International Journal of Bifurcation and Chaos

The first step in investigating the dynamics of a continuous-time system described by an ordinary differential equation is to integrate to obtain trajectories. In this paper, we attempt to elucidate the dynamics of the most commonly used family of numerical integration schemes, Runge–Kutta methods, by the application of the techniques of dynamical systems theory to the maps produced in the numerical analysis.

Those classical models are reviewed that are most widely used by neurobiologists to explain the dynamics of neurons and neuron populations, and by modelers to implement artificial neural networks. Each neuron has input fibers called dendrites that integrate and an axon that transmits the output. The differing fiber architectures reflect these dissimilar dynamic operations. The basic tools to describe them are the RC model of the membrane, the core conductor model of the fibers, the Hodgkin–Huxley model of the trigger zone, and the modifiable synapse. Populations additionally require description of macroscopic state variables, the types of nonlinearity (most importantly the sigmoid curve and the dynamic range compression at the input to the cortex), and the types and strengths of connections. The properties of these neural masses can be characterized with the tools of nonlinear dynamics. These include description of point, limit cycle, and chaotic attractors for the cerebal cortex, as well as the types and mechanisms for the state transitions between basins of attraction during learning and perception.

A method for storing and retrieving information on the stable cycles of one-dimensional maps as proposed in Dmitriev [1991], Dmitriev, Panas & Starkov [1991] is considered. The applicability of this method in storing and retrieving two-dimensional pictures is demonstrated. Possible extensions by compressing information are discussed. An implementation of random access memory using a one-dimensional map is also considered.

A study is made of the control-phase portraits encountered in the twin-well Duffing oscillator, concentrating on the loss of stability of attractors confined to a single well of the potential. This loss of stability may be understood by examining the bifurcations which precede a single-well attractor touching its basin boundary. Two distinct bifurcation scenarios are described in which the basin boundary develops a fractal structure. This fractal structure accompanies the development of a homoclinic tangency between the inset and outset of the saddle whose inset determines the separatrix action. In the first scenario described, this fractal structure, or chaotic saddle, grows due to a sequence of fractal–fractal basin implosions which are caused by the completion of Smale cycles or heteroclinic chains; the subharmonics involved occur in decreasing order, after the main homoclinic tangency. The second scenario focuses on the bifurcational events which necessarily prepare the creation of a chaotic saddle; the subharmonics involved appear in increasing order, before the main homoclinic event. Both basin erosion scenarios are consistent with a geometric model of the twin-well Duffing oscillator based on a three layer template; in the first scenario, the full three-layer template participates; while in the second scenario, only two layers, a standard horseshoe, play a role.

Basin boundaries in chaotic dynamical systems can be either smooth or fractal. As a system parameter changes, the structure of the basin boundary also changes. In particular, the dimension of the basin boundary changes continuously except when a basin boundary metamorphosis occurs, at which it can change abruptly. We present numerical experiments to demonstrate such sudden dimension changes. We have also used a one-dimensional analytic calculation and a two-dimensional qualitative model to explain such changes.

We use a geometric approach to explore the dynamical structures of three-dimensional systems with two fixed points. In particular, we study the "cocoon" bifurcation, which is a series of global and local bifurcations that create an infinite number of heteroclinic orbits (which trace out a cocoon-like tangle in space). We introduce a symbolic dynamics to the intersections of a cross-sectional plane with the manifolds of the fixed points. A one-parameter system relevant to the travelling-wave solution of the Kuramoto–Sivashinsky equation is employed to illustrate this approach. The cocoon bifurcation begins with a rapidly converging sequence of heteroclinic bifurcations (principal sequence), each of which generates a pair of heteroclinic orbits and two new pieces of the two-dimensional manifolds. These bifurcations and the manifold structures are also observed with the time delay function of chaotic scattering. At the limit of the principal sequence, a saddle-node bifurcation occurs, followed by a period doubling cascade, which destroys the elliptical orbit and results in a hyperbolic invariant set. Other heteroclinic bifurcations not in the principal sequence are responsible for the folding and eventually the fractal structures of the two-dimensional manifolds. The cocoon bifurcation is completed when all intersections of the two-dimensional manifolds are transverse. The structurally stable system after the cocoon bifurcation contains a hyperbolic invariant set (a horseshoe) imbedded in two fractal sets of the two-dimensional manifolds. All three sets have the same fractal dimension. The cocoon bifurcation is a route to fully developed chaotic scattering. After a prospective effect, called manifold reconnection, the system undergoes a secondary cocoon bifurcation. Then the system becomes a hyperbolic invariant set that is characterized by a full shift of six symbols and is imbedded in two fractal sets of two-dimensional manifolds. Higher order cocoon bifurcations may become overlapped with the homoclinic bifurcation. These results are general and do not depend on the symmetry of the system. For a physical system represented by the dynamical system, it is conjectured that the structurally stable window after the cocoon bifurcation contains the physically most probable state.

