International Journal of Bifurcation and Chaos

In a recent feature article in this journal, coauthored by Gert van der Heijden, I described the static-dynamic analogy and its role in understanding the localized post-buckling of shell-like structures, looking exclusively at integrable systems. We showed the true significance of the Maxwell energy criterion load in predicting the sudden onset of "shock sensitivity" to lateral disturbances. The present paper extends the survey to cover nonintegrable systems, such as thin compressed shells. These exhibit spatial chaos, generating a multiplicity of localized paths (and escape routes) with complex snaking and laddering phenomena. The final theoretical contribution shows how these concepts relate to the response and energy barriers of an axially compressed cylindrical shell.

After surveying NASA's current shell-testing programme, a new nondestructive technique is proposed to estimate the "shock sensitivity" of a laboratory specimen that is in a compressed metastable state before buckling. A probe is used to measure the nonlinear load-deflection characteristic under a rigidly applied lateral displacement. Sensing the passive resisting force, it can be plotted in real time against the displacement, displaying an equilibrium path along which the force rises to a maximum and then decreases to zero: having reached the free state of the shell that forms a mountain-pass in the potential energy. The area under this graph gives the energy barrier against lateral shocks. The test is repeated at different levels of the overall compression. If a symmetry-breaking bifurcation is encountered on the path, computer simulations show how this can be suppressed by a controlled secondary probe tuned to deliver zero force on the shell.

During the last decade, fractal processes and chaotic systems were widely studied in many areas of research. Chaotic systems are highly dependent on initial conditions. Small changes in initial conditions can generate widely diverging or converging outcomes for both bifurcation or attraction in chaotic systems. In this work, we present a new method on how to generate a new family of chaotic attractors by combining these with a network of fractal processes. The proposed approach in this article is based upon the construction of a new system of fractal processes.

For each λ ∈ [0, 1], λ-power distributional chaos has been defined via Furstenberg families to strengthen distributional chaos. For the sake of distinguishing them, we present a class of dynamical systems, called wedge-shape systems. Through a thorough analysis of dynamical behaviors of all point-pairs, we show that wedge-shape systems can be λ-power distributionally chaotic and admit no λ′-power distributionally scrambled pairs for any λ′ ∈ [0, λ). Then we unfold a picture of distributional chaos with rich hierarchical structures, which helps to improve our comprehension of the diversity of chaos.

In this paper, analytical solutions for period-m motions in a buckled, nonlinear Jeffcott rotor system are obtained. This nonlinear Jeffcott rotor system with two-degrees of freedom is excited periodically from the rotor eccentricity. The analytical solutions of period-m solutions are developed, and the corresponding stability and bifurcation are also analyzed by eigenvalue analysis. Analytical bifurcation trees of period-1 motions to chaos are presented. The Hopf bifurcations of periodic motions cause not only the bifurcation tree but quasi-periodic motions. The quasi-periodic motion can be stable or unstable. Displacement orbits of periodic motions in the buckled, nonlinear Jeffcott rotor systems are illustrated, and harmonic amplitude spectrums are presented for harmonic effects on periodic motions of the nonlinear rotor. Coexisting periodic motions exist in such a buckled nonlinear Jeffcott rotor.

We study the peculiar dynamical features of a fractional derivative of complex-order network. The network is composed of two unidirectional rings of cells, coupled through a "buffer" cell. The network has a Z₃ × Z₅ cyclic symmetry group. The complex derivative D^α±jβ, with α, β ∈ R⁺ is a generalization of the concept of integer order derivative, where α = 1, β = 0. Each cell is modeled by the Chen oscillator. Numerical simulations of the coupled cell system associated with the network expose patterns such as equilibria, periodic orbits, relaxation oscillations, quasiperiodic motion, and chaos, in one or in two rings of cells. In addition, fixing β = 0.8, we perceive differences in the qualitative behavior of the system, as the parameter c ∈ [13, 24] of the Chen oscillator and/or the real part of the fractional derivative, α ∈ {0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, are varied. Some patterns produced by the coupled system are constrained by the network architecture, but other features are only understood in the light of the internal dynamics of each cell, in this case, the Chen oscillator. What is more important, architecture and/or internal dynamics?

