International Journal of Bifurcation and Chaos

Nonlinear dynamic systems can be considered in terms of feedback circuits (for short, circuits), which are circular interactions between variables. Each circuit can be identified without ambiguity from the Jacobian matrix of the system. Of special relevance are those circuits (or unions of disjoint circuits) that involve all the variables of the system. We call them "nuclei" because, in the same way as, in Biology, the cell nucleus contains essential genetic information, in nonlinear dynamics, nuclei are crucial elements in the genesis of steady states. Indeed, each nucleus taken alone can generate one or more steady states, whose nature is determined by the sign patterns of the nucleus. There can be up to two nuclei in 2D systems, six in 3D systems, … n! in nD systems. However, many interesting systems of high dimensionality have only two, sometimes even one, nucleus.

This paper is based on an extensive exploitation of the Jacobian matrix of systems, in order to figure the global structure of phase space. In nonlinear systems, the value, and often the sign, of terms of the Jacobian matrix depend on the location in phase space. In contrast with a current usage, we consider this matrix, as well as its eigenvalues and eigenvectors, not only in close vicinity of steady states of the system, but also everywhere in phase space. Two distinct, but complementary approaches are used here.

In Sec. 2, we define frontiers that partition phase space according to the signs of the eigenvalues (or of their real part if they are complex), and where required, to the slopes of the eigenvectors. From then on, the exact nature of any steady state that might be present in a domain can be identified on the sole basis of its location in that domain and, in addition, one has at least an idea of the possible number of steady states.

In Sec. 3, we use a more qualitative approach based on the theory of circuits. Here, phase space is partitioned according to the sign patterns of elements of the Jacobian matrix, more specifically, of the nuclei. We feel that this approach is more generic than that described in Sec. 2. Indeed, it provides a global view of the structure of phase space, and thus permits to infer much of the dynamics of the system by a simple analysis of sign patterns within the Jacobian matrix. This approach also turned out to be extremely useful for synthesizing systems with preconceived properties.

For a wide variety of systems, once the partition process has been achieved, each domain comprises at most one steady state. We found, however, a family of systems in which two steady states (typically, two stable or two unstable nodes) differ by neither of the criteria we use, and are thus not separated by our partition processes. This counter-example implies non-polynomial functions.

It is essential to realize that the frontier diagrams permit a reduction of the dimensionality of the analysis, because only those variables that are involved in nonlinearities are relevant for the partition process. Whatever the number of variables of a system, its frontier diagram can be drawn in two dimensions whenever no more than two variables are involved in nonlinearities.

Pre-existing conjectures concerning the necessary conditions for multistationarity are discussed in terms of the partition processes.

In this paper, a nonlinear system aiming at reducing the signal transmission rate in a networked control system is constructed by adding nonlinear constraints to a linear feedback control system. Its stability is investigated in detail. It turns out that this nonlinear system exhibits very interesting dynamical behaviors: in addition to local stability, its trajectories may converge to a nonorigin equilibrium or be periodic or just be random. Furthermore it exhibits sensitive dependence on initial conditions — a sign of chaos. Complicated bifurcation phenomena are exhibited by this system. After that, control of the chaotic system is discussed. All these are studied under scalar cases in detail. Some difficulties involved in the study of this type of systems are analyzed. Finally an example is employed to reveal the effectiveness of the scheme in the framework of networked control systems.

In this paper we investigate numerically chaotic behaviors of incompressible fluid flows at large Reynolds numbers in triangular driven cavities. The problem is first formulated as incompressible Navier–Stokes equations with appropriate boundary and initial conditions. The equations are then solved numerically by an exponentially fitted finite volume method for various Reynolds numbers up to 10⁷. Numerical experiments on flows in triangular cavities with different depths are performed. The numerical results show clearly that the transitions of the flows from lamina to turbulence/chaos follow some conventional routes to chaos.

When chaotic systems are realized with finite precisions in digital computers, their dynamical properties are often found to be entirely different from the original versions in the continuous setting. In the literature, there does not seem to be much work on quantitative analysis of such degradation of digitized chaos and how to reduce its negative influence on chaos-based digital systems. Focusing on 1D piecewise linear chaotic maps (PWLCM), this paper reports some findings on a new series of dynamical indicators, which can quantitatively reflect the degradation effects on a digital PWLCM realized with a fixed-point finite precision. On top of that, the paper introduces a new method for studying digital chaos from an algorithmic point of view. In addition, the theoretical results obtained in this paper should be very helpful for the consideration of reducing negative influence of dynamical degradation in real design of various digital chaotic systems. As typical examples, the proposed dynamical indicators are applied to the performance comparison of different remedies for improving dynamical degradation, cryptanalysis of digital chaotic ciphers based on 1D PWLCM, and design of chaotic pseudo-random number generators with desired characteristics.

The generic case of three-dimensional continuous piecewise linear systems with two zones is analyzed. From a bounded linear center configuration we prove that the periodic orbit which is tangent to the separation plane becomes a limit cycle under generic conditions. Expressions for the amplitude, period and characteristic multipliers of the bifurcating limit cycle are given. The obtained results are applied to the study of the onset of asymmetric periodic oscillations in Chua's oscillator.

