International Journal of Bifurcation and Chaos

Statistical and dynamical properties of bivariate (two-dimensional) maps are less understood than their univariate counterparts. This paper gives a synthesis of extended results with exemplifications by bivariate logistic maps, the bivariate Arnold cat map and a bivariate Chebyshev map. The use of synchronization from bivariate maps in communication modeling is exemplified by an embryonic chaos shift keying system.

The ability of nonlinear dynamical systems to process incoming information is a key problem of many fundamental and applied sciences. Information processing by computation with attractors (steady states, limit cycles and strange attractors) has been a subject of many publications. In this paper, we discuss a new direction in information dynamics based on neurophysiological experiments that can be applied for the explanation and prediction of many phenomena in living biological systems and for the design of new paradigms in neural computation. This new concept is the Winnerless Competition (WLC) principle. The main point of this principle is the transformation of the incoming identity or spatial inputs into identity-temporal output based on the intrinsic switching dynamics of the neural system. In the presence of stimuli the sequence of the switching, whose geometrical image in the phase space is a heteroclinic contour, uniquely depends on the incoming information. The key problem in the realization of the WLC principle is the robustness against noise and, simultaneously, the sensitivity of the switching to the incoming input. In this paper we prove two theorems about the stability of the sequential switching and give several examples of WLC networks that illustrate the coexistence of sensitivity and robustness.

Cellular neural networks have found applications both as a platform for high speed image processing, as well as a model for complex spatiotemporal dynamical phenomena. This paper describes spatial pattern formation in a CMOS VLSI implementation of a CNN. With four layers of 32×64 cells, this CNN contains the most recurrently interconnected 2D layers of any VLSI CNN implementation to date. With over 8000 cells, it is also one of the largest in terms of the total number of cells. The spatial patterns generated by this network are bands of activity with a preferred scale and orientation, both of which can be adjusted via external bias voltages. The VLSI implementation provides a physical substrate for experimenting with complex spatiotemporal phenomena in real time, while also being amenable to theoretical analysis. Our experimental characterization of the observed patterns are in excellent agreement with our theoretical predictions.

We consider equilibrium configurations of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods when subject to end loads and clamped boundary conditions. In a first paper [Neukirch & Henderson, 2002], we discussed symmetry properties of the equilibrium configurations of the center line of the rod. Here, we are interested in the set of all parameter values that yield equilibrium configurations that fulfill clamped boundary conditions. We call this set the solution manifold and we compute it using a recently introduced continuation algorithm. We then describe the topology of this manifold and how it comprises different interconnected layers. We show that the border set of the different layers is the well-known solution set of buckled rings.

Catastrophe theory is employed to classify different types of nonlinear oscillators, basing on the complication of their potentials. By using Thom's catastrophe unfoldings as oscillator potentials, we have introduced more general models to describe the dynamics of nonlinear oscillators, differing from each other by the form of their potential wells and by the possibility of escape. Spreading the investigation in the space of the parameters of the potential function, we have revealed that our examples defined via Thom's catastrophe unfoldings have some type of universal properties in the context of forced oscillations. For oscillators with nonescaping solutions, we have detected such typical bifurcation structures as crossroad areas and spring areas, and have described the universal scenario of their evolution under the forcing amplitude variation. On increasing the potential function degree, the complexity of the charts of the dynamical regimes results from the repetition of the described bifurcation scenario. For oscillators with escaping solutions, such general properties were investigated, as dependence of the charts of the dynamical regimes and the basins on the parameters of the potential function. We have observed that these properties are typical in a broad range of the control parameters.

In this paper we show that a system containing any knot or link can be directly constructed in a simple way. The system is not chaotic and can even contain wild knots.

Applying metric (Banach-like) and topological (Schauder-like) fixed-point theorems, the existence of metric and topological fractals is respectively proved as (sub)invariant subsets of the Hutchinson–Barnsley map generated by a multifunction system. Weakly contractive and compact multifunction systems are considered, but systems of more general multifunctions are discussed as well. The notions of hyperspaces and AR-spaces are employed for this goal.

