bluuebunny/crossref_metadata_2025 huggingface dataset:
Experimental Stabilization of Unstable Periodic Orbit in Magneto-Elastic Chaos by Delayed Feedback Control
Authors: Hikihara Takashi , Touno Masato , Kawagoshi Toshiaki
Published: September 2003

In our previous paper, it was confirmed that the unstable periodic orbit embedded in the chaotic attractor in magneto-elastic beam system can be stabilized by delayed feedback control experimentally. It seems an advantage that the control method does not require any exact model of the system. However, the application of the control raises the problem that we cannot predict the stabilized unstable periodic orbit until it converges. In this paper, an "onset window" is introduced to determine the onset timing for targeting the desired orbit embedded in the chaotic attractor experimentally. Moreover, the dependence of the stabilization on the delay and the gain parameters is also discussed based on the experimental results.
Robust Control of the Chaotic Lorenz System
Authors: Lenz Henning , Obradovic Dragan
Published: September 2003

This paper presents a global approach for controlling the Lorenz system. The basic idea is to partially cancel the nonlinear cross-coupling terms such that the stability of the resulting system can be guaranteed by sequentially proving the stability of each individual state. The method combines ideas from feedback linearization, classical control theory, and Lyapunov's second method. Robust behavior with respect to model uncertainties in the feedback loop is proven. The performance of partial linearization compared to input-state linearization is illustrated on tracking of several trajectories including a periodic orbit and a steady state.
Optimal Control of Chaos with Synchronization
Authors: Pan Shing , Yin Fuliang
Published: September 2003

Optimal control of chaotic systems is discussed in this paper. Since often only open-loop control can be obtained from the minimum principle, the output of the controlled system may be dramatically affected by noise. By using chaos synchronization, we successfully keep the output of the controlled chaotic system in the designed optimal trajectory even in noisy background. The numerical experiments of the Rössler system and the Hénon system are presented to demonstrate its effectiveness.
Secure Communications Based on Variable Topology of Chaotic Circuits
Authors: Alexeyev Alexander A. , Green Michael M.
Published: September 2003

A new technique for synchronization of chaotic circuits is proposed. This technique, based on varying a circuit's overall topology rather than varying a set of continuous parameters, offers a possible resolution to the tradeoff between security and synchronizability inherent in existing chaotic systems. The encryption key is represented by a mapping from a set of nodes to a set of switches in the circuit. This method significantly improves reliability and can be easily interfaced to digital control circuits.
Effect of System Noise on Chaotic Behavior in Roessler Type Nonlinear System
Authors: Ogata S. , Iwayama T. , Terachi S.
Published: September 2003

The present paper describes the effects of the system noise on the chaotic behavior of the Roessler type signal from the point of view of controlling chaos. We have investigated the system equations by a variable transformation method and have then carried out computer simulations to verify our theoretical consideration for a DC bias as system noise. These approaches reveal that the DC bias brings about the direct effect on the system parameters in the equations, while the random noise hardly affects the chaotic system.
Absolute Stability Theory and Master-Slave Synchronization
Authors: Curran P. F. , Suykens J. A. K. , Chua L. O.
Published: September 2003

In this note we indicate the manner in which synchronization criteria may be developed for master-slave connected Lur'e systems. For flexibility we incorporate linear, static state feedback. The criteria presented are based on the generation of Lur'e–Postnikov type Lyapunov functions for the error system.
Complex Oscillations in a Closed Thermosyphon
Authors: Rodríguez-Bernal Aníbal , Van Vleck Erik S.
Published: September 2003

The dynamics of a closed thermosyphon are considered. Using an explicit construction, obtained through an inertial manifold, exact low-dimensional models are derived. The behavior of solutions is analyzed for different ranges of the relevant parameters, and the Lorenz model is obtained for a range of parameter values. Numerical experiments are performed for three- and five-mode models.
Chaotic Advection in Large-Scale Convection
Authors: Joseph Binson
Published: September 2003

We study the kinematics of mixing and transport of "passive" particles in two-dimensional thermal convection using a low-order spectral model proposed by Howard and Krishnamurti [1986]. This model allows for a large-scale flow spanning the layer width. Mixing by chaos in the oscillatory flow is demonstrated with the help of numerical Poincaré maps. The chaos induced transport process is characterized from a relation of the form ΔX2(t)~ tm, for large t, where ΔX2(t) is the mean square distance traveled by a cloud of particles. It is shown that the transport process can be either shear-flow dominated (m=2) or Brownian type (m=1) or an intermediate type characterized by fractional exponents (1<m<2), depending on the external parameters in the problem. The intermediate process has been found to be intermittent in nature. The present results are compared with earlier studies of chaotic advection and also some experimental observations. Directions for future work are also pointed out.
Real Perturbation of Complex Analytic Families: Points to Regions
Authors: Peckham Bruce B.
Published: September 2003

This study provides some connections between bifurcations of one-complex-parameter complex analytic families of maps of the complex plane C and bifurcations of more general two-real-parameter families of real analytic (or Ck or C∞) maps of the real plane ℛ2. We perform a numerical study of local bifurcations in the families of maps of the plane given by [Formula: see text] where z is a complex dynamic (phase) variable, [Formula: see text] its complex conjugate, C is a complex parameter, and α is a real parameter. For α=0, the resulting family is the familiar complex quadratic family. For α≠ 0, the map fails to be complex analytic, but is still analytic (quadratic) when viewed as a map of ℛ2. We treat α in this family as a perturbing parameter and ask how the two-parameter bifurcation diagrams in the C parameter plane change as the perturbing parameter α is varied.

