International Journal of Bifurcation and Chaos

The study of soliton systems continues to be a highly rewarding exercise in nonlinear dynamics, even though it has been almost thirty years since the introduction of the soliton concept by Zabusky & Kruskal. Increasingly sophisticated mathematical concepts are being identified with integrable soliton systems, while newer applications are being made frequently. In this pedagogical review, after introducing solitons and their (2+1)-dimensional generalizations, we give an elementary discussion on the various analytic methods available for investigation of the soliton possessing nonlinear evolution equations. These include the inverse scattering transform method and its generalization, namely the d-bar approach, for solving the Cauchy initial value problem, as well as direct methods for obtaining N-soliton solutions. We also indicate how the Painlevé singularity structure analysis is useful for the detection of soliton systems.

Higher-order spectra have been used to investigate nonlinear interactions between the Fourier components of measured time series in a remarkably wide range of random processes. The basic techniques of detecting and isolating nonlinear phase coupling in observed data using higher-order spectral analysis are reviewed here. These techniques are then used to investigate nonlinear interactions in time series of voltages measured from a realization of Chua’s circuit. For period-doubled limit cycles, quadratic and cubic nonlinear interactions result in phase coupling and energy exchange between increasing numbers of triads and quartets of Fourier components as the nonlinearity of the system is increased. For circuit parameters that result in a chaotic, Rössler-type attractor, bicoherence and tricoherence spectra indicate that both quadratic and cubic nonlinear interactions are important to the dynamics. For parameters that lead to the double-scroll chaotic attractor the bispectrum is zero, but the tricoherences are high, consistent with the importance of higher-than-second order nonlinear interactions during chaos associated with the double scroll.

Evolutionary game theory can be extended to include spatial dimensions. The individual players are placed in a two-dimensional spatial array. In each round every individual “plays the game” with its immediate neighbours. After this, each site is occupied by its original owner or by one of the neighbours, depending on who scored the highest payoff. These rules specify a deterministic cellular automaton.

We find that spatial effects can change the outcome of frequency dependent selection. Strategies may coexist that would not coexist in homogeneous populations. Spatial games have interesting mathematical properties. There are static or chaotically changing patterns. For symmetrical starting conditions we find “dynamical fractals” and “evolutionary kaleidoscopes.” There is a new world to be explored.

A bimolecular catalytic surface reaction is extended to include diffusion which yields mobilized coverage on the surface. We consider the reaction occurring in a tubular reactor with a convection flow where the reactants also diffuse. An initial boundary value problem in one-dimensional reaction-diffusion-convection equations is used in describing the model. We combine singular perturbation analysis with numerical simulations in studying the solution behavior in parameter space. We track the reaction front and the cause of period-2 oscillations. Compared with the case of having no surface diffusion, we observe regular oscillations instead of irregular oscillations. Compared with the nondiffusive nonconvective model, we obtain rich spatiotemporal patterns including stationary, oscillatory reaction fronts and multiple steady states.

This paper concerns the bifurcation properties of a model of a frequency modulated CO₂ laser in the form of a two-dimensional ordinary differential equation with a parametric periodic excitation. These properties are related to the bifurcation curves organization (structure) in a parameter plane (amplitude, frequency of the modulation). Two basic bifurcation structures appear, one concerning the higher harmonic solutions, the other the subharmonic solutions. Qualitative changes of these structures are considered when a third parameter (pump parameter) is varied.

The problem of reducing noise in a time series from a nonlinear dynamical system can be formulated as a nonlinear minimisation process. This paper demonstrates that this can be easily solved using a steepest descent method without any of the stability problems that have been associated with using a Newton method [Hammel, 1990; Farmer & Sidorowich, 1991]. The optimisation function to be minimised is also shown not to contain any local minima if the trajectory is always hyperbolic. So that in this case this method will converge eventually to a purely deterministic trajectory. Finally this method is compared with a recently proposed algorithm [Schreiber & Grassberger, 1991], which can be viewed as an alternative gradient descent method.

The method of reconstruction of Fokker-Planck and Master equations for nonlinear open systems on the basis of experimental time series is considered. In the process of time evolution the entropy of a system, renormalized to the given value of the mean effective energy, increases in accordance with the kinetic equations (H-theorem). The evolution of the renormalized entropy of stationary states in the space of rule (control) parameters is also considered (S-theorem).

A nonlinear dynamo model for solar activity which includes the feedback of the helicity upon the mean magnetic field has been investigated. The qualitative behavior of a seven-dimensional system of ordinary differential equations obtained by truncation of that model has been studied numerically. It has been compared with results from a sixth order system derived from the seventh order system by a special polar coordinate transformation. Depending on characteristic parameters, the seven-dimensional model exhibits periodic, quasiperiodic (on T² and T³) and chaotic behavior where a route to chaos via the transition T²⇒T³⇒T²⇒ chaos has been found to be typical. In contrast to that, no chaotic state occurs in the reduced system due to a nonregularity of the coordinate transformation.

