International Journal of Bifurcation and Chaos

A few mathematical problems arising in the classical synchronization theory are discussed; especially those relating to complex dynamics. The roots of the theory originate in the pioneering experiments by van der Pol and van der Mark, followed by the theoretical studies by Cartwright and Littlewood. Today, we focus specifically on the problem on a periodically forced stable limit cycle emerging from a homoclinic loop to a saddle point. Its analysis allows us to single out the regions of simple and complex dynamics, as well as to yield a comprehensive description of bifurcational phenomena in the two-parameter case. Of a particular value is the global bifurcation of a saddle-node periodic orbit. For this bifurcation, we prove a number of theorems on birth and breakdown of nonsmooth invariant tori.

Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f:I→I is chaotic, then its iterates fⁿ will display heightened oscillatory behavior or profiles as n→∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of fⁿ on subintervals. There are four distinctive cases of the growth of total variations of fⁿ as n→∞:

(i) the total variations of fⁿ on I remain bounded;

(ii) they grow unbounded, but not exponentially with respect to n;

(iii) they grow with an exponential rate with respect to n;

(iv) they grow unbounded on every subinterval of I.

We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of fⁿ are more extreme, the more complex is the behavior of the system.

The work discusses the formation of Voronoi diagrams in spatially extended nonlinear systems taking experimental and theoretical results into account. Concerning experimental systems a number of chemical systems used previously as prototype chemical processors and a barrier gas-discharge system are investigated. Although the underlying microscopic processes are very different, both types of systems show self-organized Voronoi diagrams for suitable parameters. Indeed certain chemical systems exhibit Voronoi diagrams as an output state for two distinct sets of parameters one that corresponds to the interaction of stable forced trigger waves and the other that corresponds to the spontaneous initiation and interaction of waves due to point instabilities in the system. In the case of the chemical systems front initiation, propagation and interaction (annihilation) are the primary mechanisms for Voronoi diagram formation, in the case of the barrier gas-discharge system regions of vanishing electric field define the medial axes of the Voronoi diagram. On the basis of cellular automata models the general concept of the formation of Voronoi diagrams has been explained, and related mechanisms have been simulated. Another intuitive approach towards the understanding of self-organized Voronoi diagrams has been given on the basis of reaction–diffusion models explaining the formation of Voronoi diagrams as a result of the mutual interactions of trigger fronts. The variety of systems exhibiting Voronoi diagrams as stationary states indicates that Voronoi diagrams are a generic and natural pattern formation phenomenon.

This paper studies a class of coupled Van der Pol (CVDP) cellular neural networks (CNNs) that can be realized via a coupled fourth-order circuit with two synaptic currents. The local activity theory, developed by Chua in 1997, is applied to study the CVDP CNN, thereby revealing that the bifurcation diagram of the CVDP CNN has a local activity domain with an edge of chaos, as well as a one-dimensional locally passive domain. Although no chaotic phenomena have been identified in simulations, many complex dynamical behaviors have been observed, such as the co-existence of one-periodic, divergent, and convergent orbits, at the edge of chaos.

Of concern is a two-dimensional map T of the form T(x,y)=(y,F(y)-bx). Here F is a three-piece linear map. In this paper, we first prove a theorem which states that a semiconjugate condition for T implies the existence of Smale horseshoe. Second, the theorem is applied to show the spatial chaos of one-dimensional Cellular Neural Networks. We improve a result of Hsu [2000].

This paper formulates the model and then studies its dynamics of a system of linearly and diffusively coupled identical delayed neural networks (DNNs), which is generalization of delayed Hopfied neural networks (DHNNs) and delayed cellular neural networks (DCNNs). In particularly, a simple yet generic sufficient condition for global synchronization of such coupled DNNs is derived based on the Lyapunov functional methods and Hermitian matrix theory. It is shown that global synchronization of coupled DNNs is ensured by a suitable design of the coupling matrix and the inner linking matrix. Furthermore, the result is applied to some typical chaotic neural networks. Finally, numerical simulations are presented to demonstrate the effectiveness of the approach.