The most striking phenomenon that appears as α is varied is that bifurcation points in the C plane for the quadratic family (α=0) evolve into fascinating bifurcation regions in the C plane for nonzero α. Such points are the cusp of the main cardioid of the Mandelbrot set and contact points between "bulbs" of the Mandelbrot set. Arnold resonance tongues are part of the evolved scenario.

We also provide sufficient conditions for more general perturbations of complex analytic maps of the plane of the form: [Formula: see text] to have bifurcation points for α=0 which evolve into nontrivial bifurcation regions as α grows from zero.
The Dynamics of Planar Quadratic Maps with Nonempty Bounded Critical Set
Authors: Nien Chia-Hsing
Published: September 2003

If the critical set of a planar quadratic map is bounded and nonempty, then the range is doubly covered except the image of the critical set and the region bounded by it (Fig. 1). For a map of this type, we showed that infinity is a sink and the set of bounded orbits is nonempty hence its boundary may be regarded as a generalization of Julia sets of complex quadratic maps. In this paper, we also provided evidences to conjecture that maps of this type have infinitely many periodic points.
About Convergence of Numerical Approximations to Homoclinic Twist Bifurcation Points in Z2-Symmetric Systems in ℝ4
Authors: Van Gils S. A. , Tchistiakov V.
Published: September 2003

An algorithm to detect homoclinic twist bifurcation points in Z2-symmetric autonomous systems of ordinary differential equations in ℝ4 along curves of symmetric homoclinic orbits to hyperbolic equilibria has been developed. We show convergence of numerical approximations to homoclinic twist bifurcation points in such systems. A test function is defined on the homoclinic solutions, which has a regular zero at the codimension-two bifurcation points. This codimension-two singularity can be continued appending the test function to a three-parameter vector field. We demonstrate the use of the test function on an example of two-coupled Josephson junctions.
Numerical Continuation at Double Bifurcation Points of a Reaction–Diffusion Problem
Authors: Chien C. S. , Mei Z. , Shen C.-L.
Published: September 2003

We study double bifurcations of a reaction–diffusion problem, and numerical methods for the continuation of bifurcating solution branches. To ensure a correct reflection of the bifurcation scenario in discretizations and to reduce imperfection of bifurcations, we consider a preservation of multiplicities of the bifurcation points in the discrete problems. A continuation-Arnoldi algorithm is exploited to trace the solution branches, and to detect secondary bifurcations. Numerical results on the Brusselator equations confirm our analysis.
Ginzburg-Landau Equation for Steps on Creep Curve
Authors: Bekele Mulugeta , Ananthakrishna G.
Published: September 2003

We consider a model proposed by us earlier for describing a form of plastic instability found in creep experiments. The model consists of three types of dislocations and some transformations between them. The model is known to reproduce a number of experimentally observed features. The mechanism for the phenomenon has been shown to be Hopf bifurcation with respect to physically relevant drive parameters. Here, we present a mathematical analysis of adiabatically eliminating the fast mode and obtaining a Ginzburg-Landau equation for the slow modes associated with the steps on creep curve. The transition to the instability region is found to be one of subcritical bifurcation over a major part of the interval of one of the parameters while supercritical bifurcation is found in a narrow mid-range of the parameter. This result is consistent with experiments. The dependence of the amplitude and the period of strain jumps on stress and temperature derived from the Ginzburg-Landau equation are also consistent with experiments. On the basis of detailed numerical solution via power series expansion, we show that high order nonlinearities control a large portion of the subcritical domain.
Application of Bifurcation Theory to Subsynchronous Resonance in Power Systems
Authors: Nayfeh Ali H. , Harb Ahmad M. , Chin Char-Ming , Hamdan Anan M. A. , Mili Lamine
Published: September 2003

A bifurcation analysis is used to investigate the complex dynamics of a heavily loaded single-machine-infinite-busbar power system modeling the characteristics of the BOARDMAN generator with respect to the rest of the North-Western American Power System. The system has five mechanical and two electrical modes. The results show that, as the compensation level increases, the operating condition loses stability with a complex conjugate pair of eigenvalues of the Jacobian matrix crossing transversely from the left- to the right-half of the complex plane, signifying a Hopf bifurcation. As a result, the power system oscillates subsynchronously with a small limit-cycle attractor. As the compensation level increases, the limit cycle grows and then loses stability in a secondary Hopf bifurcation, resulting in the creation of a two-period quasiperiodic subsynchronous oscillation, a two-torus attractor. On further increases of the compensation level, the quasiperiodic attractor collides with its basin boundary, resulting in the destruction of the attractor and its basin boundary in a bluesky catastrophe. Consequently, there are no bounded motions. The results show that adding damper windings may induce subsynchronous resonance.
Experimental Implementation of Migrations in Multiple-Attractor Systems
Authors: Weigel R. , Jackson E. Atlee
Published: September 2003