We propose a pseudo-arclength continuation algorithm for computing paths of Z₂-symmetry-breaking bifurcation points for two-parameter nonlinear elliptic problems. The algorithm consists of an Euler predictor step and a solution step composed of a sequence of Newton iterations. This work generalizes the algorithm of Werner and Spence for locating a one-parameter symmetry-breaking bifurcation point by using the approach of Keller and Fier for following a (two-parameter) path of limit points (a "fold"). By repeated use of the bordering algorithm, we solve linear systems whose matrix is the "symmetric" Jacobian or "antisymmetric" Jacobian, thus fully exploiting any (block tridiagonal) structures present. We give numerical results for the steady, axisymmetric flow between rotating coaxial cylinders (Taylor–Couette flow). For finite cylinders, we compute the fold curve and path of symmetry-breaking bifurcation points for small aspect ratios, and illustrate a new method to accurately locale the two Z₂-symmetric codimension one singularities. For infinite cylinders, we show the projections on the (aspect ratio, Reynolds number) plane of the folds and bifurcation point paths in the neighborhood of the two-cell/four-cell neutral curve crossing. We numerically verify a conjecture of Meyer–Spasche and Wagner concerning the connection of two neutral-curve crossings by a path of secondary subharmonic bifurcations.

We investigate a reaction-diffusion-convection model describing a catalyzed surface reaction. The model consists of parabolic partial differential equations involving non-diffusive and non-convective variables where the steady state is the solution of a system of ordinary differential algebraic equations. We analyze solution behavior through perturbation methods and exhibit a linear instability induced by the algebraic equations. Also we find multiple front solutions resulting from a jump discontinuity in the algebraic components. Numerical simulations are performed on the original system as well as the reduced system resulting from the singular perturbation technique. In addition to the stationary front solutions and multiple steady states, we find oscillations which closely relate to the stability at the equilibrium of the algebraic manifold of solutions. Experimentally we observe a regular oscillation when there is one Hopf point on the algebraic manifold, a period-doubling when two Hopf points bifurcate from one Hopf point and a chaotic oscillation when two Hopf points separate (in this last case an unstable segment occurs between the Hopf points).

Phenomena caused by low velocity impacts in a class of forced impact oscillators are studied. It is shown that such impacts play an important role in the dynamics of general impacting systems. Features observed in numerical simulations of different oscillator models are associated with previous theoretical work on grazing impact. The Poincaré mapping geometry near points leading to low velocity impacts is shown, as well as bifurcations of types not found in nonimpacting systems. The consequences of not having a completely rigid constraint and the connection with the limiting case of a pure constraint are examined. In addition, low velocity impact in a Hamiltonian system is shown to have considerable effect on the break up of invariant tori.

The interaction between a moving ice sheet and an elastic structure is studied using the analogue model of Matlock, et al. [1969]. The ice sheet is represented by a series of teeth with bilinear, discontinuous stiffness. A global analysis of the resulting dynamical system is performed. Using a combination of analytical and numerical techniques, periodic solutions are determined and basin boundaries of the Poincaré map identified. While the total system dynamics are quite complex, two types of threshold solutions are found, each necessary but not sufficient in defining local separatrices.

In this paper, we report on the influence of an external periodic signal on the familiar double-scroll chaotic attractor of Chua's autonomous circuit. As the external forcing parameters are varied, quasiperiodicity, devil's staircase structures, re-emerging double-scroll and double-hook type attractors have been observed. In addition, period-adding and crisis phenomena have also been observed for this circuit.

This paper is devoted to the problem of synchronization of dynamical systems in chaotic oscillations regimes. The authors attempt to use the ideas of synchronization and its mechanisms on a certain class of chaotic oscillations. These are chaotic oscillations for which one can pick out basic frequencies in their power spectra.

The physical and computer experiments were carried out for the system of two coupled auto-oscillators. The experimental installation permitted one to realize both unidirectional coupling (external synchronization) and symmetrical coupling (mutual synchronization). An auto-oscillator with an inertial nonlinearity was chosen as a partial subsystem. It possesses a chaotic attractor of spiral type in its phase space. It is known that such chaotic oscillations have a distinguished peak in the power spectrum at the frequency f₀ (basic frequency). In the experiments, one could make the basic frequencies of partial oscillators equal or different. The bifurcation diagrams on the plane of control parameters "detuning" and "coupling" were constructed and analyzed.

