International Journal of Bifurcation and Chaos

Chaos is a physical and mathematical phenomenon discovered by E. Lorenz in 1963. The first simple electronic implementation had been invented by L. O. Chua in 1984. This electronic circuit, called Chua's circuit was designed for ease of implementation. In the current brief we will explain chaos by building Chua's chaotic circuit using our Chua's circuit kit with inexpensive components. For readers without access to an oscilloscope, this paper proposes the use of a laptop/Personal Computer to capture the voltage waveforms generated from the circuit and plot the waveforms on a computer screen using a virtual oscilloscope software provided by the authors. The kit is available, the software is downloadable.

In this tutorial paper, we present a history of Smale horseshoe and an overview of the progress of topological horseshoe theory. Then we offer a pedagogical exposition of elements of topological horseshoe theory with a lot of examples. Finally we demonstrate some typical applications of topological horseshoe theory to practical dynamical systems.

In this paper, we show that the mathematical classification of one-dimensional cellular automata given by Dubois–Violette and Rouet [1987] does not truly represent the different behaviors possible. In fact we show that many rules of their type O have totally different behavior. A better classification is based on Smale's basic sets and nonwandering points.

Network architecture can lead to robust synchrony in coupled maps and to codimension one bifurcations from synchronous fixed-points at which the associated Jacobian is nilpotent.

We discuss the codimension one synchrony-breaking period-doubling bifurcations for three-cell coupled maps. Interesting phenomena occur for all these coupled maps — a branch of period-2 points with amplitude growing as |λ|^⅙ for coupled networks of feed-forward type, as well as multiple (two) branches of period-2 points with amplitude growing as |λ|^½ for coupled networks of feed-forward type.

We also discuss how some results related to patterns of synchrony that are valid for coupled vector fields are also valid for coupled maps.

This paper proposes a new technique for generating random-looking binary digits based on an irregularly decimated chaotic map. We present a class of irregularly decimated chaos-based keystream generators, related to the shrinking generator, for the generation of binary sequences. Each generator consists of two subsystems: a control subsystem and a generating subsystem, where each subsystem is based on a single chaotic map. This chaotic map is presented as a 1-D piecewise chaotic map related to the chaotic logistic map. We conduct an analysis of the dynamical behavior of the proposed map to integrate it as a component in the proposed generators subsystems. The output bits of these keystream generators are produced by applying a threshold function to convert the floating-point iterates of the irregularly decimated map into a binary form. The generated keystream bits are demonstrated to exhibit high level of security, long period length, high linear complexity measure and random-like properties at given certain parameter values. Standard statistical tests on the proposed generators, as well as other keystream generators, are performed and compared.

We present an algorithm, completely random in nature, that, without invoking a fitness function or purposeful design, produces symmetric replicas in a population of so-called cellicules. Cellicules consist of "cells" arranged in a structure with geometric symmetry S. Each cell has one of two possible states, thus defining a state-configuration pattern on a cellicule. The algorithm acts recurrently on a population of cellicules, possibly randomly initialized, through a random "copying interaction" between two randomly selected cellicules that first undergo a random reorientation in accordance with the symmetry S. The dynamics of the algorithm is analyzed in detail for several symmetries. This shows that it is a random walk with absorbing states which correspond to a population in which all cellicules have an identical S-symmetric configuration pattern. We discuss some aspects concerning the evolution of cellicule-populations under mixing and mutation, and some variations on the basic algorithm.

This paper is Part II of an earlier paper dealing with the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, and an external periodical excitation of period τ = 2π/ω (amplitude E). In the absence of this excitation, this equation of Duffing type does not give rise to self-oscillations. Part I was essentially devoted to analyze the harmonics behavior of period τ solutions, more precisely the behavior of rank-p harmonics according to the points of the parameter plane (ω,E). The present Part II deals with period kτ solutions related to a cascade of closed fold bifurcation curves related to fractional harmonics p/k, k = 3, p = 3,4,…. With respect to the organization of bifurcation curves associated with rank-p harmonics of the basic period τ, this study shows that the situation is a lot more complex for the sequence of bifurcation curves related to rank-p/3 harmonics.

