International Journal of Bifurcation and Chaos

Heart rhythms exhibit the following interesting phenomena: First, rhythmic cells such as sinus node cells cease their rhythmic activity when a single brief current pulse of the right magnitude is given to the cell at the right time. Second, a premature pulse can initiate a tachycardia (i.e., reentrant arrhythmia) from a quiescent atrial tissue. This tachycardia can be sustained for hours until another brief current pulse is given. Third, ECG recording sometimes shows alternans in electrical activity prior to ventricular tachycardia. Some suggested that this alternans is the first bifurcation in a period doubling sequence. Fourth, a certain class of antiarrhythmic drugs may increase the chance of sudden cardiac death for patients with a history of myocardial infarction. Fifth, when an "adjustable" tricuspid ring is shortened, the conduction velocity of a reentrant wave becomes oscillatory. When the ring size is shortened further, reentry terminates after several oscillatory cycles. Sixth, sustained reentry arises in a tricuspid ring when Class IC drugs (i.e., which exclusively block the sodium channel) are added. On the other hand, Class III (which exclusively blocks the potassium channel) can terminate sustained reentry in the tricuspid ring. In this tutorial review, we explain how one can utilize a bifurcation analysis to explain all these interesting phenomena involved in heart rhythms.

Spiral patterns have been observed experimentally, numerically, and theoretically in a variety of systems. It is often believed that these spiral wave patterns can occur only in systems of reaction–diffusion equations. We show, both theoretically (using Hopf bifurcation techniques) and numerically (using both direct simulation and continuation of rotating waves) that spiral wave patterns can appear in a single reaction–diffusion equation [in u(x, t)] on a disk, if one assumes "spiral" boundary conditions (u_r = mu_θ). Spiral boundary conditions are motivated by assuming that a solution is infinitesimally an Archimedian spiral near the boundary. It follows from a bifurcation analysis that for this form of spirals there are no singularities in the spiral pattern (technically there is no spiral tip) and that at bifurcation there is a steep gradient between the "red" and "blue" arms of the spiral.

Reaction–diffusion and cell-chemotaxis mechanisms have been widely used as models for biological pattern formation. Applications require the formation of specific patterns. Both mechanisms involve local reaction dynamics and diffusion. In addition, the cell-chemotaxis mechanism involves chemotactic movement in response to an external gradient. It has been considered that chemotaxis is like negative diffusion which has a destabilizing effect. In this paper, we use a general weakly nonlinear analysis and numerical simulations to divide the parameter space into different domains giving rise to specific classes of spatial patterns, namely, spots and stripes, compare the robustness and sensitivity of the different models, and discuss spatial patterns intermediate between stripes and spots. We show that chemotaxis is not always a destabilizing factor — it can also have positive diffusion effect.

In this paper, we investigate via physical experiments and computer simulations the response of Chua's circuit under periodic external forcing. The amplitude and frequency of the external forcing form two additional control parameters, and we present bifurcation diagrams on the plane of these two parameters. We investigate the behavior of Chua's circuit due to external forcing when the unforced circuit is in various dynamical regimes. We finally compare our results with those obtained from a nonautonomous oscillator with an inertial nonlinearity.

This paper investigates how the choice of the sampling time affects the identification of nonlinear models. It is shown how the choice of the sampling interval can affect structure selection and parameter estimation in different ways so that a compromise is required to properly select an appropriate sampling time. The use of a nonlinear correlation function to aid in the choice of the sampling rate is investigated. Several numerical examples are included to illustrate the main points of the paper.

The extent to which daily return data from the Athens' Stock Exchange Index exhibits non-linear and chaotic characteristics is investigated by employing multiple tests from both the Economics' and the Natural Sciences' fields. The BDS test and Rescale Range (R/S) analysis provide evidence for non-linearity and fractality due to noisy chaos, respectively, while methods of chaotic dynamics, like correlation dimension and related tests, as well as, Lyapunov exponents, give consistent results, which do not rule out the possibility of deterministic chaos. The occurrence of a simple low-dimensional attractor, is not supported. However, noise filtered data by the use of SVD analysis and FIR filters, gives reliable evidence for the existence of an underlying dynamical system with a limited number of degrees-of-freedom.

The statistical properties of a wide class of 1D piecewise linear Markov maps are compiled. The method used enables one to address analytically the inverse problem of designing a map with a prescribed correlation function. This class of piecewise linear maps is then used as a system of reference to analyze non-Markov piecewise linear maps and to design maps with given invariant measure and correlation function.

We study the spectral statistics of three triangular quantum billiards. We show that for an ergodic billiard the nearest neighbor spacings distribution in the low energy regime is strongly influenced by a neighboring integrable domain (equilateral triangle). With increasing energy we observe a transition towards a more chaotic-like pattern. We also show that for all three triangles the properties of the spectral distributions depend heavily on the energy range considered. This is specially true for the spectral rigidity, because of its essentially local character. The features of the distributions do not depend strongly on the genus of classical billiards.

We consider one-dimensional two state cellular automata and their orbital patterns. We are particulary interested in initial states and their orbital pattern which are invariant under a certain coarse-graining operation. We show that invariant initial states are automatic. Moreover, we numerically study the complex nature of the initial states and their invariant orbits.

Lorenz type attractors are constructed by singular perturbations. The technique is based on a new type of fast switching subsystem and the method of Deng [1994]. New attractors of the Lorenz type are found when parameters in the system are varied. The singular limit attractors are also presented in terms of one-dimensional mappings.

