THAT MAY OR MAY NOT RESULT IN PERIOD-DOUBLING
Authors: SOLIMAN MOHAMED S.
Published: October 2004

Super-critical flip bifurcations are continuous and determinate in that there is always period-doubling of the fundamental attracting solution. In the case of sub-critical flip bifurcations if the fundamental solution is located on a highly intertwined basin boundary when it loses its stability, we cannot predict whether the system period-doubles or not.
ON THE COMPLEX BIFURCATION SET FOR A SYSTEM WITH SIMPLE DYNAMICS
Authors: SHASHKOV MIKHAIL V. , TURAEV DMITRY V.
Published: October 2004

Bifurcations of a heteroclinic contour composed of two equilibrium points of saddle-focus type and two heteroclinic orbits are considered. The case is selected where dynamics of the system is simple, i.e., no more than one periodic orbit is born at bifurcations in a small neighborhood of the contour. In spite of the simplicity of dynamic behavior, the structure of the bifurcation set corresponding to multi-round heteroclinic orbits is shown to be rather complicated. The complete bifurcation analysis is done under some conditions of a general position.
COMPLEXITY OF HOMOCLINIC BIFURCATIONS AND Ω-MODULI
Authors: GONCHENKO S. V. , STEN’KIN O. V. , TURAEV D. V.
Published: October 2004

Bifurcations of two-dimensional diffeomorphisms with a homoclinic tangency are studied in one-and two-parameter families. Due to the well-known impossibility of a complete study of such bifurcations, the problem is restricted to the study of the bifurcations of the so-called low-round periodic orbits. In this connection, the idea of taking Ω-moduli (continuous invariants of the topological conjugacy on the nonwandering set) as the main control parameters (together with the standard splitting parameter) is proposed. In this way, new bifurcational effects are found which do not occur at a one-parameter analysis. In particular, the density of cusp-bifurcations is revealed.
COALESCENCE OF REVERSIBLE HOMOCLINIC ORBITS CAUSES ELLIPTIC RESONANCE
Authors: FIEDLER BERNOLD , TURAEV DMITRY
Published: October 2004

Reversible flows can possess a robust homoclinic orbit to a saddle equilibrium: the orbit is preserved under small perturbations that do not destroy the reversibility of the system. Such a homoclinic orbit is a limit of a unique one-parameter family of periodic orbits. All these orbits are saddles if the equilibrium state is a saddle. There are both saddle and elliptic periodic orbits in this family if the equilibrium state is a saddle-focus. In the present paper, we study the coalescence of two such homoclinic orbits in a one-parameter family of reversible flows. We show that, even in the case where all eigenvalues of the corresponding equilibrium are real, a family of elliptic periodic orbits arises at this bifurcation.
ON CODING OF SERIES OF HOMOCLINIC LOOPS
Authors: ELEONSKY V. M. , KOROLEV V. G.
Published: October 2004

Three possible general types of origin and coding of series of complicated homoclinic saddle loops in dynamical systems with two degrees-of-freedom are pointed out.

As an example of a physically important application, series of complicated vector solitons in birefringent (or bimodal) fiber governed by the system of two coupled nonlinear Schrödinger equations are studied. In that system, all the three types of series are found and investigated.
HYPERBOLIC HOMOCLINIC POINTS OF ℤd-ACTIONS IN LATTICE DYNAMICAL SYSTEMS
Authors: AFRAIMOVICH V. S. , CHOW SHUI-NEE , SHEN WENXIAN
Published: October 2004

We study ℤd action on a set of equilibrium solutions of a lattice dynamical system, i.e., a system with discrete spatial variables, and the stability and hyperbolicity of the equilibrium solutions. Complicated behavior of ℤd-action corresponds to the existence of an infinite number of equilibrium solutions which are randomly situated along spatial coordinates. We prove that the existence of a homoclinic point of a ℤd-action implies complicated behavior, provided the hyperbolicity of the homoclinic solution with respect to the lattice dynamical system (this is a generalization of the previous work of the first two authors). Similar result holds for hyperbolic partially homoclinic and heteroclinic points. We show the equivalence of stability for any equilibrium solutions and the equivalence of hyperbolicity for homoclinic points under various norms.
WAVE-SPLITTING IN THE BISTABLE GRAY-SCOTT MODEL
Authors: RASMUSSEN K. E. , MAZIN W. , MOSEKILDE E. , DEWEL G. , BORCKMANS P.
Published: October 2004

The Gray-Scott model describes a chemical reaction in which an activator species grows autocatalytically on a continuously fed substrate. For certain feed rates and activator life times the model allows for the coexistence of two different homogeneous steady states, the red state in which the conditions correspond to those of the pure substrate, and the blue state where the activator concentration is relatively high and limited by substrate depletion. The blue state may again undergo a Hopf bifurcation leading to self-sustained oscillations and, for a spatially extended system, this state may also exhibit a subcritical Turing bifurcation capable of producing global as well as localized, stationary spatial structures. Interactions between these instabilities can lead to a variety of different phenomena including Turing-Hopf mixed modes and spatio-temporal chaos. The paper presents the results of a computer simulation study of some of these far-from-equilibrium phenomena. Special emphasis is given to the propagation, collision and splitting of the traveling pulses which can develop in response, for instance, to strong local perturbations of the red state in a region of parameter space where the blue state is Turing and Hopf unstable.
NORMALIZATION IN THE SYSTEMS WITH SMALL DIFFUSION
Authors: KASCHENKO S. A.
Published: October 2004

