International Journal of Bifurcation and Chaos

We investigate nonlinear dynamical systems from the mode competition point of view, and propose the necessary conditions for a system to be chaotic. We conjecture that a chaotic system has at least two competitive modes (CM's). For a general nonlinear dynamical system, we give a simple, dynamically motivated definition of mode suitable for this concept. Since for most chaotic systems it is difficult to obtain the form of a CM, we focus on the competition between the corresponding modulated frequency components of the CM's. Some direct applications result from the explicit form of the frequency functions. One application is to estimate parameter regimes which may lead to chaos. It is shown that chaos may be found by analyzing the frequency function of the CM's without applying a numerical integration scheme. Another application is to create new chaotic systems using custom-designed CM's. Several new chaotic systems are reported.

This work presents a linguistic approach to the understanding of chaos. The idea comes from our work on translating into sounds and music the complexity of chaotic systems, as with Chua's attractors. Therefore, working with sounds, we have used a standardization criterion in order to detect only some of the features of the richness of chaos. This method involves the selection of a limited number of sounds, which, in turn, allow for the creation of other components of the system at different levels of organization, as with natural language. Thus, phonetic, morphological, syntactic and semantic levels, linked by a grammar of an "artificial chaotic language" are defined. Each linguistic unit of the artificial language presents networks of interrelated phenomena since for each of them, it is possible to detect its path length, tracing a graph of the mutual relationships of the system's components. We found that there is interaction among the different levels of the chaotic language. Furthermore, some traits of the dynamics of the evolution of language in human infants are found in the main routes to chaos. In this emergent dynamics, mutations, struggle for the fittest, natural selection, and the relative distribution of linguistic entities in genetic landscapes are observed. On the basis of this, a possible bridge of connection between the physical and the mental worlds may be developed.

Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetry-breaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Γ of permutations of the nodes ("cells"). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals — ones that are not fixed-point spaces — can occur for such networks. We also prove the "folk theorem" that in any Γ-equivariant dynamical system on R^k the only flow-invariant subspaces are the fixed-point spaces of subgroups of Γ.

This paper shows how the appearance and disappearance of high-frequency oscillations in a batch reactor can be predicted by means of a minimal model describing the fast dynamics of the reaction. The analysis is based on the bifurcations of the model with respect to the slowly varying species, namely product and reactants. The method of analysis can, in principle, be applied to any slow-fast system provided the evolution of the slow variables is monotonic.

The dynamics of propagating interfaces in a bistable neural network is investigated. We consider the network composed of two coupled 1D lattices and assume that they interact in a local spatial point (pin contact). The network unit is modeled by the FitzHugh–Nagumo-like system in a bistable oscillator mode. The interfaces describe the transition of the network units from the rest (unexcited) state to the excited state where each unit exhibits periodic sequences of excitation pulses or action potentials. We show how the localized inter-layer interaction provides an "excitatory" or "inhibitory" action to the oscillatory activity. In particular, we describe the interface propagation failure and the initiation of spreading activity due to the pin contact. We provide analytical results, computer simulations and physical experiments with two-layer electronic arrays of bistable cells.

This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (also known as C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a nongeneric way, such as grazing contact. Several such codimension-one events have recently been identified, causing for example, period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighborhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the grazing cycle is itself degenerate (e.g. nonhyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that with discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

We consider several discrete dynamical systems for which some invariants can be found. Our study includes complex Möbius transformations as well as the third-order Lyness recurrence.

Two-dimensional (Z₁–Z₃–Z₁) maps are such that the plane is divided into three unbounded open regions: a region Z₃, whose points generate three real rank-one preimages, bordered by two regions Z₁, whose points generate only one real rank-one preimage. This paper is essentially devoted to the study of the structures, and the global bifurcations, of the basins of attraction generated by such maps. In particular, the cases of fractal structure of such basins are considered. For the class of maps considered in this paper, a large variety of dynamic situations is shown, and the bifurcations leading to their occurrence are explained.