We study the bifurcation of limit cycles in the neighborhood of a 2-polycycle through two hyperbolic saddles with both hyperbolicity ratios being 1. The cyclicity is expressed as an explicit function of saddle quantities. Our results can be applied to the case when both connections are broken under perturbations.

A method for finding reduced-order approximations of turbulent flow models is presented. The method preserves bounds on the production of turbulent energy in the sense of the norm of perturbations from a notional laminar profile. This is achieved by decomposing the Navier–Stokes system into a feedback arrangement between the linearized system and the remaining, normally neglected, nonlinear part. The linear system is reduced using a method similar to balanced truncation, but preserving bounds on the supply rate. The method involves balancing two algebraic Riccati equations. The bounds are then used to derive bounds on the turbulent energy production. An example of the application of the procedure to flow through a long straight pipe is presented. Comparison shows that the new method approximates the supply rate at least as well as, or better than, canonical balanced truncation.

This paper shows the effects of optimal control techniques on pattern formation in a three-dimensional Rayleigh–Bénard problem with nonhomogeneous heating from below. In particular, we consider a cylindrical domain with a heating profile at the bottom localized around the origin. An axisymmetric basic state appears as soon a nonzero horizontal temperature gradient is imposed. The basic state may bifurcate to different solutions as spiral waves, stationary patterns, etc., depending on the vertical and lateral temperature gradients and on the shape of the heating function. An optimal control problem that determines the thermal boundary condition minimizing the enstrophy is proposed. The control boundary condition obtained changes the leading mode in the bifurcation and allows to eliminate the instabilities for some values of the parameters.

The existence of breaking wave solutions of the second class of singular nonlinear wave equations is proved by methods from the dynamical systems theory. For the second class of singular nonlinear traveling wave equations, dynamical behaviors of the traveling wave solutions are completely classified and thoroughly discussed. Corresponding to some bounded orbits of the traveling systems, exact parametric representations of traveling wave solutions are derived within different parameter regions of the parameter space.

In this paper, we investigate the problem of identifying or modeling nonlinear dynamical systems undergoing periodic and period-like (recurrent) motions. For accurate identification of nonlinear dynamical systems, the persistent excitation condition is normally required to be satisfied. Firstly, by using localized radial basis function networks, a relationship between the recurrent trajectories and the persistence of excitation condition is established. Secondly, for a broad class of recurrent trajectories generated from nonlinear dynamical systems, a deterministic learning approach is presented which achieves locally-accurate identification of the underlying system dynamics in a local region along the recurrent trajectory. This study reveals that even for a random-like chaotic trajectory, which is extremely sensitive to initial conditions and is long-term unpredictable, the system dynamics of a nonlinear chaotic system can still be locally-accurate identified along the chaotic trajectory in a deterministic way. Numerical experiments on the Rossler system are included to demonstrate the effectiveness of the proposed approach.

This paper studies the security of a chaotic cryptosystem based on Chua's circuit and implemented with State Controlled Cellular Neural Networks (SC-CNN). Here, we prove that the plaintext can be retrieved by bandpass filtering of the ciphertext or by using an imperfect decoder with wrong receiver parameters. In addition, we find that the key space of the system can be reduced notably, and the required resolution of the parameter values to recover a meaningful plaintext is as coarse as 5%, easing a brute-force attack. The system parameters can be determined with high precision through the analysis of the decoding error produced by the mismatch between the parameters of receiver and transmitter.