In this paper the local dynamics of systems of nonlinear PDEs with small diffusion is studied. The main feature of these systems lies in the fact that the dimension of a critical case in the stability problem for an equilibrium state is equal to infinity. Algorithms that reduce the initial problem to the analysis of nonlocal dynamics of special evolution equations playing the role of normal forms are developed.
MANY-DIMENSIONAL LORENTZ CELLULAR AUTOMATA AND TURING MACHINES
Authors: BUNIMOVICH LEONID A.
Published: October 2004

We study the class of cellular automata that generalizes the Lorentz lattice gases in statistical mechanics, the models of industrious ants in the theory of an artificial life and the so-called Tur-mites (many-dimensional Turing machines). We prove that on the square lattice ℤd, d = 2, the existence of a bounded orbit of a particle (ant, machine) determines all nondegenerate local scattering rules (states of a machine). For higher dimensional (d ≥ 3) cubic lattices we show that under some natural conditions all possible bounded orbits (vortices) can live only in some “vortex sheets” that have a dimension strictly less than d.
SHILNIKOV’S SADDLE-NODE BIFURCATION
Authors: GLENDINNING PAUL , SPARROW COLIN
Published: October 2004

In 1969, Shilnikov described a bifurcation for families of ordinary differential equations involving n≥2 trajectories bi-asymptotic to a non-hyperbolic stationary point. At nearby parameter values the system has trajectories in correspondence with the full shift on n symbols. We investigate a simple (piecewise-smooth) example with an infinite number of homoclinic loops. We also present a smooth example which shows how Shilnikov’s mechanism is related to the Lorenz bifurcation by considering the unfolding of a previously unstudied codimension two bifurcation point.
EXPONENTIAL EXPANSION WITH PRINCIPAL EIGENVALUES
Authors: DENG BO
Published: October 2004

The formulation of exponential expansion is generalized to equilibrium points having arbitrary principal eigenvalues. It is established by transforming the coordinates into admissible ones and some earlier results on exponential bounds for Shil’nikov’s solutions.
MINIMAL AND STRANGE ATTRACTORS
Authors: GORODETSKI A. , ILYASHENKO Yu.
Published: October 2004

A general concept going back to Kolmogorov claims that if a dynamical system has a complicated attracting set then its behavior has not a deterministic, but rather probabilistic character. This concept was not formalized up to now. Even the definition of attractor has a lot of different versions. This paper presents an attempt to give some definitions and results formalizing this heuristic ideas. It contains a definition of a minimal attractor, modifying the one given in Ilyashenko [1991]. The actual minimality of the attractor is discussed. The principal result is the Triple Choice Theorem. It claims that the existence of a strange minimal attractor implies some mild form of chaos for the map itself or for a nearby one. The program of further investigation is proposed as a chain of problems at the end of the paper.
SNOWFLAKE HARMONICS AND COMPUTER GRAPHICS: NUMERICAL COMPUTATION OF SPECTRA ON FRACTAL DRUMS
Authors: LAPIDUS MICHEL L. , NEUBERGER J.W. , RENKA ROBERT J. , GRIFFITH CHERYL A.
Published: October 2004

In this work, we study the steady-states vibrations of the “Koch snowflake drum”, both numerically and by means of computer graphics. In particular, we approximate the smallest 50 eigenvalues (or “frequencies” of the drum), along with the corresponding eigenfunctions (called “snowflake harmonics”) of the Dirichlet Laplacian on the Koch snowflake domain. We describe the numerical methods used in the computations, and we display graphical representations of a selected set of the eigenfunctions (as well as of their gradients). In the case of the first harmonic, the graphical results agree with mathematically derived results (by Lapidus and Pang) concerning gradient behavior (“blow up” or “infinite stress” of the membrane) on the boundary and suggest further conjectures regarding the higher eigenfunctions.

