International Journal of Bifurcation and Chaos

The most commonly used mapping to illustrate the phenomenon of chaos is the map x → 2x mod(1). This map is known as the 'unilateral shift' because, in the binary number system this map shifts all digits to the left by one decimal place, and truncates the integer. The second most commonly used paradigm of chaos is the Smale horseshoe whose complexity is essentially the bilateral shift obtained when we simply shift without truncation in some symbol system. Neither of these paradigms fully explains chaos since shifts cannot generate complex orbits from simple (rational) initial conditions. How chaos generates complexity from simplicity is an essential part that needs explanation. Providing this explanation is the objective of this paper.

The problem of the system behavior after annihilation of the resonant attractor in the region of the nonlinear resonance hysteresis is considered. The sequences of global bifurcations, in connection with the associated metamorphoses of basins of attraction of coexisting attractors, are examined. The study allows one to reveal the mechanism that governs the phenomenon of the post crisis ensuing transient trajectory to settle onto one or another remote attractor. The problem is studied in detail for the twin-well potential Duffing oscillator. The boundary which splits the considered region of system parameters into two subdomains, where the outcome is unique or the two outcomes are possible, is defined.

The dynamics of one-dimensional iterative maps in the regime of fully developed chaos is studied in detail. Motivated by the observation of dynamical structures around the unstable fixed point we introduce the geometrical concept of a turning point which represents a local minimum or maximum of the trajectory. Following we investigate the highly organized and structured distribution of turning points. The turning point dynamics is discussed and the corresponding turning point map which possesses an appealing asymptotic scaling property is investigated. Strong correlations are shown to exist for the turning point trajectories which contain the information of the fixed points as well as the stability coefficients of the dynamical system. For the more specialized case of symmetric maps which possess a symmetric density we derive universal statistical properties of the corresponding turning point dynamics. Using the turning point concept we finally develop a method for the analysis of (one-dimensional) time series.

Recently multilevel subdivision techniques have been introduced in the numerical investigation of complicated dynamical behavior. We illustrate the applicability and efficiency of these methods by a detailed numerical study of Chua's circuit. In particular we will show that there exist two regions in phase space which are almost invariant in the sense that typical trajectories stay inside each of these sets on average for quite a long time.

In order to predict the long term behavior of nonlinear dynamical systems the generalized cell mapping is an efficient and powerful method for numerical analysis. For this reason it is of interest to know under what circumstances dynamical quantities of the generalized cell mapping (like persistent groups, stationary densities, …) reflect the dynamics of the system (attractors, invariant measures, …). In this article we develop such connections between the generalized cell mapping theory and the theory of nonlinear dynamical systems. We prove that the generalized cell mapping is a discretization of the Frobenius–Perron operator. By applying the results obtained for the Frobenius–Perron operator to the generalized cell mapping we outline for some classes of transformations that the stationary densities of the generalized cell mapping converges to an invariant measure of the system. Furthermore, we discuss what kind of measures and attractors can be approximated by this method.

A first harmonic approach (describing function method) is used in this work to study the behavior of linear control systems with a saturating high-gain linear feedback. It is shown that when the eigenvalues of the closed-loop system are located deeper in the left-half complex plane, several unstable periodic orbits shrink to the origin, thus leading to an unstable equilibrium point. This dynamical behavior is interpreted as the vanishing of the region of attraction of the origin when a saturating high-gain feedback is used.

This paper presents experimental results of wire-line and wireless analog information transmission in RF band via chaos using standard signal formation and reception methods.

A fast numerical scheme based on the model of Barkley [Physica 49D (1991), 61] is extended to three space dimensions (3D). The original time-stepping scheme is improved to provide greater accuracy and a 19-point approximation for the Laplacian operator in 3D is shown to have significant advantages over the commonly used 7-point formula. Simulations are coupled to a state-of-the-art surface rendering algorithm such that the combined code allows real-time interactive simulations of 3D waves on a desktop workstation. Results are presented from simulations over a range of spatio-temporal resolutions, from coarse cellular-automaton type simulations to fully resolved simulations of the underlying partial differential equations.

We present a new numerical method for the efficient computation of periodic solutions of nonlinear systems of Delay Differential Equations (DDEs) with several discrete delays. This method exploits the typical spectral properties of the monodromy matrix of a DDE and allows effective computation of the dominant Floquet multipliers to determine the stability of a periodic solution. We show that the method is particularly suited to trace a branch of periodic solutions using continuation and can be used to locate bifurcation points with good accuracy.

