International Journal of Bifurcation and Chaos

This paper contains a survey of numerical methods for connecting orbits in discrete dynamical systems. Special emphasis is put on degenerate cases where either the orbit loses transversality or one of its endpoints loses hyperbolicity. Numerical methods that approximate the connecting orbits by finite orbit sequences are described in detail and theoretical results on the error analysis are provided. For most of the degenerate cases we present examples and numerical results that illustrate the applicability of the methods and the validity of the error estimates.

In this paper, we carefully reexamine the chaotic RF model, first studied by Rabinovich and Fabrikant [1979], and have found many new and rich complex dynamics of the model that were mostly not reported before. The chaotic RF model has proved to be a great challenge to classical numerical methods, in the sense that most classical numerical methods have not been very successful in the study of complex dynamics of this special RF model. Therefore, in this paper, we present and apply a special numerical method, the Local Iterative Linearization (LIL) method, along with a special Turbo Pascal code based on this accurate LIL algorithm, for a careful numerical study of this complex RF model. Many interesting new findings are summarized and reported in this paper.

The low-frequency portion of the power spectrum of human heartbeat intervals exhibits a 1/f-like scaling behavior. Despite its clinical significance, the mechanism remains unknown. By recording heartbeat intervals, renal sympathetic nerve activity (SNA), and blood pressure (BP) in conscious rats with normal or high BP, we reported that the scaling slope of heartbeat intervals is steeper in the rats with high BP, and that SNA leads to heartbeat interval and BP changes, and also the dynamics of these three variables, heartbeat intervals, BP, and SNA, results from a low-dimensional chaos. This hemodynamics is modeled excellently by modification of a known chaotic electrical circuit, Chua circuit. Increasing the resistive element between SNA and BP in the circuit increases the scaling slope and abolishes the low-dimensional chaos. Therefore, sensitivity of BP to sympathetic control is likely to determine the scaling slopes of heartbeat intervals in health and disease.

We compare the iterative root-finding methods of Newton and Halley applied to cubic polynomials in the complex plane. Of specific interest are those "bad" polynomials for which a given numerical method contains an attracting cycle distinct from the roots. This implies the existence of an open set of initial guesses whose iterates do not converge to one of the roots (i.e. the numerical method fails). Searching for a set of bad parameter values leads to Mandelbrot-like sets and interesting figures in the parameter plane. We provide some analytic and geometric arguments to explain the contrasting parameter plane pictures. In particular, we show that there exists a sequence of parameter values λ_n for which the corresponding numerical method has a superattracting n cycle. The λ_n lie at the centers of a converging sequence of Mandelbrot-like sets.

Chaotic behavior for a periodically forced system is considered. Under certain parameter conditions, the fact that some combination-type of chaotic motions and all possible types of subharmonic solutions occur simultaneously can be proved. The dynamics of stochastic and resonant layer have been discussed.

In anti-control of bifurcations, it is common to create different types of bifurcations by adjusting the control parameters. For maps, the type of bifurcation is determined by the eigenvalue assignment on the unit circle at the bifurcation parameter point. Thus, an unavoidable problem in the creation of bifurcations is to desirably assign some eigenvalues at the specified locations on the unit circle, and the others inside the unit circle. However, for relatively complicated and high dimensional maps, the explicit expressions of eigenvalues are usually not available so that the implementation of the eigenvalue assignment becomes very difficult. To solve this problem, we proposed the new criteria of eigenvalue assignment without using eigenvalues. The criteria give implicit conditions to specify the eigenvalue assignments in terms of some simple algebraic equalities and inequalities associated with the elements of Jacobian matrix, i.e. eventually associated with the control parameters. Bifurcation occurs with another critical condition, the transversality condition. The computation of the transversality condition is usually nontrivial in high dimensional maps because it is related to the partial differentiation of the eigenvalues on the unit circle. We also present the implicit expression of the transversality condition in the form of the derivative of the Jacobian matrix and its eigenvectors that are computable at the bifurcation point. The proposed criteria cover most known types of bifurcations in four-dimensional maps and serve as the preferable methods for designing the critical bifurcation conditions in anti-control of bifurcations. The application to a modified Hénon map is illustrated in conjunction with the use of the delayed-feedback control and the washout-filter-aided feedback control.