According to earlier work by the physicist Sapoval and his collaborators, this research may help better understand the formation and “stabilization” of fractal structures (e.g., coastlines, trees and blood vessels) in nature.
THE INFLUENCE OF HOPF BIFURCATION DEGENERACIES ON THE DYNAMIC BEHAVIOR OF A NON-IDEAL PI CONTROLLED CSTR
Authors: POSSIO CRISTINA TABLINO , PELLEGRINI LAURA
Published: October 2004

A detailed bifurcation analysis of a non-ideal Proportional-Integral controlled Continuous Stirred Tank Reactor (CSTR) has been performed taking advantage of the application of the high codimension bifurcation theory in order to point out all possible peculiar patterns of nonlinear dynamic behavior. In particular, Type I Hopf degeneracies, typically induced by integral controller introduction, reveal themselves to be a considerable source of such richness.
PERIOD-DOUBLING GEOMETRY OF THE BELOUSOV-ZHABOTINSKY REACTION DYNAMICS
Authors: WANG JICHANG , HYNNE F. , SØRENSEN P. GRAAE
Published: October 2004

The intricate geometry of stable and unstable manifolds of a saddle cycle arising from a super-critical period-doubling bifurcation is explored experimentally for the Belousov-Zhabotinsky (BZ) reaction in a CSTR (Continuous flow Stirred Tank Reactor). We find clear experimental evidence that the stable manifold winds round the stable period-doubled orbit arising at the bifurcation. The stable manifold is probed experimentally by perturbations from the period-doubled limit cycle. The method provides a model-independent experimental test for species essential for the complexity, as well as quantitative information about the geometry of the limit cycles, the associated manifolds, and their embedding in the concentration space.

The results are supported by simulations of the experiments with a four-dimensional model of the BZ reaction. Here stable manifold has several branches showing some tendency of curling. However the branches may end on the boundary of the positive orthant of the concentration space.
CHAOTIC ATTRACTORS AND WAVES IN A ONE-DIMENSIONAL ARRAY OF MODIFIED CHUA’S CIRCUITS
Authors: NEKORKIN VLADIMIR I. , KAZANTSEV VICTOR B. , CHUA LEON O.
Published: October 2004

In this paper, we investigate the possible propagation of travelling waves of a chaotic profile in an unbounded one-dimensional array of inductively-coupled Modified Chua’s Circuits. We show that the basic unit (cell) of our array is a relaxation-like chaotic oscillator, and its dynamics can be modeled by a two-dimensional system with hysteresis. This hysteresis system is studied via an associated 1D point map, and the existence of various distinct chaotic attractors is proved.
FEEDBACK CONTROL OF LYAPUNOV EXPONENTS FOR DISCRETE-TIME DYNAMICAL SYSTEMS
Authors: CHEN GUANRONG , LAI DEJIAN
Published: October 2004

A simple, yet mathematically rigorous feedback control design method is proposed in this paper, which can make all the Lyapunov exponents of the controlled system strictly positive, for any given n-dimensional dynamical system that could be originally nonchaotic or even asymptotically stable. The argument used is purely algebraic and the design procedure is completely schematic, with no approximations used throughout the derivation. This is a rigorous and convenient technique suggested as an attempt for anticontrol of chaotic dynamical systems, with explicit computational formulas derived for applications.
ON THE DYNAMICS OF COUPLED JOSEPHSON JUNCTION CIRCUITS
Authors: FINGER L.
Published: October 2004

In this paper, we consider coupled Josephson junction circuits, so-called SQUIDs. We investigate by means of analytical and numerical methods the nonlinear behaviour of two circuits, the ideal and a real model. We show that the solutions for SQUID systems occur in a cylindrical state space which make methods of the Nonlinear Dynamics applicable and lead to a general description for these systems. Moreover, methods of the Nonlinear Dynamics extend the static characteristics. Finally the behaviour is analysed with methods of the Stochastic Dynamics under influence of thermal noise and several occurring phenomena are explained.
ON A VARIATION OF THE HUBERMAN-LUMER ADAPTIVE SCHEME
Authors: WU CHAI WAH , CHUA LEON O.
Published: October 2004

In this letter, we present a variation of the Huberman-Lumer scheme for adaptive systems. We show that this scheme has the advantage that for nonlinear systems with linear parameters, the synchronization error can be proven to approach zero. We also present an adaptive scheme for nonlinear systems with multiplicative parameters. Examples using Chua's oscillators are presented.
CLUSTERING IN A COUPLED MAP LATTICE
Authors: SBITNEV VALERY I.
Published: October 2004

Normal chaotic activity and anomalous chaotic activity have been observed in a uniform coupled map lattice (CML). The former is stable providing a space-time complexity by intermittency of different primitive clusterings. The latter is metastable being a synchronous chaotic activity throughout the CML. Small disturbance destroys it and the CML returns itself to the normal chaotic activity. A transition to the anomalous chaotic activity is possible via a saddle-node bifurcation from a checkerboard state, i.e., from a coherent ensemble of 2-periodic oscillating sites throughout the lattice with the neighboring ones opposite in phases.
A FAST-MANIFOLD APPROACH TO MELNIKOV FUNCTIONS FOR SLOWLY VARYING OSCILLATORS
Authors: CHEN SHYH-LEH , SHAW STEVEN W.
Published: October 2004

A new approach to obtaining the Melnikov function for homoclinic orbits in slowly varying oscillators is proposed. The present method applies the usual two-dimensional Melnikov analysis to the “fast” dynamics of the system which lie on an invariant manifold. It is shown that the resultant Melnikov function is the same as that obtained in the usual way involving distance functions in three dimensions [Wiggins and Holmes, 1987]. This alternative derivation provides some useful insight into the structure of the dynamical system.
DETERMINATION OF THE FREQUENCY VECTOR IN THE FOUR-DIMENSIONAL STANDARD MAPPING
Authors: CELLETTI ALESSANDRA , FROESCHLÉ CLAUDE , LEGA ELENA
Published: October 2004

