International Journal of Bifurcation and Chaos

I review the early (1885–1975) and more recent history of dynamical systems theory, identifying key principles and themes, including those of dimension reduction, normal form transformation and unfolding of degenerate cases. I end by briefly noting recent extensions and applications in nonlinear fluid and solid mechanics, with a nod toward mathematical biology. I argue throughout that this essentially mathematical theory was largely motivated by nonlinear scientific problems, and that after a long gestation it is propagating throughout the sciences and technology.

This article presents a unified state-space model for ship maneuvering, station-keeping, and control in a seaway. The frequency-dependent potential and viscous damping terms, which in classic theory results in a convolution integral not suited for real-time simulation, is compactly represented by using a state-space formulation. The separation of the vessel model into a low-frequency model (represented by zero-frequency added mass and damping) and a wave-frequency model (represented by motion transfer functions or RAOs), which is commonly used for simulation, is hence made superfluous.

This paper gives a short overview of various applications of stabilization by vibration, along with the exposition of the geometrical mechanism of this phenomenon. More specifically, the following observation is described: a rapidly vibrated holonomic system can be approximated by a certain associated nonholonomic system. It turns out that effective forces in some rapidly vibrated (holonomic) systems are the constraint forces of an associated auxiliary nonholonomic constraint. In particular, we review a simple but remarkable connection between the curvature of the pursuit curve (the tractrix) on the one hand and the effective force on the pendulum with vibrating support. The latter observation is a part of a recently discovered close relationship between two standard classical problems in mechanics: (1) the pendulum whose suspension point executes fast periodic motion along a given curve, and (2) the Chaplygin skate (known also as the Prytz planimeter, or the "bicycle"). The former is holonomic, the latter is nonholonomic. The holonomy of the skate shows up in the effective motion of the pendulum. This relationship between the pendulum with a twirled pivot and the Chaplygin skate has somewhat unexpected physical manifestations, such as the drift of suspended particles in acoustic waves. Finally, a higher-dimensional example of "geodesic motion" on a vibrating surface is described.

A number of physical mechanisms give rise to confined linear wave systems whose spatial structure is governed by a hyperbolic equation. These lack the discrete set of regular eigenmodes that are found in classical wave systems governed by an elliptic equation. In most 2D hyperbolic cases the discrete eigenmodes are replaced by a continuous spectrum of wave fields that possess a self-similar spatial structure and have a (point, line or planar) singularity in the interior. These singularities are called wave attractors because they form the attracting limit set of an iterated nonlinear map, which is employed in constructing exact solutions of this hyperbolic equation. While this is an inviscid, ideal fluid result, observations support the physical relevance of wave attractors by showing localization of wave energy onto their predicted locations. It is shown that in 3D, wave attractors may co-exist with a regular kind of trapped wave. Wave attractors are argued to be of potential relevance to fluids that are density-stratified, rotating, or subject to a magnetic field (or a combination of these) all of which apply to geophysical media.

It is a rule of thumb that time delay tends to destabilize any dynamical system. This is not true, however, in the case of delayed oscillators, which serve as mechanical models for several surprising physical phenomena. Parametric excitation of oscillatory systems also exhibits stability properties sometimes defying our physical sense. The combination of the two effects leads to challenging tasks when nonlinear dynamic behaviors in these systems are to be predicted or explained as well. This paper gives a brief historical review of the development of stability analysis in these systems, induced by newer and newer models in several fields of engineering. Local and global nonlinear behavior is also discussed in the case of the most typical parametrically excited delayed oscillator, a recent model of cutting applied to the study of high-speed milling processes.

Strong high-frequency excitation (HFE) may change the "slow" (i.e. effective or average) properties of mechanical systems, e.g. their stiffness, natural frequencies, equilibriums, equilibrium stability, and bifurcation paths. This tutorial describes three general HFE effects: Stiffening — an apparent change in the stiffness associated with an equilibrium; Biasing — a tendency for a system to move towards a particular state which does not exist or is unstable without HFE; and Smoothening — a tendency for discontinuities to be apparently smeared out by HFE. The effects and a method for analyzing them are introduced, first in terms of simple physical examples, and then in generalized form for mathematical models covering broad classes of discrete and continuous mechanical systems. Several application examples are summarized. Three mathematical tools for analyzing HFE effects are described and compared: The Method of Direct Separation of Motions, the Method of Averaging, and the Method of Multiple Scales. The tutorial concludes with a suggestion that more vibration experts, researchers and students should be aware of HFE effects, for the benefit of general vibration troubleshooting, and also for furthering the creation of innovative technical devices and processes utilizing HFE effects.

Robust attracting heteroclinic cycles have been found in many models of dynamics with symmetries. In all previous examples, robust heteroclinic cycles appear between a number of symmetry broken equilibria. In this paper we examine the first example where there are robust attracting heteroclinic cycles that include the origin, i.e. a point with maximal symmetry. The example we study is for vector fields on ℝ³ with (ℤ₂)³ symmetry. We list all possible generic (codimension one) local and global bifurcations by which this cycle can appear as an attractor; these include a resonance bifurcation from a limit cycle, direct bifurcation from a stable origin and direct bifurcation from other and more familiar robust heteroclinic cycles.