In this paper, the problem of chaotifying the continuous-time fuzzy hyperbolic model (FHM) is studied. By tracking the dynamics of a chaotic system, a controller based on inverse optimal control and adaptive parameter tuning methods is designed to chaotify the FHM. Simulation results show that for any initial value the FHM can track a chaotic system asymptotically.

To suppress undesirable noise (jamming) associated with signals is important for many applications. Here we explore the idea of jamming suppression with realistic, aperiodic signals by stochastic resonance. In particular, we consider weak amplitude-modulated (AM), frequency-modulated (FM), and chaotic signals with strong, broad-band or narrow-band jamming, and show that aperiodic stochastic resonance occurring in an array of excitable dynamical systems can be effective to counter jamming. We provide formulas for quantitative measures characterizing the resonance. As excitability is ubiquitous in biological systems, our work suggests that aperiodic stochastic resonance may be a universal and effective mechanism for reducing noise associated with input signals for transmitting and processing information.

In this paper, the bifurcation method of planar systems and simulation method of differential equations are employed to investigate the bounded traveling waves of the Camassa–Holm equation. Some new bounded traveling waves are found and their implicit expressions are obtained. Both qualitative and numerical results show that they possess the properties of compactons or generalized kink waves.

We study a finite uni-directional array of "cascading" or "threshold coupled" chaotic maps. We describe some of the attractors for such systems and prove general results about their basins of attraction. In particular, we show that the basins of attraction have infinitely many path components. We show that these components always accumulate at the corners of the domain of the system. For all threshold parameters above a certain value, we show that they accumulate at a Cantor set in the interior of the domain. For certain ranges of the threshold, we prove that the system has many attractors.

We have found a formalism that lets us present generalizations of several stability theorems (see Chua & Roska, 1990; Chua & Wu, 1992; Gilli, 1993; Forti, 2002] on Multi-Layer Cellular Neural/Nonlinear Networks (MLCNN) formerly claimed for Single-Layer Cellular Neural/Nonlinear Networks (CNN). The theorems were selected with special regard to usefulness in engineering applications. Hence, in contrast to many works considering stability on recurrent neural networks, the criteria of the new theorems have clear indications that are easy to verify directly on the template values. Proofs of six new theorems on 2-Layer CNNs (2LCNN) related to symmetric, τ-symmetric, nonsymmetric, τ-nonsymmetric, and sign-symmetric cases are given. Furthermore, a theorem with a proof on a MLCNN with arbitrary template size and arbitrary layer number in relation to the sign-symmetric theorem is given, along with a conjecture for the one-dimensional, two-layer, nonreciprocal case.

We study multigrid methods in the context of continuation methods for reaction–diffusion systems, where the Bi-CGSTAB and GMRES methods are used as the relaxation scheme for the V-cycle, W-cycle and full approximation schemes, respectively. In particular, we apply the results of Brown and Walker [1997] to investigate how the GMRES method can be used to solve nearly singular systems that occur in continuation problems. We show that for the sake of switching branches safely, one would rather solve a perturbed problem near bifurcation points. We propose several multigrid-continuation algorithms for curve-tracking in nonlinear elliptic eigenvalue problems. Our numerical results show that the algorithms proposed have the advantage of being robust and easy to implement.

It is said that the numerical generation of exact chaotic time series by iterating, for example, the logistic map, will be impossible, because chaos has a high dependency on initial values. In this letter, an algorithm to generate them without the accumulation of inevitable round-off errors caused by the iteration is proposed, where rational numbers are introduced. Also, it is shown that the period of the chaotic time series depends on the rational numbers including large prime numbers, which are fundamentally related to the Mersenne and the Fermat prime ones. Since the time series are numerically regenerated by the proposed algorithm in an usual computer environment, it could be applied to cryptosystems which do not need the synchronization, and have a large key-space by using large prime numbers.

Symmetric block encryption schemes, designed on invertible two-dimensional chaotic maps on a torus or a square, prove feasible and secure for real-time image encryption according to the commonly used criteria given in the literature. In this paper, a typical map of this kind, namely, the baker map, is further extended to be three-dimensional and then used to speed up image encryption while retaining its high degree of security. The proposed algorithm is described in detail, along with its security analysis and implementation. Experimental results show that this three-dimensional baker map is 2–3 times faster than the two-dimensional one, showing its great potential in real-time image encryption applications.