In this paper we consider an approximate one-dimensional model for describing the non-linear dynamics of the Colpitts oscillator. It has already been shown that this model preserves the qualitative dynamical behavior of the Colpitts oscillator. Using the theory of snap-back repellors we prove that the one-dimensional model is chaotic for certain values of its parameters.

An intermittency of spatiotemporal chaos and checkerboard spiral waves is observed in a 2D coupled sigmoid map lattice. A rotating arm of such a spiral consists of sites oscillating in opposite phases with respect to nearest neighbors. A region of the spiral waves exhibited in the control parameter space is presented. The spatiotemporal intermittency stems from a crisis-induced intermittency of chaotic wanderings and zigzag burstings that is shown in this work.

We report the verification of the prediction of chaos in the Portevin–Le Châtelier effect or the jerky flow by analyzing the stress signals obtained from samples of polycrystalline Al–Mg alloys subjected to a constant strain rate test. Particular care is taken to obtain reasonably long and accurate stress signals. The analysis of these signals is carried out by using several complementary methods such as calculation of correlation dimension, singular value decomposition and the spectrum of Lyapunov exponents. The analysis shows the existence of a finite correlation dimension and a positive Lyapunov exponent. Using the existence of a positive Lyapunov exponent and finite correlation dimension as a discriminator, we also carry out a surrogate analysis of the time series to ascertain that the signals are not from a power law stochastic process. The analysis provides an unambiguous support for the existence of chaos in Portevin–Le Châtelier effect thus verifying the prediction of the model. Further, from the analysis we find that the minimum number of variables required for a dynamical description of the jerky flow appears to be four or five consistent with the model.

An iterative algorithm is proposed for constructing synchronizing receivers for a chosen class of chaotic transmitter systems. A complete knowledge of the transmitter system is not required. Identical synchronization is ensured with the transmitted signal in the constructed receivers. Potential applications of the method are discussed for communications.

This paper investigates the identification of global discrete nonlinear models from chaotic data. It is shown that when chaotic data from a nonlinear system do not contain enough information about all the fixed points, the usual model selection procedures can select different local nonlinear models.

We investigate second order lossless digital filters with two's complement overflow. We numerically approximate the fractal set D of points that iterate arbitrarily close to the discontinuity. For the case of eigenvalues of the associated linear map of the form e^iθ with θ/π ∉ Q we present evidence that D has positive two dimensional Lebesgue measure. For θ/π ∈ Q we confirm that D has Lebesgue measure zero. As a by-product we get estimates of the exterior dimension of D. These results imply that if such filters are realized using finite-precision arithmetic then they will have a sizeable fraction of orbits that are periodic with high period overflows.

Minimal models composed of two ordinary differential equations are considered in this paper to mimic the dynamics of the feelings between two persons. In accordance with attachment theory, individuals are divided into secure and non-secure individuals, and synergic and non-synergic individuals, for a total of four different classes. Then, it is shown that couples composed of secure individuals, as well as those composed of non-synergic individuals can only have stationary modes of behavior. By contrast, couples composed of a secure and synergic individual and a non-secure and non-synergic individual can experience cyclic dynamics. In other words, the coexistence of insecurity and synergism in the couple is the minimum ingredient for cyclic love dynamics. The result is obtained through a detailed local and global bifurcation analysis of the model. Supercritical Hopf, fold and homoclinic bifurcation curves are numerically detected around a Bogdanov–Takens codimension-2 bifurcation point. The existence of a codimension-2 homoclinic bifurcation is also ascertained. The bifurcation structure allows one to identify the role played by individual synergism and reactiveness to partners love and appeal. It also explains why ageing has a stabilizing effect on the dynamics of the feelings. All results are in agreement with common wisdom on the argument. Possible extensions are briefly discussed at the end of the paper.

Markman and Bar-Eli has studied a periodically forced Oregonator numerically and found a parameter range with the following properties: (1) Only periodic solutions are found in frequency-locked steps, each with a certain pattern of large and small oscillations (2) Between any two steps there is a step with the period being the sum of the two periods and the concatenation of the two patterns (3) Certain scaling properties as the period tends to infinity. We show that such behavior occurs if the dynamics of the system is governed by a family of diffeomorphisms of a circle with a Devil's staircase. Using invariant manifold theory we argue that an invariant circle must indeed exist when, as in the present case, the unforced system is close to a saddle-loop bifurcation. Generalizations of the results are briefly discussed.