Advances in technology have produced more and more intricate industrial systems, such as nuclear power plants, chemical centers and petroleum platforms. Such complex plants exhibit multiple interactions among smaller units and human operators, rising potentially disastrous failure, which can propagate across subsystem boundaries. This paper analyzes industrial accident data-series in the perspective of statistical physics and dynamical systems. Global data is collected from the Emergency Events Database (EM-DAT) during the time period from year 1903 up to 2012. The statistical distributions of the number of fatalities caused by industrial accidents reveal Power Law (PL) behavior. We analyze the evolution of the PL parameters over time and observe a remarkable increment in the PL exponent during the last years. PL behavior allows prediction by extrapolation over a wide range of scales. In a complementary line of thought, we compare the data using appropriate indices and use different visualization techniques to correlate and to extract relationships among industrial accident events. This study contributes to better understand the complexity of modern industrial accidents and their ruling principles.

Responses and bifurcations of a forced triple-well potential system with fuzzy uncertainty are studied by means of the Fuzzy Generalized Cell Mapping (FGCM) method. A rigorous mathematical foundation of the FGCM is established as a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. The FGCM offers a very effective approach for solutions to the fuzzy master equation based on the min–max operator of fuzzy logic. A fuzzy response is characterized by its topology in the state space and its possibility measure of membership distribution functions (MDFs). A fuzzy bifurcation implies a sudden change both in the topology and in the MDFs. The response topology is obtained based on the qualitative analysis of the FGCM involving the Boolean operation of 0 and 1. The MDFs are determined by the quantitative analysis of the FGCM with the min–max calculations. With an increase of the intensity of fuzzy noise, noise-induced escape from each of the potential wells defines two types of bifurcations, namely catastrophe and explosion. This paper focuses on the evolution of transient and steady-state MDFs of the fuzzy response. As the intensity of fuzzy noise increases, steady-state MDFs cover a bigger area in the state space with higher membership values spreading out to a larger area. The previous conjectures are further confirmed that steady-state MDFs are dependent on initial possibility distributions due to the nonsmooth and nonlinear nature of the min–max operation. It is found that as time goes on, transient MDFs spread around three potential wells. The evolutionary orientation of transient MDFs aligns with unstable invariant manifolds leading to stable invariant sets. Two examples of additive and multiplicative fuzzy noise are given.

In this work, we design a kind of linear state feedback control, via the roots connecting-curve, for a class of nonlinear systems which permits to control the Hopf bifurcation. An illustrative example is given.

In this paper, we deal with the following differential system

This article investigates the design of a series of new chaotic attractors. A switching control with different switching surfaces is designed to link two systems of linear integer order differential equations. Under such control, the linked systems have rich dynamical behaviors such as chaos. We also investigate the dynamical behaviors of the corresponding linear fractional order differential equation systems with switching controls. It is shown that such fractional order systems have chaotic behaviors as well.

Recently Braga and Mello conjectured that for a given n ∈ ℕ there is a piecewise linear system with two zones in the plane with exactly n limit cycles. In this paper, we prove a result from which the conjecture is an immediate consequence. Several explicit examples are given where location and stability of limit cycles are provided.

This paper focuses on chaos in periodic discrete systems, whose state space may vary with time. Some close relationships between some chaotic dynamical behaviors of a periodic discrete system and its autonomous induced system are given. Based on these relationships, several criteria of chaos are established and some sufficient conditions for no chaos are given for periodic discrete systems. Further, it is shown that a finite-dimensional linear periodic discrete system is not chaotic in the sense of Li–Yorke or Wiggins. In particular, an interesting problem of whether nonchaotic rules may generate a chaotic system is studied, with some examples provided, one of which surprisingly shows that a composition of globally asymptotically stable maps can be chaotic. In addition, some properties of sign pattern matrices of non-negative square matrices are given for convenience of the study.