A chaotic encryption method is proposed for digital communication using a pseudo-random switching key in which parameters irregularly adjust to a rule related to chaotic variables. The scheme derived two features — pseudo-random false switching events are densely produced to mask up binary switching and the transmitted signal possesses a transient nature thus diffusing any deterministic characteristics used by interceptors. The robustness of synchronization is discussed when the mismatch of parameters or keys occurs under erratic switching events.

In this Letter, we apply combined linear detector/parallel interference cancellation (PIC) detectors to jointly decode symbols in a multiple access chaotic-sequence spread-spectrum communication system. In particular, three different types of linear detectors, namely single-user detector, decorrelating detector and minimum mean-square-error detector, are used to estimate the transmitted symbols at the first stage of the PIC detector. The technique for deriving the approximate bit error rate (BER) is described and computer simulations are performed to verify the analytical BERs.

In this paper, bifurcations of limit cycles close to certain singularities of the vector fields are explored using an algorithm based on the harmonic balance method, the theory of nonlinear feedback systems and the monodromy matrix. Period-doubling, pitchfork and Neimark–Sacker bifurcations of cycles are detected close to a Gavrilov–Guckenheimer singularity in two modified Rössler systems. This special singularity has a zero eigenvalue and a pair of pure imaginary eigenvalues in the linearization of the flow around its equilibrium. The presented results suggest that the proposed technique can be promising in analyzing limit cycle bifurcations arising in the unfoldings of other complex singularities.

This letter presents an experimental confirmation of controlling the chaotic behavior of a target unstable periodic orbit when the periodically switched nonlinear circuit has a chaotic attractor. The pole assignment for the corresponding discrete system derived from such a nonautonomous system via Poincaré mapping works effectively, and the control unit is easily realized by the window comparator, sample-hold circuits, and so on.

Departing from the OGY method reported in 1990, many methodologies for chaos control have been proposed. A major criticism is that most of them are merely straightforward applications of methodologies borrowed from feedback control theory. In fact, an authentic chaos control methodology should rely on the underlying structure (e.g. topology, dynamics, etc.) of chaotic behavior in order to provide simple and successful feedback control algorithms. This brief paper focuses on this objective; namely, to exploit chaos structure to design feedback controllers aimed to eliminate chaos. The well-known Duffing oscillator is used to show how the mechanisms leading to chaotic behavior can be destroyed by means of a simple feedback control. Specifically, it is shown that injected damping, via feedback control, is able to eliminate the transverse homoclinic orbit responsible for the chaotic behavior of the Duffing oscillator.

A second-order nonlinear differential equation with an aftereffect for the density of a thin homogeneous layer on a liquid and vapor interface is considered. The acts of evaporation and condensation of molecules, which are regarded as periodic "impacts", excite the layer. The mentioned NDE is integrated over a finite time interval to find a 2D (two-dimensional) mapping whose numerical solution describes the chaotic dynamics of density and pressure in time. The algorithms of constructing bifurcation diagrams, Lyapunov's exponents and Kolmogorov's entropy for systems with first-order, second-order phase transitions and Van der Waals' systems were elaborated. This approach allows to associate such concepts as phase transition, deterministic chaos and nonlinear processes. It also allows to answer a question whether deterministic chaos occurs in systems with phase transitions and how fast the information about starting conditions is lost within them.

The systematic study of application of Cohen's class time-frequency representations in the analysis of periodic BZ oscillations generated in batch reactor has been presented. Several distributions belonging to Cohen's class have been applied in the analysis of selected signal being the register of bromide selective electrode. Among them were Wigner–Ville, Choi–Williams and cone-shaped distributions. The application of mentioned methods allow instantaneous power spectra to be obtained and simultaneously give the possibility of observing evolution of frequency composition of investigated oscillations. The systematic filtering of signal, using so-called kernel functions, allows to eliminate undesirable cross terms and finally leads to the selection of a suitable method for chemical oscillations decomposition. The results presented in the form of three-dimensional pictures, illustrate the decrease in frequency of oscillations of exponential character. On the basis of the obtained results the cone-shaped distribution for kernel function parameter α=1 has been selected as the best method for Belousov–Zhabotinsky oscillations analysis in joint time-frequency domain.