It was shown by Jackson and Grosu [1995a] that entrainment control of a complex dynamic system could be achieved by using an open-plus-closed-loop (OPCL) control action. Jackson [1995] showed that, for multiple-attractor systems, it is also possible to transfer the dynamics of a system between arbitrary attractors, using this control only for a limited period of time ("migration control"). In the case of the Chua system [1986], Jackson and Grosu [1995b] showed that this migration action can be achieved using only a limited pre-recorded signal of one variable of the "target" attractor. Here we study the experimental implementation of this migration action on a Chua circuit with five attractors (fixed points, a limit cycle, and chaotic attractors) [Wu & Pivka, 1994]. It is shown that reliable migrations are possible with a pre-recorded signal of one target variable, provided that the signal is longer than a minimal time, dependent on an adjustable control strength. The minimal time for successful migration is studied for all distinct migration processes. It is also shown that this action is robust to the limited accuracy of the recorded data.
Detecting Determinism in Human Posture Control Data
Authors: Cao L. Y. , Kim B. G. , Kurths J. , Kim S.
Published: September 2003

In this paper, determinism in human posture control data is investigated using the approach of nonlinear prediction. We first comment that one should be cautious of using some statistical methods to analyze nonstationary time series. Then we test the predictability of the human posture control data with different prediction techniques, and investigate how nonstationarity and noise affect the prediction results. Different time series are tested, including data from healthy and ill persons, and different predictabilities are found in different time series.
On the Application of Geometric Singular Perturbation Theory to Some Classical Two Point Boundary Value Problems
Authors: Hayes Michael , Kaper Tasso J. , Kopell Nancy , Ono Kinya
Published: September 2003

In this tutorial, we illustrate how geometric singular perturbation theory provides a complementary dynamical systems-based approach to the method of matched asymptotic expansions for some classical singularly-perturbed boundary value problems. The central theme is that the criterion of matching corresponds to the criterion of transverse intersection of manifolds of solutions. This theme is studied in three classes of problems, linear: ∊y″+αy′+βy=0, semilinear: ∊y″+αy′+f(y)=0, and quasilinear: ∊y″+g(y) y′+f(y)=0, on the interval [0,1], where t∈[0,1], ′=d/dt, 0<∊≪1, and general boundary conditions y(0)=A, y(1)=B hold. Chosen for their relatively simple structure, these problems provide a useful introduction to the methods of geometric singular perturbation theory that are now widely used in dynamical systems, from reaction-diffusion equations with traveling waves to perturbed N-degree-of-freedom Hamiltonian systems, and in applications to a variety of fields.
Lyapunov Graph for Two-Parameters Map: Application to the Circle Map
Authors: de Figueiredo J. C. Bastos , Malta C. P.
Published: September 2003

In a Lyapunov graph the Lyapunov exponent, λ, is represented by a color in the parameter space. The color shade varies from black to white as λ goes from -∞ to 0. Some of the main aspects of the complex dynamics of the circle map (θn+1=θn+Ω+(1/2π)K sin (2πθn)( mod 1)), can be obtained by analyzing its Lyapunov graph. For K>1 the map develops one maximum and one minimum and may exhibit bistability that corresponds to the intersection of topological structures (stability arms) in the Lyapunov graph. In the bistability region, there is a strong sensitivity to the initial condition. Using the fact that each of the coexisting stable solution is associated to one of the extrema of the map, we construct a function that allows to obtain the boundary separating the set of initial conditions converging to one stable solution, from the set of initial conditions converging to the other coexisting stable solution.
Passivity Based Design of Synchronizing Systems
Authors: Pogromsky A. Yu.
Published: September 2003

A particular problem of controlled synchronization of nonlinear systems is considered. Minimum conditions for producing such synchronization are presented and it is shown that these conditions also provide the feedback passivity of the overall system. It is also demonstrated that the conditions of feedback passivity allow one to design an adaptive synchronizing control law which ensures global synchronization in the cases when the systems to be synchronized have different parameters. Additionally it is demonstrated that the approach presented for the design of synchronizing systems allows one to cope with external disturbances in order to protect the systems from loss of synchrony.
Analytic Solutions of the Cvitanović–Feigenbaum and Feigenbaum–Kadanoff–Shenker Equations
Authors: Briggs K. M. , Dixon T. W. , Szekeres G.
Published: September 2003