Different numerical methods, for the computation of frequency versus initial conditions in the four-dimensional standard mapping, are proposed. The first method relies on the analytic properties of invariant rotational tori. A purely numerical technique was derived from an extension of a method suggested by M. Hénon for the two-dimensional mapping. A refinement of the last technique can be obtained implementing the frequency analysis introduced by J. Laskar. These methods agree up to some values of the coupling and perturbing parameters, close to the breakdown threshold for the onset of chaos. The transition to the chaotic regime is further confirmed by the computation of Lyapunov exponents and by suitable slice projections of the original mapping.
ANALYSIS OF THE RECOVERED SIGNAL IN SYNCHRONIZED CHUA’S COMMUNICATION SYSTEM
Authors: CHEN YANQING , PEI LIUQING
Published: October 2004

The recovered message signal in the synchronized Chua’s system, which is proposed for secure communications, is analyzed by means of both numerical simulations and analytical methods. The results show that the characteristic of “frequency response” of the system is independent of the amplitude of the input message signal, and the amplitude of the recovered signal is suppressed by the system in the lower frequency range and amplified in the higher frequency range. Specifically, we find that, as the frequency of the input signal increases continuously, there is a band of frequency over which a resonance type of phenomenon occurs in the system whereby the input message’s amplitude is amplified, but the location of this band lies beyond the range of the main spectrum of the chaotic driving signal. The phase difference between input signal and output signal is also obtained. Furthermore based on numerical simulations, the characteristic frequency response of a synchronized Lorenz’s system is given, which also shows a band of resonance in the lower range of frequencies.
AN ALGORITHMIC MODEL OF THE MAMMALIAN HEART: PROPAGATION, VULNERABILITY, RE-ENTRY AND FIBRILLATION
Authors: HOLDEN A.V. , POOLE M.J. , TUCKER J.V.
Published: October 2004

The theory of synchronous concurrent algorithms is outlined and applied to the construction of an anisotropic coupled map lattice model of electrical activity in the mammalian cardiac ventricles, derived from anatomical measurements of geometry and fibre orientation. The model is used to illustrate propagation, vulnerability, re-entry and fibrillation.
PATTERNS, DEFECTS, AND EVOLUTION OF BÉNARD–MARANGONI CELLS
Authors: BRAGARD J. , PONTES J. , VELARDE M.G.
Published: October 2004

We consider a thin fluid layer of infinite horizontal extent, confined below by a rigid plane and open above to the ambient air, with surface tension linearly depending on the temperature. The fluid is heated from below. First we obtain the weakly nonlinear amplitude equations in specific spatial directions. The procedure yields a set of generalized Ginzburg–Landau equations. Then we proceed to the numerical exploration of the solutions of these equations in finite geometry, hence to the selection of cells as a result of competition between the possible different modes of convection.
NATURAL TOLERANCE AS A FUNCTION OF NETWORK CONNECTIVITY
Authors: CALENBUHR V. , VARELA F.J. , BERSINI H.
Published: October 2004

This article investigates the following basic question: in the relatively stable molecular environment of a vertebrate body, can a dynamic idiotypic immune network develop a natural tolerance to endogenous components? Our approach is based on stability analysis and computer simulation using a model that takes into account the dynamics of two agents of the immune system, namely, B-lymphocytes and antibodies. We investigate the behavior of simple immune networks in interaction with an Ag whose concentration is being held constant as a function of the connectivity matrix of the network. The latter is characterized by the total number of clones, N, and the number of clones, C, with which each clone interacts. The idiotypic network models typically become unstable in the presence of this type of Ag. We show that idiotypic networks that can be found in particular connected regions of NC-space show tolerance towards auto-Ag without the need for ad hoc mechanisms that prevent an immune response. These tolerant network structures provide dynamical regimes in which the clone which interacts with the auto-Ag is suppressed instead of being excited such that an unbounded immune response does not occur. Possible implications for the future treatment of auto-immune disease such as IvIg-treatment are discussed in the light of these results. Moreover, we propose an experimental approach to verify the results of the present theoretical study.
ORIENTATION-INDUCED PATTERN FORMATION: SWARM DYNAMICS IN A LATTICE-GAS AUTOMATON MODEL
Authors: DEUTSCH ANDREAS
Published: October 2004

Swarming patterns might arise not just at organismic levels (bird and fishes exhibiting particularly striking examples) but even at cellular and intracellular scales whenever “collective motion” of biological or chemical entities is involved. Examples are the swarming of myxobacteria and ants, aggregation and slug pattern formation of the slime mold Dictyostelium discoideum, or intracellular network dynamics of actin filaments.