A nonlinear differential equation of fractional order μ between two and three is studied. Existence, uniqueness and regularity are proved. For μ=3 the equation represents a chaotic dynamical system. As μ varies from 2 through 3 the dynamical system experiences period doublings and triplings and for two ranges of μ becomes chaotic. The attractor is represented by a trajectory which is dense in it and is studied by numerical estimates of its dimension and Lyapunov exponents. The infinite dimension of the state space (which is the history space) is taken into account.

Bonhöffer–van der Pol(BVP) oscillator is a classic model exhibiting typical nonlinear phenomena in the planar autonomous system. This paper gives an analysis of equilibria, periodic solutions, strange attractors of two BVP oscillators coupled by a resister. When an oscillator is fixed its parameter values in nonoscillatory region and the others in oscillatory region, create the double scroll attractor due to the coupling. Bifurcation diagrams are obtained numerically from the mathematical model and chaotic parameter regions are clarified. We also confirm the existence of period-doubling cascades and chaotic attractors in the experimental laboratory.

The spatiotemporal symmetry properties of a periodically forced network of three Continuous Stirred Tank Reactors are assessed. The symmetry is induced to the system by a cyclic switching of the feed and discharge positions. The symmetry properties imply that the Poincaré map is the third iterate of another nonstroboscopic map. This feature is here used to characterize the symmetry of the regime solutions, and to carry out bifurcation analysis. Possible bifurcation scenarios and, in particular, symmetry-breaking bifurcations are discussed. As the switch time is varied, different transitions have been identified: among the others, an important role in the birth of asymmetric regimes is played by frequency-locking phenomena. In addition, different routes to chaotic regimes (both symmetric and asymmetric) are reported.

The output tracking for a general family of nonlinear systems presents formidable technical challenges. In this paper, we present a novel scheme for tracking control of a class of affine nonlinear systems with multi-inputs. This effective procedure is based on a new sliding mode design for tracking control of such nonlinear systems. The construction of an optimal sliding mode is a difficult problem and no systematic and efficient method is currently available. Here, we develop an innovative approach that utilizes a chaotic optimizing algorithm, which is then successfully applied to obtain the optimal sliding manifold. The existing efficient reaching law approach is then utilized to synthesize the sliding mode control law. The sliding mode control scheme proposed here is particularly appropriate for robust tracking of the chaotic motion trajectory.

This paper is concerned with chaos synchronization of discrete-time systems for communication purposes. The information to be masked is embedded in the chaotic dynamics of the transmitter and acts as an external input. It cannot be transmitted to the receiver for the sake of security. Hence, the receiver system must be designed such that the information can be unmasked, given the only available output data consisting of a function of the state vector. This problem is solved with an Unknown Input Observers based approach which achieves an Input Independent Global Synchronization. The conditions of existence of such kind of observers along with the systematic procedure to design the gains of the observers are based on the notion of polyquadratic stability and expressed in terms of Linear Matrix Inequalities.

The current feedback op amps (CFOAs), with some significant advantages over the conventional op amps, have been used instead of voltage op amps (VOAs) in new implementations of Chua's circuit. In our previous study, after providing a comparative investigation of CFOA-based realizations of Chua's circuit in the literature, we have also presented an alternative inductorless CFOA-based realization of Chua's circuit, and the circuit's chaotic behavior by Pspice simulations. In this paper, we investigate CFOA-based Chua's circuit by constructing an experimental setup, and testing the performance of the proposed implementation at different frequencies. Its excellent high frequency performance was experimentally verified.