The Cvitanović–Feigenbaum (CF) equation arising in the universal scaling theory of iterated maps of the real line has strong links with the classical Schröder and Abel functional equations. This link is exploited to obtain information about the analytic solutions, and specifically the singular solution, of the CF equation, providing an alternative description of the latter to that of Eckmann and Wittwer. We obtain an accurate numerical approximation to this singular solution, using special techniques to handle the divergent series. This accuracy is a substantial improvement on previous estimates of the solution, and of the associated asymptotic feigenvalues α and δ. The solutions of the Feigenbaum–Kadanoff–Shenker equation for universal scaling in circle maps are shown to yield to the same analysis, producing accurate numerical values for the associated α and δ.
Homoclinic and Heteroclinic Bifurcations Close to a Twisted Heteroclinic Cycle
Authors: Zimmermann Martín G. , Natiello Mario A.
Published: September 2003

We study the interaction of a transcritical (or saddle-node) bifurcation with a codimension-0/codimension-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reaction–diffusion equation. The manifold organization is such that two branches of homoclinic orbits to each fixed point are created when varying the two parameters controlling the codimension-2 loop. It is shown that the homoclinic orbits may become degenerate in an orbit-flip bifurcation. We establish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurcate in an inclination-flip bifurcation, where a Smale's horseshoe is found.
Bifurcations and Chaos in a Model Biochemical Reaction Pathway
Authors: Mohan T. R. Krishna
Published: September 2003

Control mechanisms in the form of positive and negative feedback loops are responsible for the sensitivity and stability in the coherent behavior of the spatio-temporal organization in living cells. Models of these networks involving such feedback mechanisms have been shown to exhibit a rich spectrum of dynamical behaviors. A network involving both positive and negative feedbacks was earlier investigated by Sinha and Ramaswamy [1987]. We obtain a phase diagram of the possible dynamical behaviors for this model. Further, we investigate the origin and properties of the complex oscillations in the model. A simpler system is derived and shown to possess similar dynamical behaviors. Avenues for further investigation of the system with respect to relevant variations in some of the parameter values are suggested.
Restricting Chaos to a Small Region in the Phase Space
Authors: Wu Zhihua , Zhu Zhaoxuan , Zhang Chengfu
Published: September 2003

The idea of restricting chaos in dissipative systems to a small region in the phase space is proposed. The possibility of realization of this idea is demonstrated by applying a simple method summed up from computer simulations successfully to three different dynamical systems. It is found that not only does the trajectory of the controlled system occupy a region smaller than that of the uncontrolled chaotic system, the corresponding attractor of the Poincaré map is also smaller than that of the uncontrolled system. In addition, but also the maximum Lyapunov characteristic exponent of the system is greatly lowered.
Absolute Controllability Condition for Probabilistic Control of Chaos
Authors: Antoniou I. , Basios V. , Bosco F.
Published: September 2003

We derive an absolute condition for probabilistic control [Antoniou et al., 1996] of the unstable dynamics of the β-adic Renyi map. The controllability condition over the parameters of the probabilistic dynamical system is valid for a wide class of observables and is obtained from the fixed point solution of the associated Markov operator for measures.
An Algorithm for the Computation of Preimages in Noninvertible Mappings
Authors: Nien Chia-Hsing , Wicklin Frederick J.
Published: September 2003

For discrete dynamical systems generated by iterating a diffeomorphism, every point in the phase space has a unique preimage and it is straightforward to compute geometric structures such as inverse orbits and one-dimensional stable manifolds of periodic points. For noninvertible mappings, however, some points have multiple preimages; others may have no preimages. This makes the computation of inverse orbits difficult, because accurate computations require global knowledge about the way the mapping folds and pleats phase space.

In this article we use ideas from singularity theory to examine the geometry of noninvertible mappings. We use the geometry to derive a computational algorithm for efficiently computing preimages in noninvertible mappings.
Chaotic Vibrations of the One-Dimensional Wave Equation due to a Self-Excitation Boundary Condition.
Authors: Chen Goong , Hsu Sze-Bi , Zhou Jianxin
Published: September 2003

The nonlinear reflection curve due to a van der Pol type boundary condition at the right end becomes a multivalued relation when one of the parameters (α) exceeds the characteristic impedance value (α=1). From stability and continuity considerations, we prescribe kinematic admissibility and define hysteresis iterations with memory effects, whose dynamical behavior is herein investigated. Assume first that the left end boundary condition is fixed. We show that asymptotically there are two types of stable periodic solutions:

(i) a single period-2k orbit, or

(ii) coexistence of a period-2k and a period-2(k+1) orbits,

where as the parameter α increases, k will also increase and assume all positive integral values. Even though unstable periodic solutions do appear, there is obviously no chaos.