Chua's circuit is very suitable as a programmable chaos generator because of its robust nonlinearity. In addition to exhibiting a rich variety of bifurcation and chaos phenomenon, this circuit can be modeled and realized with a fixed main system block and many different nonlinear function blocks such as piecewise-linear function, cubic-like function, piecewise-quadratic function and other trigonometric functions. This paper presents a FPAA (Field Programmable Analog Array) based programmable implementation of Chua's circuit. Nonlinear function blocks used in Chua's circuit are modeled with an FPAA and hence a model can be rapidly changed for realization of other nonlinear functions. In this study, four FPAA-based reconfigurable implementations of Chua's circuit have been realized. Experimental results agree with numerical simulation and results obtained from discrete electronic implementations of Chua's circuit.

In this paper we investigate, via numerical simulations, the parameter space of the set of autonomous differential equations of a Chua oscillator, where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode was replaced by a cubic polynomial. As far as we know, we are the first to report that this parameter-space presents islands of periodicity embedded in a sea of chaos, scenario typically observed only in discrete-time models until recently. We show that these islands are self-similar, and organize themselves in period-adding bifurcation cascades.

A simplified Lorenz system with one bifurcation parameter is investigated by a detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations and routes to chaos. The results show that this system has complex dynamics with interesting characteristics.

In this paper, we consider a planar dynamical system with a piecewise linear function containing two dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system, with five singular points, can have at most four limit cycles, three of which surround the foci one by one and the fourth limit cycle surrounds all of the singular points of this system.

A logistic equation with piecewise constant arguments is investigated. Firstly, the linear stability of the model is studied. It is found that there exists a Hopf bifurcation when the parameter passes a critical value. Then the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution is derived by using the normal form method and center manifold theorem. Finally, computer simulations are performed to illustrate the analytical results found.

An interesting phenomenon in billiards was found. If the midpoints of every segment of a straight line in trajectory of a billiard were taken, they will form an interesting diagram which we call midpoint diagram. After research on several kinds of billiards, it was discovered that midpoint diagram which is made up of several discrete points corresponds to the periodic behavior of the billiard, and the midpoint diagram made up of lines corresponds to quasi-periodic behavior, while the midpoint diagram made up of large numbers of dispersion points corresponds to chaotic behavior. The midpoint diagram can be used to describe the dynamical behavior of billiards.

Let X be a dendrite and f : X → X be a continuous map. Denote by R(f) and P(f) the sets of recurrent points and periodic points of f respectively. In this paper we show that, if the cardinal number Card(End(X)) of the set of endpoints of X is less than the cardinal number c of the continuum, then . From this we derive that, if Card(End(X)) < c, then f is chaotic in the sense of Devaney if and only if f is transitive.

It has been suggested that the properties of "integration" and "differentiation" are necessary for the emergence of consciousness. We present a dynamical system model that is based on these two conditions. The collection of neurons are partitioned into clusters on which we define a map that reflects the communication between clusters. Such a map displays the forward and backward circuitry between clusters in a probabilistic manner. The presence of "re-entry" guarantees that the map is sufficiently complex, that is, nonlinear and chaotic, to possess numerous invariant sets of clusters, which are referred to as agglomerations. We suggest that an agglomeration that is mixing characterizes a conscious state. The model establishes a theoretical framework that may structure and encourage experimental work.

Traditional Boolean networks consist of nodes within a single network, each updating synchronously, although asynchronous versions have also been presented. In this paper the dynamics of two, mutually coupled traditional networks are investigated. In particular, the effects of varying the degree and type of intra-network connectivity are explored. The effects from different inter-network evolution rates are then considered, i.e. asynchronousity at the network level is examined. Finally, state memory is included within the nodes of coupled networks and shown to alter the dynamics of the networks under certain circumstances.

In this paper, a time-delayed feedback controller is applied to a fluid flow model of Internet congestion control system in order to control Hopf bifurcation. It has been shown that the system without control loses stability and a Hopf bifurcation occurs when the bifurcation parameter, which is the communication delay of the model, passes through a critical value. Therefore, a control approach based on delayed feedback is proposed to postpone the onset of undesired Hopf bifurcation. Theoretical analysis and numerical simulations confirm that the controller is efficient in bifurcation control of the Internet congestion control system.