An experimental method for generating homoclinic oscillations using two nonidentical Chua's oscillators coupled in unidirectional mode is described here. A homoclinic oscillation is obtained at the response oscillator in the weaker coupling limit of phase synchronization. Different phase locking phenomena of homoclinic oscillations with external periodic pulse have been observed when the frequency of the pulse is close to the natural frequency of the homoclinic oscillation or its subharmonics.

A contrast enhancement and image inverting tool using a lattice of uncoupled nonlinear oscillators is proposed. We show theoretically and numerically that the gray scale picture contrast is strongly enhanced even if this one is initially very small. An image inversion can be also obtained in real time with the same Cellular Nonlinear Network (CNN) without reconfiguration of the network. A possible electronic implementation of this CNN is finally discussed.

Recently, we have investigated a new chaotic system of three-dimensional autonomous quadratic ordinary differential equations, and found that the system visually displays a four-scroll chaotic attractor confirmed by both numerical simulations and circuit implementation. In this paper, we further study the following question: Is it really true that this system can generate a four-scroll chaotic attractor, or is it only a numerical artifact? By a more careful theoretical analysis along with some further numerical simulations, we conclude that the four-scroll chaotic attractor of this system, which we observed on both computer and oscilloscope, cannot actually exist in theory. The fact is that this system has two co-existing two-scroll chaotic attractors that are arbitrarily close in the phase space for some system parameters, therefore extremely tiny numerical round-off errors or signal fluctuations will nudge the system orbit to switch from one attractor to another, thereby forming the seemingly single four-scroll chaotic attractor on screen display.

In this paper, a neuron is introduced for chaotifying nonchaotic discrete-time systems with arbitrary dimensions. By modeling the neuron with a hyperbolic tangent activation function, a scalar feedback signal expressed in a linear combination of the neuron outputs is used. Chaos can then be generated from the controlled discrete-time system. The existence of chaos is verified by both theoretical proof and computer simulations.

Dynamical properties of a chaotic neural network model in a chaotically wandering state are studied with respect to sensitivity to weak input of a memory fragment. In certain parameter regions, the network shows weakly chaotic wandering, which means that the orbits of network dynamics in the state space are localized around several memory patterns. In the other parameter regions, the network shows highly developed chaotic wandering, that is, the orbits become itinerant through ruins of all the memory patterns. In the latter case, once the external input consisting of a memory fragment is applied to the network, the orbit quickly moves to the vicinity of the corresponding memory pattern including the memory fragment within several iteration steps. Thus, chaotic dynamics in the model is effective for instantaneous search among memory patterns.

The main characteristic of a forced single-degree-of-freedom weakly nonlinear system is determined by its primary, super- and sub-harmonic resonances. A nonlinear parametric feedback control is proposed to modify the steady-state resonance responses, thus to reduce the amplitude of the response and to eliminate the saddle-node bifurcations that take place in the resonance responses. The nonlinear gain of the feedback control is determined by analyzing the bifurcation diagrams associated with the corresponding frequency-response equation, from the singularity theory approach. It is shown by illustrative examples that the proposed nonlinear feedback is effective for controlling three kinds of resonance responses.

This letter proposes a new global synchronization theorem for a class of chaotic systems. Specially, in the synchronization theorem neither the linear part nor the nonlinear part of the chaotic system requires the special structure indicated in [Wang & Wang, 1998]. We take a linear combination of the original system state variables as the scale-driving signal. We prove that the global synchronization can be attained through the simple linear output error feedback. The linear output error feedback gain is a function of a free parameter. We use Chua's chaotic oscillator and Saito's hyperchaotic system to illustrate the applicability of our synchronization scheme and discuss the issues concerning robustness.

In this letter, we reconsider the problem of controlling chaos in the well-known Lorenz system. Firstly, the difficulty in controlling the Lorenz system is discussed in the general strict-feedback form. Then, singularity-free adaptive control is presented for the Lorenz system with three key parameters unknown by exploiting the physical property of the system using decoupled backstepping design. The proposed controller guarantees the asymptotic convergence of the output and the boundedness of all the signals in the closed-loop system. Simulation results are conducted to show the effectiveness of the approach.