The dynamics of two unidirectionally coupled period-doubling systems is investigated depending on three relevant parameters (control parameters of subsystems and coupling). There is a hierarchy of critical behavior types. Feigenbaum’s critical surfaces existing in the parameter space are bounded by tricritical lines and intersect along the bicritical line. These lines, in turn, intersect at a new multicritical point BT. Universality and scaling properties for all the critical situations are discussed, and the table of critical indices is given.

Dynamic and stochastic effects of the interaction of individual localized structures and lattices of structures of a nonlinear field are considered within one-dimensional large-box Ginzburg-Landau and Swift-Hohenberg models. The criterion of soliton “survival” in competition with counterpropagating modes is derived analytically for a quasigradient case. The criterion of linear stability of a spatially homogeneous regime, that is similar to the Benjamin-Feir condition, is obtained for coupled Ginzburg-Landau equations. The relation between the correlation dimensions of the space series of counterpropagating modes and the dimension of the time series is investigated. It is shown that within a quasigradient model of two weakly coupled counterpropagating modes, the field can be represented as a superposition of two fixed stochastic spatial lattices which correspond to these modes and move through one another at constant velocity. The dimension of the time series at a given point in space is close to the sum of dimensions of the spatial distributions of counterpropagating modes. With the increase of coupling, the interaction of coupled modes “loosens” the equilibrium state corresponding to fixed lattices and the dimension of the time series grows.

We consider a perturbed twist map when the perturbation is big enough to destroy the invariant rotational curve (IRC) with a given irrational rotation number. Then an invariant Cantorian set appears. From another point of view, the destruction of the IRC is associated with the appearance of heteroclinic connections between hyperbolic periodic points. Furthermore the destruction of the IRC is also associated with the existence of non-Birkhoff orbits. In this paper we relate the different approaches.

In order to explain the creation of non-Birkhoff orbits, we provide qualitative and quantitative models. We show the existence of elliptic non-Birkhoff periodic orbits for an open set of values of the perturbative parameter. The bifurcations giving rise to the elliptic non-Birkhoff orbits and other related bifurcations are analysed. In the last section, we show a celestial mechanics example displaying the described behavior.

Critical curves are the natural two-dimensional extension of the notion of critical points in one-dimensional endomorphisms. They play a fundamental role in determining the dynamical properties and their bifurcations. This letter demonstrates such a role for two new behaviors.

The problem of the global behavior of the system with 3/2 degrees of freedom is considered in this paper. The results of the theoretical investigation of the case when the systems approach a nonlinear Hamiltonian system are presented. Using, for example, the Duffing-van der Pol equation, computer analysis gives good agreement with theory when the external force has moderate amplitude values (“miracle of a small parameter”). The bifurcations of the invariant torus and resonances are analyzed as a function of the system parameters.

In this paper we present a simple piecewise-linear circuit which exhibits a chaotic attractor similar to that observed from the Lorenz equation. Whereas the nonlinearities in the Lorenz equation consists of two product terms between two state variables, the nonlinearities in our circuit consists of two piecewise-linear terms.

We investigate the one-dimensional motion of a traveling wave in a fluctuating medium. Two cases are studied; namely, fluctuating barriers with memory and continuously fluctuating medium. We have shown that as a result of the combination of noise and the spreading dynamics associated with traveling wave propagation, a signal propagates faster, even for values of coupling greater than the critical value associated with the propagation failure phenomena.

In this letter we show how to use a new form of integration, called dynamical integration, that utilizes the dynamics of a system defined by an ODE to construct a map that is in effect a one-step integrator. This method contrasts sharply with classical numerical methods that utilize polynomial or rational function approximations to construct integrators. The advantages of this integrator is that it uses only one step while preserving important dynamical properties of the solution of the ODE: First, if the ODE is conservative, then the one-step integrator is measure preserving. This is significant for a system having a highly nonlinear component. Second, the one-step integrator is actually a one-parameter family of one-step maps and is derived from a continuous transformation group as is the set of solutions of the ODE. If each element of the continuous transformation group of the ODE is topologically conjugate to its inverse, then so is each member of the one-parameter family of one-step integrators. If the solutions of the ODE are elliptic, then for sufficiently small values of the parameter, the one-step integrator is also elliptic. In the limit as the parameter of the one-step family of maps goes to zero, the one-step integrator satisfies the ODE exactly. Further, it can be experimentally verified that if the ODE is chaotic, then so is the one-step integrator. In effect, the one-step integrator retains the dynamical characteristics of the solutions of the ODE, even with relatively large step sizes, while in the limit as the parameter goes to zero, it solves the ODE exactly.

We illustrate the dynamical, in contrast to numerical, accuracy of this integrator with two distinctly different examples: First we use it to integrate the unforced Van der Pol equation for large ∊, ∊≥10 which corresponds to an almost continuous square-wave solution. Second, we use it to obtain the Poincaré map for two different versions of the periodically forced Duffing equation for parameter values where the solutions are chaotic. The dynamical accuracy of the integrator is illustrated by the reproduction of well-known strange attractors. The production of these attractors is eleven times longer when using a conventional fourth-order predictor-corrector method. The theory presented here extends to higher dimensions and will be discussed in detail in a forthcoming paper. However, we caution that the theory we present here is not intended as a line of research in numerical methods for ODEs.