Spiking and bursting observed in nerve membranes seem to be important when we investigate information representation model in the brain. Many topologically different bursting responses are observed in the mathematical models and their related bifurcation mechanisms have been clarified. In this paper, we propose a design method to generate bursting responses in FitzHugh–Nagumo model with a simple periodic external force based on bifurcation analysis. Some effective parameter perturbations for the amplitude of the external input are given from the two-parameter bifurcation diagram.

The fold-flip bifurcation occurs if a map has a fixed point with multipliers +1 and -1 simultaneously. In this paper the normal form of this singularity is calculated explicitly. Both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf bifurcation of ODEs. Two examples are presented, the generalized Hénon map and an extension of the Lorenz-84 model. In the latter example the first-, second- and third-order derivatives of the Poincaré map are computed using variational equations to find the normal form coefficients.

Hopf bifurcations lead to limit circles that exhibit oscillatory behaviors in nonlinear dynamical systems. Building a specified limit circle into nonlinear systems by control enables us to achieve preferred dynamical performance in the systems. This paper reports a general control method for generating a variety of specified Hopf bifurcations at a desired parameter location. The proposed control law is applicable to nonresonance, weak resonance, strong resonance, or degenerate cases. It allows us to flexibly operate and manipulate the control of Hopf bifurcations. The discrete washout filters are used for the feedback controller. The feedback control gains are derived from the critical conditions and stability conditions of Hopf bifurcations by means of the center manifold method and normal form technique. Numerical experiments indicate that the control method is effective.

Tracking a periodic orbit of a chaotic attractor has been a considerable challenge since it often encounters numerical sensitivity. In this paper, a lifted prediction-based low-gain digital controller obtained from a predesigned high-gain analog controller is developed and applied to the hybrid chaotic system. The new proposed digital tracker with a relatively longer sampling time is insensitive to numerical errors and performs better than the one developed via the existing prediction-based digital redesign method. The Chen and Chua attractors are used as illustrative examples to demonstrate the effectiveness of the proposed lifted digital redesign method.

In this paper the anti-control technique of chaos to systems continuous with respect to the state variable using a time-delay feedback technique introduced by Wang, Chen and Yu, is adapted to a class of dynamical systems discontinuous with respect to the state variable. The considered discontinuous initial value problem is transformed into a differential inclusion using the Filippov regularization. Then, certain results on existence and uniqueness of solutions to differential inclusions are used to define our class of discontinuous dynamical systems. Afterwards, introducing an adequate concept of derivative for the considered discontinuous functions, we show that the algorithm for continuous dynamical systems can be adapted to our class of discontinuous problems. Three examples are given.

In this paper, the dynamics of maps representing classes of controlled sampled systems with backlash are examined. First, a bilinear one-dimensional map is considered, and the analysis shows that, depending on the value of the control parameter, all orbits originating in an attractive set are either periodic or dense on the attractor. Moreover, the dense orbits have sensitive dependence on initial data, but behave rather regularly, i.e. they have quasiperiodic subsequences and the Lyapunov exponent of every orbit is zero. The inclusion of a second parameter, the processing delay, in the model leads to a piecewise linear two-dimensional map. The dynamics of this map are studied using numerical simulations which indicate similar behavior as in the one-dimensional case.

We consider the behavior of piecewise isometries in Euclidean spaces. We show that if n is odd and the system contains no orientation reversing isometries then recurrent orbits with rational coding are not expected. More precisely, a prevalent set of piecewise isometries do not have recurrent points having rational coding. This implies that when all atoms are convex no periodic points exist for almost every piecewise isometry.

By contrast, if n≥2 is even then periodic points are stable for almost every piecewise isometry whose set of defining isometries are not orientation reversing. If, in addition, the defining isometries satisfy an incommensurability condition then all unbounded orbits must be irrationally coded.

The generalized asymptotic inertial manifold (GAIM) of the damped driven Sine–Gordon equation is put forward to govern the long-term behavior of the full partial differential equation (PDE). We then study qualitatively the ordinary differential equation (ODE) by the singular perturbation-theory, which results from restricting the damped driven Sine–Gordon equation to its GAIM. Firstly an analytical criterion for the existence of the homoclinic orbit resulting in chaos is given. Further, the existence of the pulse orbits is showed under the same parametric values as those used in the previous numerical experiments. In our viewpoint these results reflect just the breather competition behavior observed numerically in the Sine–Gordon equation. By comparing with the earlier results obtained in the two-mode Fourier truncation system of the damped driven Sine–Gordon equation, we think that a reasonable discretization reduction from PDE to ODE is very important in the study of dynamics in the infinite dimensional dynamical systems.