The paper considers the existence of quasi-periodic solutions in three-dimensional systems. Since these solutions commonly arise as a consequence of a Neimark–Sacker bifurcation of a limit cycle, a fairly general relation connected to this phenomenon is pointed out as the main result of the paper. Then, the application of harmonic balance techniques makes possible to exploit such a relation. In particular, a simplified condition denoting the quasi-periodicity onset can be derived, in making evident the main elements for this transition in terms of structure and parameters, and hence some remarks on the features of the interested systems. Several examples show the application of the above condition to detect "tori" in the state space in a qualitative (not simply numerical) way. They consider classical systems — Rössler, where such behavior seems to be unknown, Chua, forced Van der Pol — and new quadratic systems.

This paper is devoted to the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, submitted to an external periodical excitation of period τ = 2π/ω , having an amplitude E. In absence of this excitation the equation of Duffing type does not give rise to self oscillations. It may be a model of a series R-L-C electrical circuit with validity conditions in the parameter space. The purpose is essentially an analysis of the harmonics behavior of the period τ solutions according to points of the parameter plane (ω, E). Let r be the place occupied by a rank-m harmonic from an ordering based on line amplitudes of a frequency spectrum in descending order. Domains of the (ω, E) plane, for which the amplitude of the rank-m harmonic has the place r in the ordering mentioned above, are defined from properties of their boundaries. When r = 2 they are regions of predominance for the rank-m harmonic. When r = 1 they are regions of full predominance for the harmonic m, m = 2,3,4,…, which contain a set of points leading to a resonance for this harmonic. These regions fulfil the following important property: each of them is directly related to a fold bifurcation structure, called rank-mlip (two folds curves joining at two cusps), associated with a well-defined rank-m harmonic. The set of such structures, with m = 2,3,4,…, constitutes an isoordinal lips cascade. As an opening to Part II (to be published) period kτ solutions related to fractional harmonics behavior are also considered for k = 2,3.

The actuator saturation effects on the global state space structure of a rotating arm with a Hamiltonian control law are studied via bifurcation analysis. In addition to other well-known codimension-two bifurcation points, some saturation-induced "fan-like" bifurcation points are detected. The dynamical complexity found in this seemingly simple example is remarkable.

The transition from periodicity to chaos in a DC-DC Buck power converter is studied in this paper. The converter is controlled through a direct Pulse Width Modulation (PWM) in order to regulate the error dynamics at zero. Results show robustness with low output error and a fixed switching frequency. Furthermore, some rich dynamics appear as the constant associated with the first-order error dynamics decreases. Finally, a transition from periodicity to chaos is observed. This paper describes this transition and the bifurcations in the converter. Chaos appears in the system with a stretching and folding mechanism. It can be observed in the one-dimensional Poincaré map of the inductor current. This Poincaré map converges to a tent map with the variation of the system parameter k_s.

In modeling nonlinear dynamical systems, error growth can arise both because of dynamical error in the model, and observational error in measurements of the underlying system. Errors thus introduced can be further amplified by the system dynamics. This paper develops a method to approximate error growth, based on the model drift, for the general case when the model is given by a (possibly large) set of ordinary differential equations. The ability of the model to shadow (stay close to) the true system is analyzed, and a criterion for model quality based on the error dynamics is proposed. Examples are given with a range of models, which are chosen to be representative of different types of model error.

In this paper we describe several new types of invariant sets that appear in the Julia sets of the complex exponential functions E_λ(z) = λe^z where λ ∈ ℂ in the special case when λ is a Misiurewicz parameter, so that the Julia set of these maps is the entire complex plane. These invariant sets consist of points that share the same itinerary under iteration of E_λ. Previously, the only known types of such invariant sets were either simple hairs that extend from a definite endpoint to ∞ in the right half plane or else indecomposable continua for which a single hair accumulates everywhere upon itself. One new type of invariant set that we construct in this paper is an indecomposable continuum in which a pair of hairs accumulate upon each other, rather than a single hair having this property. The second type consists of an indecomposable continuum together with a completely separate hair that accumulates on this continuum.

In this paper, the generalized nonlinear Schrödinger equation (GNLS) is studied. The bifurcation of solitary waves of the equation is discussed first, by using the bifurcation theory of planar dynamical systems. Then, the respective numbers of solitary waves are derived under different conditions on the equation parameters. Exact solutions of smooth solitary waves are obtained in the explicit form of a(ξ)e^{i(ψ(ξ)-ωt)}, ξ = x - vt by qualitatively seeking the homoclinic and heteroclinic orbits for a class of Liénard equations. Finally, nonsmooth solitary wave solutions of the GNLS are investigated.