The results of investigations permit one to conclude that classical ideas of synchronization can be applied to chaotic systems of the mentioned type. Two mechanisms of chaos synchronization were established: 1) basic frequency locking and 2) basic frequency suppression. The bifurcational background of these mechanisms was created using numerical analysis on a computer. This allowed one to analyze the evolution of different oscillation characteristics under the influence of synchronization.

We study self-synchronization of digital phase-locked loops (DPLL's) and the chaotic synchronization of DPLL's in a communication system which consists of three or more coupled DPLL's. Triangular wave signals, convenient for experiments, are employed. Numerical and experimental studies of two loops are in good agreement, giving bifurcation diagrams that show quasiperiodic, locked, and chaotic behavior. The approach to chaos does not show the full bifurcation sequence of sinusoidal signals. For studying synchronization to a chaotic signal, the chaotic carrier is generated in a subsystem of two or more self-synchronized DPLL's where one of the loops is stable and the other is unstable. The receiver consists of a stable loop. We verified numerically and experimentally that the receiver may synchronize with the transmitter if the stable loop in the transmitter and receiver are nearly identical and the synchronization degrades with noise and parameter variation. We studied the phase space where synchronization occurs, and quantify the deviation from synchronization using the concept of mutual information.

Recent work on the synchronizing of chaotic systems raises the possibility of finding ways to apply chaos to real problems. Studying these applications requires that real physical systems be used, as well as mathematical models. There are no general rules for designing or building such physical systems. We present here a circuit that is useful for studying applications of synchronized chaotic systems and discuss some of the considerations that went into designing this circuit. We also show how this circuit is used for studying cascaded synchronized chaotic systems.

In a case of two dissipative coupled electronic circuits with possible chaotic dynamics, results of experimental, analytical and computer studies are provided concerning bifurcations on the boundaries of the synchronization regime of chaotic self-oscillations.

We will say that Nature reads an object as fractal when the object's physical properties depend upon its – not necessarily integral – fractal dimension d_f. We will review a number of problems in physics in which an object is thus recognized as a fractal, annotating the corresponding laws L=L(d_f) of dependence on d_f. Next we will find that, although these problems are obviously not related, many such laws L(d_f) have a certain common structure, which we will identify, and denote with L_df. Finally, we will solve some problems in pure mathematics – P.D.E. in regions with fractal boundaries – finding again and again, our function L_df. Again, these pure-math problems are not related… what is the significance of L_df? Our conjecture: such strong formulistic connections are due to profound underlying yet-to-be-studied relationships – for which L_df is the clue – among the experimental problems, among the various pure math problems, and between the latter and the former.

We present a new type of strange attractors generated by an odd-symmetric three-dimensional vector field with a saddle-focus having two homoclinic orbits at the origin. This type of attractors is intimately related to the double scroll. We present the mathematical properties which prove rigorously the chaotic nature of this strange attractor to be different from that of a Lorenz-type attractor or a quasiattractor.

In particular, we proved that for certain nonempty intervals of parameters, our two-dimensional map has a strange attractor with no stable orbits. Unlike other known attractors, this strange attractor contains not only a Cantor set structure of hyperbolic points typical of horseshoe maps, but also unstable points (i.e., stable in reverse time). This implies that the points from the stable manifolds of the hyperbolic points must necessarily attract the unstable points.

Several recent papers have investigated the feasibility of synchronization of chaotic systems. Experimentally one of the easiest systems to control and synchronize is the electronic circuit. This paper examines synchronization in Chua's Circuit, proven to be the simplest electronic circuit to exhibit chaotic behavior.

Secure communications via chaotic synchronization is experimentally demonstrated using Chua's circuit. In our experiment the energy lost in the information-bearing signal is approximately 0.6 dBV. The reduction in chaotic signal after the recovery process is between -40 and -50 dBV.

In the 1970's it was found that; for low frequencies (<10 Hz), speech is scaling: it has no characteristic time scale. Now such scale invariance is associated with multiscaling statistics, and multifractal structures. Just as Gaussian noises frequently arise because they are generically produced by sums of many independent noise processes, scaling noises have an analogous universal behavior arising from nonlinear mixing of processes. We show that low frequency speech is consistent with these ideas, and use the measured parameters to produce stochastic speech simulations which are strikingly similar to real speech.

It is shown that, what so far were considered to be purely spatiotemporal, phenomena in a model for a Coupled Map Lattice can be obtained through a newly introduced Time-Delayed Map with only one chaotic element, thereby effectively eliminating the spatial dimension.