Here a stochastic process — discrete in space and time — is developed, the “swarm lattice-gas automaton”. For some lattice-gas models (in physics and chemistry) it was demonstrated that the limit behavior resembles known master equations by means of expectation values of suitably chosen microscopic variables. In particular, for the Navier–Stokes equation the derivation of a continuous macroscopic description from discrete microdynamic equations was shown.

The “swarm lattice-gas automaton” possesses a non-local integral-like interaction operator. Particles (cells, organisms) are assigned some orientation (and fixed absolute velocity) which might change by means of interaction with other members of the swarm within a given “region of perception”. The corresponding microdynamical equation is given and results of numerical experiments are shown.

Simulations exhibit a variety of aggregation patterns which are distinguished by means of microscopic and macroscopic variables. The influence of a sensitivity parameter and particle density on pattern formation is examined systematically.
SUPEREXCITABILITY INDUCED SPIRAL BREAKUP IN EXCITABLE SYSTEMS
Authors: GIAQUINTA A. , BOCCALETTI S. , ARECCHI F.T.
Published: October 2004

We introduce a 2D coupled map lattice model which, besides simulating the two variable FitzHugh–Nagumo reaction diffusion mechanism, accounts also for a superexcitable period. Superexcitability in the threshold dynamics of excitable media has been recently observed in experiments on cardiac tissues. By this model, we can reproduce the transition from normal cardiac behavior toward fibrillating processes in a 2D assembly of cardiac cells. The role of superexcitability results in producing two states of wave propagation and a spiral breakup mechanism in qualitative agreement with the experimental evidence of coarse and fine fibrillation in human hearts.
SYNCHRONIZATION IN SMALL ASSEMBLIES OF VAN DER POL–DUFFING CHAOTIC OSCILLATORS
Authors: GÜÉMEZ J. , MATÍAS M.A.
Published: October 2004

The aim of the present contribution is to present a preliminary report on the design of small arrays of chaotic circuits connected by using a recently introduced method that yields chaos synchronization through the use of one-way connections. This method works by driving the response such that the driving signal enters at a particular place in the dynamical equations of the response, instead of suppressing the evolution of the driving signal in the response. This method allows the set up networks with a much higher number of possible connections, including the possibility of a number of simultaneous ones. Several connections are considered for the case of four van der Pol–Duffing circuits, illustrating several features of the method.
IS THE DISCRETE COUPLING OF CARDIAC CELLS REFLECTED IN THE WAVEFRONT OF EXTRACELLULAR POTENTIALS? — AN EXPERIMENTAL APPROACH
Authors: HOFER ERNST , MOHR GÜNTHER , JORGE ANA CLAUDIA , PLATZER DIETER , SCHAFFERHOFER INGRID
Published: October 2004

In recent years there has been a remarkable progress in the knowledge of the microstructure of the cardiac tissue and its influence on the conduction of the cardiac impulse. The tissue domain can be thought as a discrete network of cells coupled electrically at stochastically distributed sites. During in-vitro experiments the tissue is surrounded by a conducting superfusate which represents a continuous domain. With electrode arrays, electrograms can be recorded simultaneously at many sites in this volume conductor and the spatio-temporal distribution of potentials can be obtained. In this paper we show, that despite the fact that the sensors were placed in a continuous medium, we were able to detect microscopic discontinuities of propagation with appropriate techniques.
GAP SOLITONS, RESONANT KINKS, AND INTRINSIC LOCALIZED MODES IN PARAMETRICALLY EXCITED DIATOMIC LATTICES
Authors: HUANG GUOXIANG , LOU SEN-YUE , VELARDE MANUEL G.
Published: October 2004

The dynamics of localized nonlinear excitations of resonant frequencies ωj (j = 0, 1, 2, 3) and carrier wave frequency frequency ωe≈ωj in a damping and parametrically driven lattice system is considered. The excitations are created in a one-dimensional nonlinearly coupled diatomic pendulum lattice which is subjected to a vertical oscillation of frequency 2ωe. The recent experimental observation of gap solitons, resonant kinks, and intrinsic localized modes in the diatomic pendulum lattice system are explained by using an extended nonlinear Schrödinger theory after neglecting the nonuniformity of the pendulums and the small periodic modulations of the amplitudes of the excitations.
REPRESENTATION OF SINO-ATRIAL NODE DYNAMICS BY CIRCLE MAPS
Authors: KREMMYDAS G.P. , HOLDEN A.V. , BEZERIANOS A. , BOUNTIS T.
Published: October 2004

We characterise a model of the sinoatrial node, the pacemaker of the heart, by its response to brief current pulses by a family of phase transition curves. These circle maps are used to describe the effects of acetylcholine on the sensitivity of the pacemaker to perturbations.
FAST DYNAMIC COUPLING FROM SLOW NEURONAL OSCILLATORS
Authors: LE VAN QUYEN MICHEL , SCHUSTER HEINZ G. , VARELA FRANCISCO J.
Published: October 2004