In this paper we start with two-dimensional (2D) linear cellular automata (CA) in relation with basic mathematical structure. We investigate uniform linear 2D CA over ternary field, i.e. ℤ₃. Present work is related to theoretical and imaginary investigations of 2D linear CA. Even though the basic construction of a CA is a discrete model, its macroscopic level behavior at large times and on large scales could be a close approximation to a continuous system. Considering some statistical properties, someone may also study geometrical aspects of patterns generated by cellular automaton evolution. After iteratively applying the linear rules, CA have been shown capable of producing interesting complex behaviors. Some examples of CA produce remarkably regular behavior on finite configurations. Using some simple initial configurations, the produced pattern can be self-replicating regarding some linear rules. Here we deal with the theory 2D uniform periodic, adiabatic and reflexive boundary CA (2D PB, AB and RB) over the ternary field ℤ₃ and the applications of image processing for patterns generation. From the visual appearance of the patterns, it is seen that some rules display sensitive dependence on boundary conditions and their rule numbers.

A delay predator–prey model is formulated with continuous threshold prey harvesting and Holling response function of type III. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. The positive invariance of the non-negative orthant is proved and the uniform boundedness of the trajectories. Stability of equilibria is investigated and the existence of some local bifurcations is established: saddle-node bifurcation, Hopf bifurcation. We use optimal control theory to provide the correct approach to natural resource management. Results are also obtained for optimal harvesting. Numerical simulations are given to illustrate the results.

Surrogate data methods have been widely applied to produce synthetic data, while maintaining the same statistical properties as the original. By using such methods, one can analyze certain properties of time series. In this context, Theiler's surrogate data methods are the most commonly considered approaches. These are based on the Fourier transform, limiting them to be applied only on stationary time series. Consequently, time series including nonstationary behavior, such as trend, produces spurious high frequencies with Theiler's methods, resulting in inconsistent surrogates. To solve this problem, we present two new methods that combine time series decomposition techniques and surrogate data methods. These new methods initially decompose time series into a set of monocomponents and the trend. Afterwards, traditional surrogate methods are applied on those individual monocomponents and a set of surrogates is obtained. Finally, all individual surrogates plus the trend signal are combined in order to create a single surrogate series. Using this method, one can investigate linear and nonlinear Gaussian processes in time series, irrespective of the presence of nonstationary behavior.

This paper discusses a simple yet interesting three-dimensional autonomous quadratic chaotic system, which has two fixed points of saddle-focus type but does not belong to the class of Šil'nikov type since there are neither homoclinic nor heteroclinic orbits. Dynamical behaviors of this chaotic system are investigated in detail through analyzing bifurcation routes, Lyapunov exponents and Poincaré mappings. These investigations partially reveal the mechanism of how chaos in non-Šil'nikov type of systems is generated.

We give the normal forms of all polynomial differential systems in ℝ³ which have a nondegenerate or degenerate quadric as an invariant algebraic surface. We also characterize among these systems those which have a Darboux invariant constructed uniquely using the invariant quadric, giving explicitly their expressions. As an example, we apply the obtained results in the determination of the Darboux invariants for the Chen system with an invariant quadric.

In this paper, we consider a model which is a modulated equation in a discrete nonlinear electrical transmission line. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we derive all explicit exact parametric representations of solutions (including smooth solitary wave solutions, smooth periodic wave solutions, peakons, compactons, periodic cusp wave solutions, etc.) under different parameter conditions.

This paper studies the existence of bistable states and control strategies to induce state transitions of a delay differential equation model of neutrophil dynamics. We seek the conditions that a stable steady state and an oscillatory state coexist in the neutrophil dynamical system. Physiologically, stable steady state represents the healthy state, while oscillatory state is usually associated with diseases such as cyclical neutropenia. We study the control strategies to induce the transitions from the disease state to the healthy state by introducing temporal perturbations to system parameters. This study is valuable in designing clinical protocols for the treatment of cyclical neutropenia.