International Journal of Bifurcation and Chaos

This paper is meant to serve as a tutorial describing the link between symbolic dynamics as a description of a chaotic attractor, and how to use control of chaos to manipulate the corresponding symbolic dynamics to transmit an information bearing signal. We use the Lorenz attractor, in the form of the discrete successive maxima map of the z-variable time-series, as our main example. For the first time, here, we use this oscillator as a chaotic signal carrier. We review the many previously developed issues necessary to create a working control of symbol dynamics system. These include a brief review of the theory of symbol dynamics, and how they arise from the flow of a differential equation. We also discuss the role of the (symbol dynamics) generating partition, the difficulty of finding such partitions, which is an open problem for most dynamical systems, and a newly developed algorithm to find the generating partition which relies just on knowing a large set of periodic orbits. We also discuss the importance of using a generating partition in terms of considering the possibility of using some other arbitrary partition, with discussion of consequences both generally to characterizing the system, and also specifically to communicating on chaotic signal carriers. Also, of practical importance, we review the necessary feedback-control issues to force the flow of a chaotic differential equation to carry a desired message.

Basins generated by a noninvertible mapping formed by two symmetrically coupled logistic maps are studied when the only parameter λ of the system is modified. Complex patterns on the plane are visualized as a consequence of the basins' bifurcations. According to the already established nomenclature in the literature, we present the relevant phenomenology organized in different scenarios: fractal islands disaggregation, finite disaggregation, infinitely disconnected basin, infinitely many converging sequences of lakes, countable self-similar disaggregation and sharp fractal boundary. By use of critical curves, we determine the influence of zones with different number of first rank preimages in the mechanisms of basin fractalization.

We study the set of periods of tree maps f : T → T which are monotone between any two consecutive points of a fixed periodic orbit P. This set is characterized in terms of some integers which depend only on the combinatorics of f|_P and the topological structure of T. In particular, a typep ≥ 1 of P is defined as a generalization of the notion introduced by Baldwin in his characterization of the set of periods of star maps. It follows that there exists a divisor k of the period of P such that if the set of periods of f is not finite then it contains either all the multiples of kp or an initial segment of the _kp≥ Baldwin's ordering, except for a finite set which is explicitly bounded. Conversely, examples are given where f has precisely these sets of periods.

Stability of steady-state solutions of 1-dim coupled map lattices is studied. The stability is determined by the spectrum of linear operators on two-sided sequences of vectors in arising as a linearization of the corresponding nonlinear evolution operators. Theoretical results are applied to several examples.

This paper deals with the identification and the qualitative analysis of a dynamic model of a shallow lagoon. The model describes the relations between biotic (phytoplankton, zooplankton) and abiotic (oxygen, nutrients) components of a lagoon. The first step of the paper is to derive estimates for the model parameters through an identification procedure using real data. The second step is to perform a qualitative analysis of the model dynamics, via the introduction of a parameterized reduced order model. The main contribution of the paper is to make an effort in the direction of synergyzing the identification stage with the qualitative analysis of the model dynamics, in order to gain a better understanding of the system behavior and obtain more reliable estimates for the model parameters and exogenous inputs.

We study the properties of a n²-dimensional Lotka–Volterra system describing competition among n different species with adaptive skills, i.e. whose interaction coefficients are time averages of the species level of interaction over their past.

Starting by the case of adaptive competition among species all having the same carrying capacities, we focus our attention on the model obtained on perturbing the carrying capacity of a fixed species, which is made more or less disadvantaged.

We prove the existence of a certain class of invariant subspaces and introduce a seven-dimensional reduced model, where n appears as a parameter, which gives full account of existence and stability of equilibria in the system. The relevance of this reduced model to the complete one has also been found when the time dependent regimes have been investigated.

Ecologically relevant questions, i.e. species survival and the time dependent behavior of the system have also been analyzed focusing on the role of behavioral adaptation. In particular, we have found that competitive exclusion cannot occur but coexistence is possible in various forms (i.e. competitive stable equilibria and different periodic oscillations).

We study the dynamics of a rotating asymmetric body under the influence of an aerodynamic drag. We assume that the drag torque is proportional to the angular velocity of the body. Also we suppose that one of the moments of inertia of the body is a periodic function of time and that the center of mass of the body is not modified. Under these assumptions, we show that the system exhibits a transient chaotic behavior by means of a higher dimensional generalization of the Melnikov's method. This method give us an analytical criterion for heteroclinic chaos in terms of the system parameters. These analytical results are confirmed by computer numerical simulations of the system rotations.

From the point of information theory, the baseband binary pulse amplitude modulated (PAM) signals transmission, via tuning the nonlinear receiver's parameters, is studied over an additive white Gaussian noise (AWGN) channel. It is demonstrated that the channel capacity of baseband binary communication systems, for a given signal added noise, can be maximized by optimal designed receivers. This new form of stochastic resonance (SR) is referred to as parameter-induced stochastic resonance (PSR) in a broad sense. PSR effect does not require that the input signals are subthreshold. The characteristic behavior of channel capacity versus transmission bit rate is also studied. For reproducing the original binary signals more correctly, time scale transformation method and the approach of ensemble average probability of error bits are introduced. We observe a marked enhancement of the channel capacity for binary PAM signals transmission by PSR in numerical simulations. With this theory and method, the practical applications are considered.