We focus here on the possible role of the widely distributed slow (4–10 Hz) intrinsic bursting neurons in the establishment of long range spacetime coherent subgroups of neurons at multiple locations. We show that a dynamic coupling of these relaxation oscillators provides a simple mechanism whereby an excitatory interconnected network can give rise to coexisting synchronous assemblies of slow neurons in a phase ordered relationship. In the model, this sequential phase locking is dependent on fast inhibitory feedbacks. As a consequence, global rhythms of a broad frequency range spontaneously arise from the regular succession of synchronous assemblies.
CELLULAR AUTOMATON MODEL AND MEASUREMENTS OF AUTOWAVE SPLITTING
Authors: MUÑUZURI ALBERTO P. , MARKUS MARIO
Published: October 2004

A cellular automaton model — based on a semi-random grid to prevent numerical anisotropies — is adapted to calculate the response of the ruthenium-catalyzed Belouzov–Zhabotinsky reagent to a light pulse. In the automaton, increased light intensity is associated with increased excitation threshold and decreased recovery time. Simulations predict splitting (backfiring) of autowaves, which are experimentally confirmed. Optimization of automaton parameters leads to satisfactory agreement with measurements. In particular, splitting of a spiral leads to its destruction, as it generates and then collides with a spiral of opposite topological charge. Furthermore, splitting leads to soliton-like behavior (conservation) of colliding planar waves.
SPATIAL DISORDER AND WAVES IN A RING CHAIN OF BISTABLE OSCILLATORS
Authors: NEKORKIN VLADIMIR I. , MAKAROV VALERI A. , VELARDE MANUEL G.
Published: October 2004

Using analytical and numerical methods we analyze the phenomenon of spatial disorder and also travelling wave propagation in a ring of coupled bistable oscillators. We show the existence of space-homogeneous and space-inhomogeneous waves. We also provide the linear stability of these solutions. Moreover, we discuss the role of the bistability of the unit oscillator and point out a specific property, genuine of the discreteness of the chain.
NUMERICAL STUDY OF PATTERNS AND THEIR EVOLUTION IN FINITE GEOMETRIES
Authors: PONTES J. , CHRISTOV C.I. , VELARDE M.G.
Published: October 2004

Pattern formation in a finite layer of fluid induced either by buoyancy or by a surface-tension gradient is considered. The fluid is confined between poor conducting horizontal boundaries, leading to patterns with a characteristic horizontal scale much larger than the fluid depth. The evolution of the system is studied by numerical integration of the (1+2)D equation introduced by Knobloch [1990]: [Formula: see text] Here µ is the scaled bifurcation parameter, κ=1, and a represents the effects of a heat transfer finite Biot number. The coefficients β, δ and γ do not vanish when the boundary conditions at top and bottom are not identical (β≠0, δ≠0) or when non-Boussinesq effects are taken into account (γ≠0). When the conductive state becomes unstable due to surface-tension inhomogeneities, it is shown that the system evolves towards a stationary pattern of hexagons with up or down flow depending on the relative value of the coefficients β and δ. In the case of buoyancy-driven convection (β=δ≠0), the system displays a tesselation of steady squares. Knobloch’s equation also describes time-dependent patterns at high thermal gradients, including spatio-temporal chaos, due to the non-variational character of the equation.
OPTICAL POTENTIAL MAPPING HELPS TO REVEAL DISCRETE-NATURAL-PHENOMENA IN CARDIAC MUSCLE
Authors: WINDISCH H. , MÜLLER W. , AHAMMER H. , SCHAFFER P. , DAPRA D. , HARTBAUER M.
Published: October 2004

The cardiac muscle is composed of electrically coupled cells which form a complex, three-dimensional structure. Depending on many physiological parameters (e.g. age), the amount of electrical coupling between the cells varies, which leads to nonuniformities in the propagating wavefronts. Optical potential mapping allows one to monitor all kinds of excitation phenomena in various size scales at many sites simultaneously, implying the whole heart the same as isolated cardiomyocytes. The spatial and temporal resolution requirements depend on the specimen under study and can be increased to about 10 µm and to excitation delays of some µs respectively. An optical mapping system with 256 measuring spots, built in our laboratory, allowed us to clearly demonstrate the discrete and discontinuous nature of propagation in rat cardiac tissues. As a unique application, the optical method allows field stimulation studies which mimic defibrillation-like conditions. Recent results obtained in such experiments did not confirm calculations from a one-dimensional strand computer-model of discretely coupled cells, which predicted spatial alternating of membrane potentials within each cell during the imposed stimulus. With a high resolution system built around an inverted microscope, we were able to monitor excitation phenomena in single cells under various experimental conditions, and to show the high amount of electrical coupling within the isolated cardiomyocyte. In more recent studies in other laboratories, using optical potential monitoring, microscopic discontinuities of excitation were assessed in synthetic strands of neonatal rat myocytes with cellular and even subcellular resolution.
TOWARD AN UNDERSTANDING OF STOCHASTIC HOPF BIFURCATION
Authors: ARNOLD LUDWIG , SRI NAMACHCHIVAYA N. , SCHENK-HOPPÉ KLAUS R.
Published: October 2004

In this paper, asymptotic and numerical methods are used to study the phenomenon of stochastic Hopf bifurcation. The analysis is carried out through the study of a noisy Duffing-van der Pol oscillator which exhibits a Hopf bifurcation in the absence of noise as one of the parameters is varied.