When the left end boundary condition is energy-injecting, however, we show that for a certain parameter range a shift sequence of subintervals of an invariant interval can be constructed and, therefore, chaos appears. Numerical simulations of chaotic and nonchaotic phenomena are also illustrated.
Analytics of Bifurcation
Authors: Phillipson Paul E. , Schuster Peter
Published: September 2003

Oscillations described by autonomous three-dimensional differential equation systems display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario, the key instability is usually an initial bifurcation from a single period oscillation to its subharmonic of period two, or the reverse. An asymptotic averaging formalism is introduced by means of an example which permits an analytic development of the bifurcation dynamics, and in particular, prediction of the onset of periods 1 ↔ 2 bifurcations in terms of the system control parameters.
Globalizing Two-Dimensional Unstable Manifolds of Maps
Authors: Krauskopf Bernd , Osinga Hinke
Published: September 2003

We present an algorithm for computing the global two-dimensional unstable manifold of a hyperbolic fixed point or a normally hyperbolic invariant circle of a three-dimensional map. The global stable manifold can be obtained by considering the inverse map. Our algorithm computes intersections of the unstable manifold with a finite number of leaves of a chosen linear foliation. In this way, we obtain a growing piece of the unstable manifold represented by a mesh of prescribed quality. The performance of the algorithm is demonstrated with several examples.
On Overparametrization of Nonlinear Discrete Systems
Authors: Mendes Eduardo M. A. M. , Billings S. A.
Published: September 2003

One of the subjects which has received a great deal of attention is the overparametrization problem. It is known that the dynamical performance of the model representations deteriorates if the respective model structure is too complex. This paper investigates the problem of model overparametrization. Two new types of overparametrization, fixed-point and dimension overparametrization, are introduced and based upon this a new procedure for improving structure detection of nonlinear models is developed. This procedure uses all the information from the cluster cancellation and the location of the fixed points. Numerous examples are given to illustrate the ideas.
Reconstructing Nonlinear Dynamics by Extended Kalman Filtering
Authors: Walker David M. , Mees Alistair I.
Published: September 2003

We investigate the use of the extended Kalman filter as a tool for the parameter estimation of radial basis function models. We show that the method is best used as an add-on to other estimation techniques such as subset selection of the centers using a minimum description length criterion rather than as a stand-alone method. We also show that the covariance matrix returned by the extended Kalman filter can be used to calculate Rissanen's minimum description length criterion more easily. Illustrative examples are given for data from the Ikeda map and two time series of real world experimental systems.
Stochastic Resonance in the Information Capacity of a Nonlinear Dynamic System
Authors: Godivier Xavier , Chapeau-Blondeau François
Published: September 2003

We consider a nonlinear bistable dynamic system governed by the quartic potential with two-state quantization at the output — the earliest system to have revealed the phenomenon of periodic stochastic resonance. We devise a scheme in which this system is used to transmit a broadband aperiodic informative signal. With this scheme, we demonstrate that the system can be operated as a memoryless symmetric binary channel, and we develop the characterization of the transmission up to the evaluation of the input–output information capacity of this channel. We show that a regime exists where the information capacity can be increased by means of noise addition, a property we interpret as a form of aperiodic stochastic resonance. In addition, we demonstrate that a positive input–output gain in the efficacy of the signal recovery can be obtained with the stochastic resonator, compared to the recovery that would directly operate on the input signal-plus-noise mixture.
Bifurcation of Locked States in Stochastic Systems
Authors: Kim Seunghwan , Park Seon Hee , Ryu Chang Su
Published: September 2003

A stochastic system with a vanishing diffusion at deterministic fixed points is analytically studied. Above the critical noise intensity, the noise stabilizes the deterministic unstable fixed points. The system is locked at one of the deterministic stable or unstable fixed points with probabilities which depend on the noise intensity and an initial condition. An example is studied numerically showing the bifurcation of locked states.
Bifurcation and Chaos Response of a Nonlinear Cracked Rotor
Authors: Jibing Zheng , Zan Mi , Guang Meng
Published: September 2003

The dynamic responses of a cracked rotor affected by nonlinear whirl speed are investigated, with particular focus on the behaviors of bifurcation and chaos. A great deal of numerical simulations show that the system has many nonlinear dynamic behaviors, such as period doubling bifurcation, quasiperiodic responses and chaotic motions at the speed ratio near 1/2Ωc and 2/3Ωc when the stiffness change ratio ΔK is large. Periodic vibrations turn to quasiperiodic motions by two different ways at speed ratio 1/2Ωc and 2/3Ωc, one bifurcates a low frequency precession component, and the other bifurcates from fraction harmonic component. With further increasing of ΔK, chaotic vibration occurs first at the speed ratio of 2/3Ωc, the response turns to chaos along various routes, it may be from period doubling to chaos, from quasiperiod to chaos, or relating to period-3 response which may turn to other kinds of periodic motions as initial values changed. When ΔK is very large, chaos from quasiperiod is found near speed ratio 1/2Ωc. The results of this paper show that some signals which were always considered as random noise and filtered off may represent the chaotic characteristics of system, and can be used in future fault diagnosing of rotating machinery.
Symbolic Computation of Secondary Bifurcations in a Parametrically Excited Simple Pendulum
Authors: Butcher Eric A. , Sinha S. C.
Published: September 2003