A numerical study of an interior crisis of a large-amplitude Alfvén wave described by the driven-dissipative derivative nonlinear Schrödinger equation, in the low-dimensional limit, is reported. An example of Alfvén interior crisis is characterized using the unstable periodic orbits and their associated invariant stable and unstable manifolds in the Poincaré plane. We suggest that this type of chaotic transition can be observed in space and laboratory plasmas.

This research is part of an ongoing effort to construct "synthetic velocity" subgrid-scale (SGS) models using discrete dynamical systems (DDSs) for use in large-eddy simulations of turbulent flows. Here we will outline the derivation of the two-dimensional (2-D) "Poor Man's Navier–Stokes" (PMNS) equation from the 2-D, incompressible Navier–Stokes equation to be used in such models and report results from subsequent numerical investigations. In our results emphasis is placed on the effects of initial conditions on the dynamics of the 2-D PMNS equation, using such modes of investigation as regime maps, basins of attraction diagrams, phase portraits, time series and power spectra. The most important findings of this investigation concern applicable ranges of bifurcation parameters, causes and effects of symmetries seen in solutions of the PMNS equation, and the suitability of the methods of investigation used here.

Nonideal mixed yet structured chemical reaction systems tend to exhibit spatial inhomogeneities in their conversion kinetics. Reactive two-phase systems invariably involve coupling phenomena between transport processes and the chemical reaction, at least in the phase boundary region. The spatiotemporal structure of the dispersion determines the contact area as well as contact times, local penetration depths, thus setting the boundary conditions for chemical conversion kinetics and mass transport processes. A monodispersive foam is a system of reduced complexity for investigating spatial and temporal reaction kinetics in conjunction with transport processes.

Producing a well-defined foam with equally sized bubbles is a difficult task. After solving this problem, we introduce chemical reactive species into the foams and observe their physicochemical behavior.

The mesoscopic dynamic behavior of the Oregonator model of the Belousov–Zhabotinsky chemical reaction is investigated as the model system experiences a supercritical Hopf bifurcation from focus to limit cycle oscillation. The study is performed by stochastically simulating the corresponding chemical master equation. Comparing the mesoscopic dynamic results with those obtained by the macroscopic dynamics, we find in the mesoscopic description a new type of oscillating state, in which large-amplitude oscillations and small-amplitude oscillations appear randomly alternately. This new state comes out spontaneously within a certain region called Hopf bifurcation range by us. In the mesoscopic description, the Hopf bifurcation point cannot be shown, being replaced by a Hopf bifurcation range. Furthermore, the applications of this new oscillating state to internal signal stochastic resonance are pointed out.

A two-variable bistable system modeling the selection between competing options is considered in the presence of additive white noise, representing environmental or internal variability. It is shown that when the strengths of the noises acting on the evolution equations of the two variables are different a symmetry-breaking process takes place, whereby the state in which the excess variable is perturbed by the strongest noise becomes predominant.

In this paper, logistic mapping with a time-dependent system-parameter (referred as "LMTD") is proposed. In various choices of time dependence, the periodic one has been tried in order to investigate the dynamical properties of LMTD. In certain parameter regions, two different attractors coexist depending on the initial values, for instances, two different chaotic attractors, two different periodic attractors, or two periodic/chaotic attractors. In addition, the whole configuration space of the initial values forms basins of attractions of which structures indicate self-similarity.

We compute explicitly the irreducible polynomial corresponding to the bifurcation point B₄ of the logistic map. This settles in the positive recent two conjectures by Bailey and Broadhurst.

This letter shows some counter-intuitive simulation results that the symbolic sequences and the state variables of a digital filter with two's complement arithmetic and arbitrary initial conditions and order will be eventually zero when all the filter parameters are even numbers, no matter the system matrix of the filter is stable or not.