In an array of coupled oscillators, synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries, and this leads to the idea of symmetry breaking bifurcations of chaotic attractors. Finally, the results and ideas are illustrated for several systems of coupled oscillators.

Two resistively coupled Chua's circuits are studied in this paper. We consider the case when the two circuits are identical, and the case when there is a detuning between the basic frequencies of the two circuits. In the first case, the multistability (i.e. the coexistence of attractors) evolution scenario is studied. In the second case, we study the influence of detuning on the multistability, and on the mutual synchronization in the chaotic regimes. We present bifurcation diagrams of the main synchronization regions. We also compare the dynamics of coupled Chua's circuits with the dynamics of other coupled systems.

A comprehensive bifurcation analysis of a low-order atmospheric circulation model is carried out. It is shown that the model admits a codimension-2 saddle-node-Hopf bifurcation. The principal mechanisms leading to the appearance of complex dynamics around this bifurcation are described and various routes to chaotic behavior are identified, such as the transition through the period doubling cascade, the breakdown of an invariant torus and homoclinic bifurcations of a saddle-focus. Non-trivial limit sets in the form of a chaotic attractor or a chaotic repeller are found in some parameter ranges. Their presence implies an enhanced unpredictability of the system for parameter values corresponding to the winter season.

We consider the problem of determining the stochasticity transition value in nearly-integrable mappings. We perform explicitly a canonical transformation, which conjugates the original mapping to an integrable one, up to a given order in the perturbing parameter. Then we derive a numerical evidence of the existence of an invariant curve associated with the transformed system and, correspondingly, to the original one. In the second part of the paper we implement a numerical method due to M. Hénon [Hénon] for the computation of the rotation number corresponding to a given initial condition. Following an idea of Laskar et al. [1992] and Laskar [1993], we determine with high accuracy the critical breakdown threshold of invariant curves for standard-mapping like systems which allows not only to test Hénon's method but also to compare our analytical results with an accurate numerical one. An application is also made about the accuracy of the leap frog method.

In this paper, a simple and direct statistical method is proposed for estimating the Lyapunov exponent of an unknown dynamic system using its time series of observation data. It is shown that the asymptotic distribution of the estimates obtained from the proposed method is normal. Monte Carlo and block bootstrap methods are used to simulate the estimation for the logistic map, in which they both provide the expectation and variance for the estimates. Computer simulations show that our estimates are very close to the true values of the exponent for the logistic map with different parameters.

The problem of reconstruction of one-dimensional potentials with a given spectrum that is strictly equidistant except for one gap of a specified size and location is solved. This problem leads to a necessity of a nonlinear generalization of spectral shift operators. The solvability condition for an equation for these operators generates a dynamical system for possible potentials.

Two classes of symmetric and regular in ℛ¹ potentials obeying this equation are found and investigated in detail. The first class is a countable set of one-parameter families of anharmonic oscillators with a spectrum as described above. The second class is a countable set of two-parameter families of complicated potentials with asymptotics that oscillate with linearly increasing frequency and amplitude. The spectrum of that class is a combination of three strictly equidistant sequences of levels with the ground states shifted arbitrarily relative to each other.

Bifurcations of those potentials under changes of structure parameters as well as bifurcations of corresponding spectra have been studied.

Stabilization of chaotic dynamical systems onto periodic orbits is considered via a control approach based on the contraction-mapping theorem. The convergence of the control scheme is guaranteed by the construction of a contraction map. The control time required to reach within a finite precision of the desired state is in a logarithmic form, which can be adjusted by setting the convergence rate. The effects of noise on the control process are investigated; in particular (a) the effect on the magnitude of control inputs, which determines the condition of a controllable noise level, and (b) the effect on the robustness of the control process. The Cubic map is used to numerically demonstrate the control process.

A case study is presented to demonstrate the discretization chaos in the switching control system with only finite switching values. It is proved that for the second order oscillator, with several classes of sampling periods, discrete switching control enables periodic motions around the desired equilibrium. The pattern of the discretized system is determined by the initial conditions as well as the sampling periods. Simulation results are presented to confirm the theoretical investigations.

This letter investigates the effect of the length of an Earth orbiting dumbbell system on the number and the Lyapunov stability of its stationary motions. This effect appears when a weak aerodynamical drag is taken into consideration and being important for low orbit satellites. The equation of motion is obtained as a Lagrange equation. Using analytical methods a critical length can be found causing essential changes in the structure of the set of stationary positions and in the stability properties, too. For longer systems the number of stationary positions is four, for shorter ones it is two. In the critical case the other two positions come together and disappear in a saddle-node bifurcation. The numerical values of the bifurcation point and of the critical length are also presented in a case described in the literature.

Jackson and Grosu developed a method for controlling complex dynamic systems that employs an open-plus-closed-loop control components (the OPCL method), and proved that it is capable of entraining the dynamics of standard dynamic systems to arbitrary goal dynamics, g(t) ∈ Rⁿ [Jackson & Grosu, 1995]. Rather than use this control for entrainment, one of us has shown [Jackson, 1995] how the OPCL can be easily employed to transfer the dynamics of a multiple-attractor system from one attractor to another ("migration controls" [Jackson, 1990]). For this purpose, a Chua-circuit model with five distinct attractors was used. In the present study we will show how to experimentally implement the OPCL migration control on such a circuit, by using prerecorded signals, which makes it possible to restrict the control action to only one dynamic component.