We analyze the behavior of some coupled chaotic systems, which are synchronization-like. This phenomenon occurs when all conditional Lyapunov exponents of a system are not negative. Recently, Shuaiet et al. [1997] observed that synchronization can be achieved even with positive conditional Lyapunov exponents. In this paper we review this observation, and, based on this observation, we will see that not only interesting synchronization behaviors occur with positive or zero conditional Lyapunov exponents, but also these behaviors depend on different eigenvalues of the linearized system describing the evolution of the difference between the pair of chaotic systems.

In this paper, a criteria of suppressing chaos for a kind of nonlinear oscillators is established by the theory of the strange attractor. The oscillators considered include Duffing, van der Pol, Duffing–van der Pol and pendulum. According to this criteria, we analyze the phase effect using two methods, one by adding the second external force term and the other by adding parametric excitation, both of which may be used to suppress chaos in the systems. Some examples are used to illustrate the validity of the criteria and the importance of phase effect in suppressing chaos.

This paper addresses the problem of reconstructing a slowly-varying information-bearing signal from a parametrically modulated, nonstationary dynamical signal. A chaotic electronic oscillator model characterized by one control parameter and a double-scroll-like attractor is used throughout the study. Wavelet transforms are used to extract features of the chaotic signal resulting from parametric modulation of the control parameter by the useful signal. The vector of feature coefficients is fed into a feed-forward neural network that recovers the embedded information-bearing signal. The performance of the developed method is cross-validated through reconstruction of randomly-generated control parameter patterns. This method is applied to the reconstruction of speech signals, thus demonstrating its potential utility for secure communication applications. Our results are validated via numerical simulations.

By developing topological shooting methods, the existence of single-humped periodic solutions and homoclinic orbit for a class of fourth-order ordinary differential equations is obtained, under some general conditions. Using these strict mathematical conclusions to a model of the deflection patterns of elastic struts resting on elastic foundations, the existence of single-humped periodic solutions which have been found by asymptotical and numerical methods is determined. An estimation of the half-period of the periodic solutions is also given.

We study the behavior of topological entropy in one-parameter families of interval maps obtained from a continuous map f by truncating it at the level depending on the parameter. When f is piecewise monotone, the entropy function has the devil's staircase structure.

This letter demonstrates the use of Shannon entropies to detect chaos exhibited in some local regions on the phase portraits. When both eigenvalues of the second-order digital filters with two's complement arithmetic are outside the unit circle, the Shannon entropies of the state variables are independent of the initial conditions and the filter parameters, except for some special values of the filter parameters. At these special values, the Shannon entropies of the state variables are relatively small. The state trajectories corresponding to these filter parameters either exhibit random-like chaotic behaviors in some local regions or converge to some fixed points on the phase portraits. Hence, by measuring the Shannon entropies of the state variables, these special state trajectory patterns can be detected. For completeness, we extend the investigation to the case when the eigenvalues of the second-order digital filters with two's complement arithmetic are complex and are inside or on the unit circle. It is found that the Shannon entropies of the symbolic sequences for the type II trajectories may be higher than that for the type III trajectories, even though the symbolic sequences of the type II trajectories are periodic and have limit cycle behaviors, while that of the type III trajectories are aperiodic and have chaotic behaviors.

Let Q denote a quadratic polynomial and A_∞ the super-attracting basin of Q at the point ∞ on the Riemann sphere . There exists a unique Riemann mapping Φ from the open disk onto A_∞ such that Φ(∞)=∞, Φ'(∞)=1 and Φ^-1 conjugates Q:A_∞→A_∞ to the squaring map S:D→D:z↦z². In this paper, we show if Q is real and infinitely renormalizable of bounded type then the continuous extension of Φ to the closed disk cannot have any Hölder continuity on the boundary circle .