Simple polynomial unimodal maps which show robust chaos, that is, a unique chaotic attractor and no periodic windows in their bifurcation diagrams, are constructed. Their invariant distributions and Lyapunov exponents are examined.

Unusual behavior of dynamic systems with a control parameter is analyzed. Bifurcation induced qualitative changes in phase portraits of such systems are quite routine and ensure efficient operation. This class of systems includes ships with high maneuvering capabilities, aircraft and controlled underwater vehicles designed to be unstable in steady-state motion that are interesting in terms of applications.

Bifurcations may generate tracks of bifurcation memory that are "indicators" of regions of reduced controllability referred to as phase spots. The transition process in the phase spot is estimated qualitatively as a universal dependence of the index of loss of controllability on the control parameter.

The proposed approach has allowed us to predict and investigate emergency manifestations of the bifurcation memory effect occurring during routine maneuvering.

An upper bound Z(2,n)≤[n/2]+6[(n-1)/2]+5 is derived for the number of zeros of Abelian integrals Ĩ(h)=∮_γ(h)g(x,y)dx-f(x,y)dy on the open interval (0,1/6), where γ(h) is an oval lying on the algebraic curve H_λ=(1/3)x³+(1/2)x²+λy³-y²=h, f(x,y), g(x,y) are polynomials of x and y, and n=max{deg f(x,y), deg g(x,y)}. The proof exploits the expansion of near λ=0.

The model of the cardiac tissue as a conductive system with two interacting pacemakers and a refractory time is proposed. In the parametric space of the model the phase locking areas are investigated in detail. The obtained results make possible to predict the behavior of excitable systems with two pacemakers, depending on the type and intensity of their interaction and the initial phase. Comparison of the described phenomena with intrinsic pathologies of cardiac rhythms is given.

This paper is concerned with synchronization of chaotic systems. We consider a drive-response type of synchronization via a scalar transmitted signal. Given some structural conditions of chaotic systems, a sliding observer-based response system is constructed to synchronize with the drive system within a finite time. Moreover, if the observer gain is judiciously chosen, robustness with respect to bounded parameter variations is guaranteed. To improve furthermore the performance of the response system, unknown parameters are adaptively estimated in conjunction with the sliding observer. To demonstrate the efficiency of the proposed approach numerical simulation results are presented.

In this Letter, we propose a new synchronization principle for a class of Lur'e systems. We design, using only a single scalar output, a possible class of observers to detect whether two dynamical systems exhibit identical oscillations. The proposed method is then applied to suggest a means to secure communications. The transmitter contains a chaotic oscillator with an input that is modulated by the information signal. The receiver is a copy of the transmitter driven by a synchronization signal. The advantage of this method over the existing one is that the synchronization time is explicitly computed. An illustrative example of the cubic Chua's circuit is given to show the effectiveness of the proposed approach.

In this paper, generalized synchronization (GS) on a linear manifold between two algebraic nonlinear systems is considered. In particular, conditions for generalized synchronization on a linear manifold are derived and a general approach to the design of a nonlinear receiver that operates under GS conditions is presented. The coupling signal is unidirectional and depends on the state variables of the drive. Moreover, a method that ensures the occurrence of linear GS is discussed and applied to standard chaotic systems (i.e. Lorenz system and Chua's circuit).

A relationship between the loop gain and nonlinear behavior of a transformer-coupled oscillator is established in this paper. With increase of the loop gain, the system undergoes a series of changes in its dynamical behavior, i.e. no oscillation, near-sinusoidal, period-doubling, chaotic and squegging oscillations. It is expected that the approach of loop gain analysis can be universally applied to investigating the nonlinear behavior, especially chaos, in other oscillators.

In this study, a chaotic circuit suitable for an integrated circuit is proposed. The circuit consists of two CMOS ring oscillators and a pair of diodes. By using a simplified model of the circuit, the mechanism of generating chaos is explained and the exact solutions are derived. The exact expressions of the Poincaré map and its Jacobian matrix make those possible to confirm the generation of chaos using the Lyapunov exponents and to investigate the related bifurcation phenomena.