A symbolic computational technique is used to study the secondary bifurcations of a parametrically excited simple pendulum as an explicit function of the periodic parameter. This is made possible by the recent development of an algorithm which approximates the fundamental solution matrix of linear time-periodic systems in terms of system parameters in symbolic form. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet transition matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation boundaries in the parameter space. Since this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. By repeating the linearization and computational procedure after each bifurcation of an equilibrium or periodic solution, it is shown how the bifurcation locations as well as the new linearized equations may be obtained in closed form as a function of the periodic parameter. Bifurcation diagrams are constructed and the results are compared with those obtained elsewhere using the point mapping method.
Experimental Results of Impulsive Synchronization Between Two Chua's Circuits
Authors: Panas Andrey I. , Yang Tao , Chua Leon O.
Published: September 2003

Impulsive synchronization of chaotic dynamic systems has some important applications to chaotic secure communication and chaotic spread-spectrum communication systems. In this paper we present some experimental results on impulsive synchronization between two Chua's circuits. In our experiments, only one synchronizing impulse sequence is transmitted. The robustness of impulsive synchronization with respect to variations in the frequency and the width of impulses is studied. Experimental results show that robust impulsive synchronization can be achieved under noisy conditions and a 2% parameter mismatch between the driving system and the driven system. We also found that an amplified impulse sequence with a gain greater than unity can make the impulsive synchronization more robust. Moreover, we found that impulsive synchronization can be achieved with very narrow impulses.
Synchronization of Chaotic Systems and Transmission of Information
Authors: Hasler Martin
Published: September 2003

An overview over the methods currently under investigation for the transmission of information hidden in a chaotic signal is given. First, the notion of synchronization is discussed, then the coupling of two systems to achieve synchronization is presented and then four methods to mix the information signal with a chaotic carrier are described. Finally, the problem to send several information bearing chaotic signals through the same communication channel is posed as a challenge.
Thermodynamic and Kinetic Description of the Second Order Phase Transitions
Authors: Klimontovich Yu. L.
Published: September 2003

A thermodynamic and kinetic description of phase transitions for the model of ferroelectrics on the basis of kinetic equations for the distribution function of values of the "order parameter", coordinates and time is discussed.

For one-domain ferroelectrics the self-consistent approximation for the first moment is used. The kinetic equation is reduced to the relaxation Ginsburg–Landau equation. The susceptibility is governed by the Curie law and there is an increase in heat capacity.

Calculations are carried out for one-domain and polydomain ferroelectrics. In the first case the self-consistent approximation for the first moment is used. In the second one the self-consistent approximation for the second moment is carried out. In the last case there is a jump in susceptibility. The heat capacity is governed by the Curie law.

It is shown also that the Ornstein–Zernike formula is valid not for the space correlator of fluctuations, but only for the temporal spectral density of the space correlator at zero frequency.

In the kinetic theory of the phase transition all physical characteristics at the critical point have finite values. Thus, the problem of "infinities" is absent.
Bifurcation Analysis of Zero-Dispersion Nonlinear Resonance
Authors: Mannella R. , Soskin S. M. , McClintock P. V. E.
Published: September 2003

The problem of zero-dispersion nonlinear resonance — a phenomenon that can occur in a periodically-driven nonlinear oscillator whose eigenfrequency as a function of energy possesses an extremum — has been formulated in general for both the dissipative and nondissipative situations. A complete bifurcation analysis and classification of period-1 orbits is presented. The significance of bifurcations for the onset of chaos in the system, and for fluctuations in the presence of external noise, is discussed.
On Bifurcation Control in Time Delay Feedback Systems
Authors: Basso M. , Evangelisti A. , Genesio R. , Tesi A.
Published: September 2003

The paper addresses bifurcations of limit cycles for a class of feedback control systems depending on parameters. A set of simple approximate analytical conditions characterizing all the generic limit cycle bifurcations is determined via a first-order harmonic balance analysis in a suitable frequency band. Based on the results of this analysis, an approach to limit cycle bifurcation control is proposed. In particular, an example concerning a biological delay model is developed, where a flip bifurcation control is designed via a modified Pyragas technique.
The Complex Lorenz Model: Geometric Structure, Homoclinic Bifurcation and One-Dimensional Map
Authors: Vladimirov A. G. , Toronov Vladislav Yu. , Derbov Vladimir L.
Published: September 2003

It is shown that the phase space of the complex Lorenz model has the geometric structure associated with a fiber bundle. Using the equations of motion in the base space of the fiber bundle the surfaces bounding the attractors in this space are found. The homoclinic "butterfly" responsible for the Lorenz-like attractor appearance is shown to correspond to a codimension-two bifurcation. One-dimensional map describing bifurcation phenomena in the complex Lorenz model is constructed.
Critical Dynamics of Pitch-Fork Bifurcation in a System Driven by a Fractal Sequence
Authors: Kuptsov P. V.
Published: September 2003