Chaotic motion of fluid particles in low Reynolds number flow between two eccentric cylinders rotating alternately is studied by investigating the tangling between invariant manifolds of the underlying dynamical system. Such tangles are revealed by a numerical method carrying out the global search of periodic points of the associated Poincaré map and the global tracking of stable and unstable manifolds to the hyperbolic points thus found. Along with such an anatomy the breakup of some elliptic orbits as predicted by Poincaré–Birkhoff theorem is clearly shown. It is found that several drastic changes either in the characteristic of ellipticity and hyperbolicity or in the pattern of homoclinic tangles take place when the cylinders' eccentricity is increased towards its maximum. The results are confirmed either by other numerical approaches or by a rigorous shadowing to a certain extent, and are expected to be of use to some geometric understanding about mixing.

The dynamic behavior of a nonlinear elastic rectangular plate of large deflection subjected to harmonic excitation is investigated. Using the Galerkin principle, a double mode model is established for the plate. The stability and bifurcation behavior of the plate is studied in detail for various loading conditions and system parameters. A set of autonomous equations is derived with the method of averaging. The bifurcation behavior is examined on the basis of the autonomous equations, and the results of theoretical bifurcation analysis are numerically verified. The chaotic response of the plate to the external excitation is also investigated. The governing equations of single as well as double modes are established and the comparison between the single and double mode models is carried out. The applicable conditions of the single mode method are provided. The results obtained show that the single mode approach usually used may lead to incorrect conclusions under certain conditions.

The standard map x′ = x + y′, y′ = y + (K/2π)sin(2πx), where both x and y are given modulo 1, becomes mostly chaotic for K ≥ 8, but important islands of stability appear in a recurrent way for values of K near K = 2nπ (groups of islands I and II), and K = (2n + 1)π (group III), where n ≥ 1. The maximum areas of the islands and the intervals ΔK, where the islands appear, follow power laws. The changes of the areas of the islands around a maximum follow universal patterns. All islands surround stable periodic orbits. Most of the orbits are irregular, i.e. unrelated to the orbits of the unperturbed problem K = 0. The main periodic orbits of periods 1, 2 and 4 and their stability are derived analytically. As K increases these orbits become unstable and they are followed by infinite period-doubling bifurcations with a bifurcation ratio δ = 8.72. We find theoretically the connections between the various families and the extent of their stability. Numerical calculations verify the theoretical results.

A simple delayed neural network model with four neurons is considered. Linear stability of the model is investigated by analyzing the associated characteristic equation. It is found that Hopf bifurcation occurs when the sum of four delays varies and passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. An example is given and numerical simulations are performed to illustrate the obtained results. Meanwhile, the bifurcation set is provided in the appropriate parameter plane.

We reinvestigate the dynamical behavior of a first order scalar nonlinear delay differential equation with piecewise linearity and identify several interesting features in the nature of the associated bifurcations and chaos as a function of delay time and external forcing parameters. In particular, we point out that the fixed point solution exhibits a stability island in the two parameter space of time delay and strength of nonlinearity. The significant role played by transients in attaining steady state solutions is pointed out. Various routes to chaos and existence of hyperchaos even for low values of time delay evidenced by multiple positive Lyapunov exponents are brought out. The study is extended to the case of two coupled systems, one with delay and the other one without delay.

Bifurcations of a general class of traveling wave solutions are analyzed. In particular, the existence of solitary wave, kink and anti-kink wave solutions, and uncountably infinite periodic wave solutions and breather solutions of a general class of traveling wave equations is proved. Also, the existence of breaking wave solution is discussed in detail. Under different parametric conditions, several sufficient conditions for the existence of these solutions are derived. Sufficient simulation results are provided to visualize the theoretical results.

A comparison between polynomial and wavelet expansions for the identification of coupled map lattice (CML) models for deterministic spatiotemporal dynamical systems is presented in this paper. The pattern dynamics generated by smooth and nonsmooth nonlinear maps in a well-known two-dimensional CML structure are analyzed. By using an orthogonal feedforward regression algorithm (OFR), polynomial and wavelet models are identified for the CML's in chaotic regimes. The quantitative dynamical invariants such as the largest Lyapunov exponents and correlation dimensions are estimated and used to evaluate the performance of the identified models.

CNN templates for image processing and pattern formation are derived from neural field equations, advection equations and reaction–diffusion equations by discretizing spatial integrals and derivatives. Many useful CNN templates are derived by this approach. Furthermore, self-organization is investigated from the viewpoint of divergence of vector fields.

The nonlinearity in Chua's oscillator is commonly implemented as a three-segment piecewise-linear resistor. The piecewise-linear nature of the element means that the implementation requires a significant amount of circuitry and the speed of operation is limited. The qualitative behavior of Chua's oscillator has also been captured using a smooth cubic nonlinearity; the implementation of the latter also requires a significant amount of circuitry and suffers from limited speed of operation.

This work describes a novel implementation of Chua's oscillator using just four transistors and a battery to produce a cubic-like nonlinearity. The circuit is simple, robust, and capable of operating at frequencies over one thousand times greater than the original Chua's oscillator.