Reaction–diffusion (RD) chemical systems are known to realize sensible computation when both data and results of the computation are encoded in concentration profiles of chemical species; the computation is implemented via spreading and interaction of either diffusive or phase waves, while a silicon RD chip is an electronic analog of the chemical RD systems where the concentration profiles of chemicals are represented by voltage distributions on the chip's surface. In this paper, we present a prototype RD chip implementing a chemical RD processor for a well-known NP-complete problem of computational geometry — computation of a Voronoi diagram. We offer experimental results for fabricated RD chips and compare the accuracy of information processing in silicon analogs of RD processors and their experimental "wetware" prototypes.

We consider a diffusion equation on a domain Ω with a cubic reaction at the boundary. It is known that there are no patterns when the domain Ω is a ball, but the existence of such patterns is still unkown in the more general case in which Ω is convex. The goal of this paper is to present numerical evidence of the existence of nonconstant stable equilibria when Ω is the unit square. These patterns are found by continuation of families of unstable equilibria that bifurcate from constant solutions.

In this work we report theoretical and numerical results on convection in a viscoelastic binary mixture under rotation. Instability thresholds for stationary convection are calculated. We obtain explicit expressions of convective thresholds in terms of the control parameters of the system for oscillatory convection. Finally, we analyze the stabilizing effect of rotation on instability thresholds for aqueous DNA suspensions.

In this letter, chaos in a current-mode controlled boost converter is studied. Firstly, the existence of chaos is proven theoretically in this system. The proof consists of showing that the dynamics of the system is semiconjugate to that of a one-sided shift map, which implies positive entropy of the system and hence chaotic behavior. The essential tool is the horseshoe hypotheses proposed by Kennedy and Yorke, which will be reviewed prior to the discussion of the main finding. Then, the existence of chaos is illustrated in the light of homoclinic connection. Furthermore, global chaos resulting from homoclinic intersection of stable and unstable manifolds are illustrated numerically.

Various forms of chaotic synchronization have been proposed as ways of realizing associative memories and/or pattern recognizers. To exploit this kind of synchronization phenomena in temporal pattern recognition, a chaotic dynamical system representing the class of signals that are to be recognized must be established. As shown recently [De Feo, 2003], this system can be determined by means of identification techniques where chaos emerges by itself to model the diversity of nearly periodic signals. However, the emerging chaotic behavior is subharmonic, i.e. period doubling-like, and therefore, as explained in [De Feo, 2004a, 2004b], it is not suitable for a synchronization-based pattern recognition technique. Nevertheless, as shown here, bifurcation theory and continuation techniques can be combined to modify a subharmonic chaotic system and drive it to homoclinic conditions; obtaining in this way a model suitable for synchronization-based pattern recognition.

In this letter, we investigate the cross-coupled BVP oscillators. A single BVP oscillator has two terminals which can extract an independent state variable, so in the preceding works, several coupling systems have been studied. Synchronization modes and chaos in these systems are classified as results of bifurcation problems. We revisit a coupled oscillator, and clarify new results which have not been reported before, i.e. stable tori and its breakdown, and chaotic motions. Also classification of synchronized periodic solutions is done by a bifurcation diagram.

In this letter, a simple nonlinear state feedback controller is designed for generating hyperchaos from a three-dimensional autonomous chaotic system. The hyperchaotic system is not only demonstrated by computer simulations but also verified with bifurcation analysis, and is implemented experimentally via an electronic circuit.

It has been established that the class of bandpass sigma delta modulators (SDMs) with single bit quantizers could exhibit state space dynamics represented by elliptic or fractal patterns confined within trapezoidal regions. In this letter, we find that elliptical fractal patterns may also occur in bandpass SDMs with multibit quantizers, even for the case when the saturation regions of the multibit quantizers are not activated and a large number of bits are used for the implementation of the quantizers. Moreover, the fractal pattern may occur for low bit quantizers, and the visual appearance of the phase portraits between the infinite state machine and the finite state machine with high bit quantizers is different. These phenomena are different from those previously reported for the digital filter with two's complement arithmetic. Furthermore, some interesting phenomena are found. A bit change of the quantizer can result in a dramatic change in the fractal patterns. When the trajectories of the corresponding linear systems converge to a fixed point, the regions of the elliptical fractal patterns diminish in size as the number of bits of the quantizers increases.

In this paper, a new adaptive switching control scheme is presented to solve control and synchronization problems. Based on Lyapunov stability theory, an adaptive control law is applied to globally stabilize chaotic systems and achieve states synchronization of two chaotic systems whose dynamics are subjected to the system disturbances and/or some unknown parameters. Simulation examples, the chaotic Chen's system and Chua's circuit, are given to show the feasibility and effectiveness of the proposed theory and method.

In practice it is very difficult to distinguish between stochastic and chaotic nonlinearity. It is also difficult to identify nonlinear structures in time series when the hidden dynamics are complex. In this paper, we provide evidence that in the presence of specific high-dimensional underlying structures, White [1989] and Theiler et al. [1992] tests fail to detect nonlinearity.

The effect of keeping memory of previous choices when the players adopt the Paulov and Anti-Paulov strategies in the continuous-valued iterated Prisoner's Dilemma is assessed.