In the first part of this paper, we present an introduction to the theory of random dynamical systems (in particular, their generation, their invariant measures, the multiplicative ergodic theorem, and Lyapunov exponents). We then present the two concepts of stochastic bifurcation theory: Phenomenological (based on the Fokker-Planck equation), and dynamical (based on Lyapunov exponents).

The method of stochastic averaging of the nonlinear system yields a set of equations which, together with its variational equation, can be explicitly solved and hence its bifurcation behavior completely analyzed. We augment this analysis by asymptotic expansions of the Lyapunov exponents of the variational equation at zero.

Finally, the stochastic normal form of the noisy Duffing-van der Pol oscillator is derived, and its bifurcation behavior is analyzed numerically. The result is that the (truncated) normal form retains the essential bifurcation characteristics of the full equation.
ANALYTICAL EXPRESSIONS FOR STABLE AND UNSTABLE MANIFOLDS IN HIGHER DEGREE OF FREEDOM HAMILTONIAN SYSTEMS
Authors: DANKOWICZ HARRY
Published: October 2004

Perturbations of completely integrable Hamiltonian systems with three or more degrees of freedom are studied. In particular, the unperturbed systems are assumed to be separable into a product of simple oscillator-type systems and a system containing homo- or heteroclinic connections consisting of stable and unstable manifolds of saddle points. Under a perturbation, the manifolds persist but separate and may no longer intersect. In this paper we show how, with proper choices for initial conditions, one may solve the variational equations to obtain analytical expressions for orbits on the perturbed manifolds in the form of expansions in the small parameter characterizing the perturbation. The derivation also shows how the distance between the manifolds can be uniquely defined, and thus provides an alternative to the traditional higher dimensional Melnikov method.

It is finally argued that the approximate knowledge of the shape and position of the perturbed manifolds could be utilized for the study of large-scale phase-space motions, such as those associated with Arnold diffusion. The theory is further illuminated in two example problems.
TIME SERIES ANALYSIS OF COMPLEX DYNAMICAL BEHAVIOR CONTAMINATED WITH OBSERVATIONAL NOISE
Authors: MIYANO TAKAYA
Published: October 2004

Diagnostic methods for discovering deterministic chaos based on the instability and the parallelness of nearby trajectories generated from a time series in phase space are applied to numerical time series contaminated with additive random noise. The diagnostic algorithm based on nonlinear forecasting is prone to be fooled when handling chaotic data including observational noise. Such a misdiagnosis can be circumvented by estimating the degrees of parallelness of neighboring trajectories in the phase space. Dynamical properties of global temperature variations and voice signals of Japanese vowel /a/ are examined by the combinational use of the diagnostic algorithms.
SUBHARMONIC-SADDLE CONNECTIONS AS A PRECURSOR TO CHAOS IN A PERIODICALLY MODULATED LASER
Authors: CARR THOMAS W. , SCHWARTZ IRA B.
Published: October 2004

We investigate the onset of chaos in a model for a periodically-forced class-B laser. By periodically modulating the laser-cavity damping rate we consider a pair of coupled nonautonomous ordinary differential equations for the laser intensity and population inversion. As the excitation is increased, the system exhibits saddle-node bifurcations to subharmonic oscillations, period-doubling bifurcations, chaos, and crises. Our investigations focus on the role of a pair of unstable orbits in the onset of the first chaotic attractor. One orbit is a subharmonic saddle while the other is a period one saddle. Upon creation of the period one saddle due to a period doubling bifurcation, its stable manifold immediately forms transverse intersections with the unstable manifold of a coexisting subharmonic saddle forming a heteroclinic crossing. These heteroclinic intersections provide a mechanism for the formation of horseshoes and the development of a chaotic attractor.
SYNCHRONOUS CHAOS IN HIGH-DIMENSIONAL MODULAR NEURAL NETWORKS
Authors: WENNEKERS THOMAS , PASEMANN FRANK
Published: October 2004

The relationship between certain types of high-dimensional neural networks and low-dimensional prototypical equations (neuromodules) is investigated. The high-dimensional systems consist of finitely many pools containing identical, dissipative and nonlinear single-units operating in discrete time. Under the assumption of random connections inside and between pools, the system can be reduced to a set of only a few equations, which — asymptotically in time and system size — describe the behavior of every single unit arbitrarily well. This result can be viewed as synchronization of the single units in each pool. It is stated as a theorem on systems of nonlinear coupled maps, which gives explicit conditions on the single unit dynamics and the nature of the random connections. As an application we compare a 2-pool network with the corresponding two-dimensional dynamics. The bifurcation diagrams of both systems become very similar even for moderate system size (N=50) and large disorder in the connection strengths (50% of mean), despite the fact, that the systems exhibit fairly complex behavior (quasiperiodicity, chaos, coexisting attractors).
NONLINEAR DYNAMICS OF MULTISTABLE CHUA’S CIRCUITS
Authors: PONOMARENKO VALERY P. , MATROSOV VALERY V.
Published: October 2004