One of the usual ways to build up mathematical models corresponding to a wide class of DC–DC converters is by means of piecewise linear differential equations. These models belong to a class of dynamical systems called Variable Structure Systems (VSS). From a classical design point of view, it is of interest to know the dynamical behavior of the system when some parameters are varied. Usually, Pulse Width Modulation (PWM) is adopted to control a DC–DC converter. When this kind of control is used, the resulting mathematical model is nonautonomous and periodic. In this case, the global Poincaré map (stroboscopic map) gives all the information about the system.

The classical design in these electronic circuits is based on a stable periodic orbit which has some desired characteristics. In this paper, the main bifurcations which may undergo this orbit, when the parameters of the circuit change, are described. Moreover, it will be shown that in the three basic power electronic converters Buck, Boost and Buck–Boost, very similar scenarios are obtained. Also, some kinds of secondary bifurcations which are of interest for the global dynamical behavior are presented. From a dynamical systems point of view, VSS analyzed in this work present some kinds of bifurcations which are typical in nonsmooth systems and it is impossible to find them in smooth systems.

In this Letter we analyze the concept of partial synchronization as observer design. Our aim is to estimate a signal, the function of the coordinates of a particular system that, when considering a specific output, is not necessarily fully observable.

We describe a method for the design of feedback stabilization control laws for nonlinear systems using the theory of normal forms and the results from optimal control theory. We show that the resulting controllers can provide a larger region of stability than local linear controllers designed to perform the same task.

A novel approach to control chaotic systems has been developed. The approach employs the technique of phase space reconstruction in nonlinear dynamical systems theory to construct a linear part in the reconstructed system and design a feedback control law. The effectiveness of the proposed approach has been demonstrated through its applications to two well-known chaotic systems: Lorenz chaos and Rössler chaos.

This paper deals with the control of a class of n-dimensional chaotic systems. The proposed method consists in a Variable Structure Control approach based on system energy consideration for both controller design and system stabilization. First, we present some theoretical results related to the stabilization of global invariant sets included in a selected two-dimensional subspace of the state space. Then, we define some conditions, involving both system definition and control law design, under which the stabilized orbits can be maintained in a neighborhood of an invariant, nondegenerate, closed conic section (i.e. an ellipse or a circle). Finally, an example related to the chaotic circuit of Chua is given.

In this paper, we present experiments using a nonlinear electrical line, modeling the FitzHugh–Nagumo equation, without recovery term. Different patterns are studied according to the parameters of this medium and initial conditions. We then propose to apply these results to the domain of signal processing. We show that erosion and dilation of a binary signal, two kinds of binarization — one depending on an amplitude threshold, the other on an energetical threshold — and nonlinear filtering allowing noise removal can be obtained in the same medium.

This Letter displays, via the numerical simulation of a real digital filter, that a finite-state machine may behave in a near-chaotic way even when its corresponding infinite-state machine does not exhibit chaotic behavior.

We describe algorithms for computing hyperbolic invariant sets of diffeomorphisms and their stable and unstable manifolds. This includes the calculation of Smale horseshoes and the stable and unstable manifolds of periodic points in any finite dimension.

In this paper, we perform our algorithm developed in [Yu & Lee, 2001] to present the entire branches of quasiperiodic solutions starting from the bifurcation points in the branches of periodic solutions in an interval of parameters for the 2-mode damped, driven sine-Gordon ODE.

We investigate one degree of freedom Hamiltonian systems when trajectories cross a separatrix. We study such systems numerically using high precision computations and compare our results with first order, analytical predictions. In all our experiments we find oscillatory behavior which is not predicted by the first order analysis.

A parallel chemical processor is a thin-layer of a reagent mixture which reacts to changes in its concentration — data configuration — in a predictable way to form a stationary pattern corresponding to the concentration of the reagent — result configuration. A computation in the chemical processor is implemented via the spreading and interaction of diffusive or phase waves. We design chemical processors that solve a classical problem of computational geometry — computation of a Voronoi diagram. Namely, we study the possibility of designing a multitasking chemical processor that independently and simultaneously computes Voronoi diagrams of two different data planar sets. We define a two-tasking chemical processor as two distinct reactant–substrate couples within a reaction–diffusion processor that solve separate tasks but share the same physical space. A micro-volume of the physical space is an elementary processor of a massively parallel chemical processor, therefore two reaction–diffusion systems occupying the same space are considered to be a single chemical processor. We found that when a single reactant is on a gel layer containing either one or two substrates the same single Voronoi diagram corresponding to the original location of the reactant drops is constructed. However, when two reactants are on a gel containing two substrates and where there is extremely limited cross reactivity between the separate reactant-substrate couples then two Voronoi diagrams of the data planar points (two sets of drops of separate reactants) are constructed; the third "complementary" pattern is also constructed. The first Voronoi diagram constructed is identical at least in position to the one constructed where one reactant was with one substrate (with the same original configuration of reactant drops). After the formation of the first diagram is completed the diffusion fronts corresponding to unlike reactants cross and are only annihilated where they meet another reactant front composed of the same reactant. The result is the computation of two additional Voronoi diagrams pertaining to the spatial positions of the two sets of reactant drops. The outcomes of this experiment albeit in a simple chemical system are significant because the system constitutes the first class of a synthetic chemical parallel processor capable of at least two computations at the same time.