We consider the system, exhibiting pitch-fork bifurcation, forced by the external perturbation with fractal properties. Developing a renormalization group approach, we show that the situation is characterized by nonclassical critical exponents. These exponents appear to depend on external influence intensity and we get the analytical expressions for them. Apparently, a new kind of strange nonchaotic attractor arises in the case under consideration.
Large Fluctuations in a Periodically Driven Dynamical System
Authors: Dykman M. I. , Smelyanskiy V. N. , Luchinsky D. G. , Mannella R. , McClintock P. V. E. , Stein N. D.
Published: September 2003

Fluctuations in a periodically driven overdamped oscillator are studied theoretically and experimentally in the limit of low noise intensity by investigation of their prehistory. It is shown that, for small noise intensity, fluctuations to points in coordinate space that are remote from the stable states occur along paths that form narrow tubes. The tubes are centered on the optimal paths corresponding to trajectories of an auxiliary Hamiltonian system. The optimal paths themselves, and the tubes of paths around them, are visualized through measurements of the prehistory probability distribution for an electronic model. Some general features of fluctuations in nonequilibrium systems, such as singularities in the pattern of optimal paths, the corresponding nondifferentiability of the generalized nonequilibrium potential, and the feasibility of their experimental investigation, are discussed.
Microscopic Models and Simulations of Local Activation Processes
Authors: Ebeling Werner , Podlipchuk Victor Yu. , Sapeshinsky Mikhail G.
Published: September 2003

The classical reaction theory of Kramers describes transitions over a potential barrier (activation processes) by Fokker–Planck equations or the corresponding Langevin equations. Kramers model is based on the assumption of uncorrelated stochastic forces. In this work simple but more realistic microscopic models for transitions are developed. The physical forces leading to transitions are studied by means of molecular dynamics simulations. The investigation is restricted to two-dimensional systems. In particular we study local energy excitations of soft molecules solved in liquids. This investigation makes use of an analogy to Toda lattices. For Toda systems it was shown in earlier work that soliton excitations and soliton fusion at the soft sites may lead to local energy spots. The basic configuration studied here is a soft reacting site imbedded into a thermal bath of hard molecules. In the first part we develop a microscopic model for Kramers bistable potential well and calculate the transition rates by MD-simulations. In the following parts we study models for the dissociation of molecules and for the fusion of molecules.
Climate Dynamics as a Nonlinear Brownian Motion
Authors: Sonechkin D. M.
Published: September 2003

Based on the heat balance equation of the global climate system the well-known surface air temperature time series of the Northern and Southern hemispheres were analyzed as realizations of a fractional Brownian motion. The technique of the so-called wavelet transform was used for this purpose. The technique easily admits splitting time series of interest to statistically stationary oscillations and a trend.

Such temperature oscillations were extracted which include within themselves almost all differences between both hemispheric time series. As a result of subtraction of the oscillations from the primary hemispheric series a residual trend-like component was evaluated. The latter evidences a single warming trend of the global climate system that was started from the early 20th century.
Flicker-Noise in Processes of Solar Activity
Authors: Kostuchenko I. G. , Timashev S. F.
Published: September 2003

The time series describing the solar activity variations in the diverse solar layers are analyzed in terms of the developed method of the flicker-noise diagnostic which allows one to extract the dynamic characteristics of the process under study. The revealed results allow us to conclude that in all studied solar processes short term variations of the observational variables (with typical time less than ~3 years) reflect the behavior of some global dynamics. The dynamic characteristics turned out to be similar for the processes which are observed at the level of the solar photosphere and gradually changes in the solar atmosphere. That is why all Sun layers can be considered as a single dynamic system. We find that the short term variations of the solar neutrino capture rate are not random but are caused by some dynamic process. The analysis of solar total irradiance time series in terms of the applied method allows us to suggest that the mentioned dynamic processes exists in the solar core.
Characteristic Scales of Reconstruction Distortions
Authors: Potapov Alexei
Published: September 2003

We consider the distortions of reconstructed chaotic attractors related with the choice of reconstruction parameters (embedding dimension m and delay τ) and characteristic scales of these distortions. The "spatial" scale may be interpreted as the upper bound for neighborhood size, which is used in many algorithms such as the calculation of Lyapunov exponents. The corresponding "temporal scale" gives the upper bound for the window length spanned by reconstructed vector w=(m-1)τ — the so-called "irrelevance time" [Casdagli et al., 1991]. The analysis enabled us to propose the new algorithm of time series processing for the direct estimation of generalized entropy from time series without calculating the correlation integral.
Computer Simulations of Neuronal Signal Transduction: The Role of Nonlinear Dynamics and Noise
Authors: Braun H. A. , Huber M. T. , Dewald M. , Schäfer K. , Voigt K.
Published: September 2003

Nonlinear ionic interactions at the nerve cell membrane can account for oscillating membrane potentials and the generation of periodic neuronal impulse activity. In combination with noise, external modulation of the endogenous oscillations allows for continuous transitions between a variety of impulse patterns. Such "noisy oscillators" afford, thereby, an important mechanism of neuronal encoding as is demonstrated here with experimental data from peripheral cold receptors and corresponding computer simulations.
Wandering Motion and Co-Operative Phenomena in a Chaotic Neural Network
Authors: Makishima Makoto , Shimizu Toshihiro
Published: September 2003