Dynamic regimes and their bifurcations in the Chua’s circuit with a modified smooth nonlinearity are investigated. The characteristic feature of this modified Chua’s circuit is that it may have more than three equilibrium states. Stability of the equilibrium states and oscillatory regimes are analyzed. The curves corresponding to the bifurcations of the regimes are plotted on the plane of two parameters where regions of chaotic motions are identified. It is found that there exist chaotic multi-spiral attractors. The evolution of dynamics and the effects accompanying the variation of the parameters of the system are considered.
SYNCHRONIZATION OF CHAOS IN A MODIFIED CHUA’S CIRCUIT USING CONTINUOUS CONTROL
Authors: YIN YUAN-ZHAO
Published: October 2004

We modify Chua’s circuit by adding an RC parallel circuit into the L-arm in Chua’s circuit. The bifurcation, control and synchronization of the modified Chua’s circuit has been studied by computer simulation.
CHAOTIC SYNCHRONIZATION OF GLOBALLY COUPLED MAPS WITH AN APPLICATION IN COMMUNICATION
Authors: OKETANI NOBUHIRO , USHIO TOSHIMITSU
Published: October 2004

First, we consider a globally coupled map (GCM) where each subsystem has different dynamics from the others, and show two GCMs are synchronized chaotically by using Wu and Chua’s method. In the synchronized GCM, only the corresponding subsystems are synchronized. Next, we propose a method for a chaos time division multiple access communication using chaotically synchronized GCMs. Each channel has different chaos dynamics, and the information signals are masked chaotically. The corresponding channel can recover the information signals while other channels cannot.
THE VARIANCE AND THE ALGORITHMIC COMPLEXITY OF HEART RATE VARIABILITY DISPLAY DIFFERENT RESPONSES TO ANESTHESIA
Authors: STORELLA R.J. , SHI Y. , WOOD H.W. , JIMÉNEZ-MONTAÑO M.A. , ALBANO A.M. , RAPP P.E.
Published: October 2004

In this study, the variance (a distribution-determined measure of disorder) and the algorithmic complexity (a sequence-sensitive measure) were used to characterize heart rate variability before, during and after cardiac surgery. While it is easy to construct mathematical examples where variance and complexity respond differently to parameter changes, this contrasts with the comparative lack of such examples from biological data. The results presented here provide an example, based on clinical data, where the pattern of changes displayed by the complexity is significantly different from the pattern seen in the variance. While complexity recovers to pre-anesthetic levels following cardiac surgery, variance does not. The details of this pattern suggest that of the two measures, complexity may be the more appropriate metric for characterizing changes in the cardiovascular system in response to anesthetics. The findings reported here demonstrate that measurement of complexity can reveal dynamical changes in biological systems not detected by distribution determined measures such as variance.
SOME PROPERTIES OF A TWO-DIMENSIONAL PIECEWISE-LINEAR NONINVERTIBLE MAP
Authors: MIRA CHRISTIAN , RAUZY CHRISTINE , MAISTRENKO YURI , SUSHKO IRINA
Published: October 2004

Properties of a piecewise-linear noninvertible map of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical point in the one-dimensional case). This map is of (Z0–Z2) type, i.e. the plane consists of a region without preimage, and a region giving rise to two rank one preimages. For the considered parameter values, the map has two saddle fixed points. The characteristic features of the “mixed chaotic area” generated by this map, and its bifurcations (some of them being of homoclinic and heteroclinic type) are examined. Such an area is bounded by the union of critical curves segments and segments of the unstable set of saddle cycles.
SYMMETRY AND BIFURCATION OF A THREE-HINGED ROD
Authors: T LEUNG A. Y. , RAJENDRAN S.
Published: October 2004

Postbuckling of a three-hinged rod under conservative compressive force is considered. The equations for the system are nonlinear and possess some interesting symmetry properties. The group theoretic methods provide powerful mathematical tools to model and exploit the symmetry properties of a problem. In this paper the symmetry properties of the three-hinged rod problem are explored, and the use of symmetry groups to reduce the problem size is demonstrated. The equations are solved for typical values of parameters and the results are presented. The symbolic computer software, Mathematica, is used for developing explicit equations and their numerical solution.
BISPECTRAL PROPERTIES OF THE SIMPLEST DISSIPATIVE NONAUTONOMOUS CHAOTIC CIRCUIT
Authors: LIPTON J. M. , DABKE K. P. , LAKSHMANAN M.
Published: October 2004

The transition from periodicity to chaos for the simplest dissipative nonautonomous circuit is examined in the bispectral domain. The bispectrum and the corresponding bicoherence spectrum demonstrate the complex nonlinear interactions that the circuit exhibits.