In this paper, a modular adaptive design is developed for the control of chaos in a Chua's circuit. The dynamics of the Chua's circuit are represented in a parametric strict feedback form. For the purpose of design, it is assumed that all the system parameters are unknown, but the sign of certain parameters and lower bounds on their absolute values are known. Then, an adaptive modular backstepping design approach is used to derive the control system for trajectory tracking of a selected output variable (node voltage). Unlike the direct adaptive controllers for the Chua's circuit available in literature, an input-to-state-stabilizing control law is used herein. The control law accomplishes input-to-state-stability with respect to parameter estimation errors and their derivatives as disturbance inputs. The control system includes a passive identifier (an observer and an adaptation law) for the parameter estimation. In the closed-loop system, control of the node voltage is accomplished, and the remaining state variables remain bounded. Simulation results are presented which show that the controller is effective in accomplishing the precise tracking of the reference trajectory and suppression of the chaotic motion.

The optimal detection of a signal of known form hidden in additive white noise is examined in the framework of stochastic resonance and noise-aided information processing. Conditions are exhibited where the performance in the optimal detection increases when the level of the additive (non-Gaussian bimodal) noise is raised. On the additive signal–noise mixture, when a threshold quantization is performed prior to the optimal detection, another form of improvement by noise can be obtained, with subthreshold signals and Gaussian noise. Optimization of the quantization threshold shows that even in symmetric detection settings, the optimal threshold can be away from the center of symmetry and in subthreshold configuration of the signals. These properties concerning non-Gaussian noise and nonlinear preprocessing in optimal detection, are meaningful to the current exploration of the various modalities and potentialities of stochastic resonance.

We study several algorithms to simulate bone mass loss in two-dimensional and three-dimensional computed tomography bone images. The aim is to extrapolate and predict the bone loss, to provide test objects for newly developed structural measures, and to understand the physical mechanisms behind the bone alteration. Our bone model approach differs from those already reported in the literature by two features. First, we work with original bone images, obtained by computed tomography (CT); second, we use structural measures of complexity to evaluate bone resorption and to compare it with the data provided by CT. This gives us the possibility to test algorithms of bone resorption by comparing their results with experimentally found dependencies of structural measures of complexity, as well as to show efficiency of the complexity measures in the analysis of bone models. For two-dimensional images we suggest two algorithms, a threshold algorithm and a virtual slicing algorithm. The threshold algorithm simulates bone resorption on a boundary between bone and marrow, representing an activity of osteoclasts. The virtual slicing algorithm uses a distribution of the bone material between several virtually created slices to achieve statistically correct results, when the bone-marrow transition is not clearly defined. These algorithms have been tested for original CT 10 mm thick vertebral slices and for simulated 10 mm thick slices constructed from ten 1 mm thick slices. For three-dimensional data, we suggest a variation of the threshold algorithm and apply it to bone images. The results of modeling have been compared with CT images using structural measures of complexity in two- and three-dimensions. This comparison has confirmed credibility of a virtual slicing modeling algorithm for two-dimensional data and a threshold algorithm for three-dimensional data.

In this paper we investigate the geometrical complexity of chaotic attractors in optically injected lasers by calculating the Kaplan–Yorke dimension (D_KY) together with the largest Lyapunov exponent (LLE). For the first time to the best of our knowledge, a color map is constructed indicating regions in the parameter space with high and low dimensionality. It is shown that as the injection power K of the master to the slave laser is increased to high values, the dimension of the attractor decreases for all ranges of the frequency detuning ω. The relation between the trace of the Jacobian matrix and the Kaplan–Yorke dimension shows that the optically injected laser is a system capable to span the dimension anywhere between 2 and 3. A qualitative analysis is finally made in order to identify the material parameters of the slave device for this coupled laser system for a result in the maximized possible dimension, leading to the need for a high electron lifetime and a high linewidth enhancement factor that occur at low temperature operation for a solitary slave semiconductor laser.

Ten years ago, Dixon et al. [1993] studied the behavior of a continuous-time system displaying erratic, apparently chaotic, dynamics. This is a paradoxical case since the system is two-dimensional, which is seemingly a violation of the Poincare–Bendixon theorem. Using numerical studies, Dixon et al. explained such a behavior from the presence of an attracting singularity, which induces arbitrarily large sensitivity to initial conditions. The aim of this letter is to use singularity regularization techniques to study the dynamics around the system singularity. The results obtained in this way explain the paradoxical situation of having continuous "chaotic" dynamics in a two-dimensional system.

We present some sufficient conditions for the existence of isochronous sections of planar differential systems. We consider isochronous sections at a critical point, a cycle, the boundary of a central or attraction region.

In this article, we present some distance estimation formulas that can be used to ray traced slices of the bicomplex Mandelbrot set and the bicomplex filled-Julia sets in dimension three. We also present a simple method to explore and infinitely approach these 3D fractals. Because of its rich fractal structure and symmetry, we emphasize our work on the generalized Mandelbrot set for bicomplex numbers in dimension three: the Tetrabrot.