A new model for associative memory with discrete time is proposed, in which chaos is taken into account and moreover the internal state of neurons depends on the past history. The network itself can search for minima of the energy successively by using the wandering motion and co-operative phenomena. By using this model we can clearly discuss the mechanism or the dynamics of the wandering motion and co-operative phenomena in a chaotic neural network to search for minima of the energy. It is shown that our model has great ability in the problem of associative memory.
A General Form for Global Dynamical Data Models for Three-Dimensional Systems
Authors: Lainscsek Claudia S. M. , Schürrer Ferdinand , Kadtke James B.
Published: September 2003

The information contained in a scalar time series and its numerical derivatives is used to construct a global model for the underlying dynamical system, using a model transformation presented previously. Here, however, we analytically determine the most general form for the transformed model in the case of a three-dimensional model ansatz. We then test this method by reconstructing global models for known chaotic dynamical systems, and comparing correlation and topological measures of the re-constructed and original systems. We also do a preliminary investigation of a real data set consisting of the sunspot number over the last 200 years.
Entropy Analysis of Noise Contaminated Sequences
Authors: Freund Jan A. , Rateitschak Katja
Published: September 2003

The entropy analysis of symbol sequences created by symbolic dynamics reveals temporal correlations present in the stationary dynamics. We investigate how noise, introduced in different ways, affects temporal correlations found in the logistic map at the period doubling accumulation point. The deterministic system under consideration yields a highly structured spectrum of conditional entropies. We will discuss how additive noise, parametric noise, a fluctuating cell boundary and a noisy transmission channel modify the unperturbed entropy spectrum.
A Geometric Approach to a Quantum Counterpart of a Saddle-Node Bifurcation in a 1:1 Resonant Perturbed Oscillator
Authors: Uwano Yoshio
Published: September 2003

In previous papers [Uwano, 1994, 1995], it was shown that a degeneracy of energy levels is a quantum counterpart of a Hamiltonian pitchfork bifurcation of periodic trajectories of a certain 1:1 resonant oscillator with two parameters. As a continuation of those papers, a quantum study is made from a geometric point of view in order to find a quantum counterpart of a saddle-node bifurcation taking place in a certain 1:1 resonant perturbed oscillator with three parameters. The torus quantization method is applied to the perturbed oscillator to show that a degeneracy of energy levels is a quantum counterpart of that bifurcation: The bifurcation set for the saddle-node bifurcation in classical theory is viewed as a "bifurcation set" for the degeneracy of energy levels in quantum theory.
A Nonhorseshoe Template in a Chaotic Laser Model
Authors: Boulant G. , Lefranc M. , Bielawski S. , Derozier D.
Published: September 2003

We have performed a topological analysis of a chaotic regime of a modulated single-mode class-B laser model, at realistic parameter values. In contrast with previous numerical and experimental studies of this type of laser, we observe a topological structure which differs from the one described by the paradigmatic horseshoe template. In view of this result, class-B lasers appear to be good candidates for the first clear characterization of a nonhorseshoe template in an experimental system.
Paired Lévy–Mandelbrot Trajectory as a Homogeneous Fractal
Authors: Uchaikin V. , Gismjatov I. , Gusarov G. , Svetukhin V.
Published: September 2003

This paper deals with the problem of using the Lévy–Mandelbrot (LM) trajectory for constructing a homogeneous random point fractal in three-dimensional space. For investigating stochastic properties of a random number of points M(R) within a radius R around an arbitrary occupied point, it is proposed to use a paired LM trajectory. It is found that for sufficiently large values of R the distribution of the random variable M(R)/ does not depend on R and can be approximated by the gamma distribution with a unique parameter depending on the fractal dimension. This property is analogous to KNO scaling of multiplicity in hadron-hadron collisions.
Game of Morphogenesis: What Can We Learn from the Pattern-Form Interplay Models?
Authors: Spirov A. V.
Published: September 2003

We approach the problem of how the information encoded in linear DNA molecule becomes translated into a three-dimensional form from position of Pattern-Form Interplay Models. The characteristic feature of these models is the existence of feedback loops from (bio)chemical pattern formation to modeling embryo form changes. In accordance with the model the system is open and changes in a pattern give rise to changes in form and these changes in form (surface geometry) cause further pattern changes, and so on.

Spontaneous pattern formation takes place in the model as primary and secondary bifurcations of nonlinear parabolic PDEs describing reaction-diffusion systems with imposed gradient. We briefly review the main results of previous works and consider the phenomenon of axis tilting as a case of symmetry breaking via secondary bifurcations. The axis tilting bifurcation occurs as a consequence of position-dependency of diffusion coefficients. The explicit demonstration of this phenomenon in Pattern-Form Interplay Models is believed to be new.