Hybrid systems are typified by strong coupling between continuous dynamics and discrete events. For such piecewise smooth systems, event triggering generally has a significant influence over subsequent system behavior. Therefore, it is important to identify situations where a small change in parameter values alters the event triggering pattern. The bounding case, which separates regions of (generally) quite different dynamic behaviors, is referred to as grazing. At a grazing point, the system trajectory makes tangential contact with an event triggering hypersurface. The paper formulates conditions governing grazing points. Both transient and periodic behaviors are considered. The resulting boundary value problems are solved using shooting methods that are applicable for general nonlinear hybrid (piecewise smooth) dynamical systems. The grazing point formulation underlies the development of a continuation process for exploring parametric dependence. It also provides the basis for an optimization technique that finds the smallest parameter change necessary to induce grazing. Examples are drawn from power electronics, power systems and robotics, all of which involve intrinsic interactions between continuous dynamics and discrete events.

In much of the previous study of switched dynamical systems, it has been assumed that switching occurs at a common border between two regions in the same space as the system trajectory crosses the border. However, models arising from this consideration cannot cover systems whose trajectories do not actually "cross" the border. A typical example is the current-mode controlled boost converter whose trajectory is "reflected" at the border. In this paper, we propose a general method to model switched dynamical systems. Also, we suggest an analytical procedure to determine periodic solutions and their stability. The method is developed in terms of solution flows, and no solution has to be explicitly written. Most practical switched dynamical systems can be modeled and analyzed by this method.

In this paper, a chaotic switched arrival system with N buffers and controlled internal connections is considered. An unstable N periodic orbit embedded in the chaotic attractor is theoretically derived. A novel method for the stabilization of the unstable N periodic orbit using the flow rates of the internal connections as control variables is presented. It is showed that the switched arrival system can be stabilized by using N - 2 internal connections. Numerical simulation results are also provided to demonstrate the theoretical analysis.

In this paper, we have experimentally investigated Current-Feedback Operational Amplifier (CFOA)-based inductorless Mixed-Mode Chaotic Circuit by constructing an experimental setup, experimentally verifying the static and dynamic performance of the proposed implementation.

The problem of making a stable discrete time system chaotic has attracted considerable interest. The definition of chaotification used in these studies is based on introducing a feedback control which establishes the existence of a scrambled set for the controlled system. In this note, we introduce a stronger and potentially more useful definition of chaotification based on the existence of an absolutely continuous invariant measure (acim). The utility of this approach is that once a desired chaotic behavior (in the form of an acim near a stable fixed point) is specified, we can determine a small feedback control (as a function of the stable system) such that the controlled dynamical system exhibits the desired behavior. An example is worked out in detail.

We consider the problem of identical synchronization (IS) of chaotic systems in two cases: (i) system uncertainty exists in the drive system, and (ii) output uncertainty exists in the drive system. No matter which case it is, the IS problem can be resolved via the combination of unknown input observers (UIOs) and the linear matrix inequalities (LMIs). Theoretical analysis and numerical simulations verify our main results in this paper.

We demonstrate experimentally the regime of ghost stochastic resonance in the response of a Monostable Schmit Trigger electronic circuit driven by noise and signals with N frequency components: kf₀+Δf, (k+1)f₀+Δf,…, k+nf₀+Δf where k is an integer greater than one. It is verified that stochastic resonance occurs at the frequency f_r = f₀ + (Δf/(k+(N-1)/2)), as predicted in the theory. At the frequency for which the resonance is maximum there is no input energy, and thus this form is called "ghost" stochastic resonance.

Following the question of Smítal and Štefánková we show that there exist continuous maps of [0,1]ⁿ, n=2,3,… possessing an ω-scrambled set of the full n-dimensional Lebesgue measure.

One of the most complex problems of nonlinear dynamics is the investigation of collective dynamics of ensembles consisting of a large number of coupled elements. A number of investigations were carried out for linear coupling of diffusion (difference) type. Recently, based on successful investigations of neurobiological objects [Abarbanel et al., 1996], an interest has grown in the study of dynamics of ensembles with nonlinear, nondiffusion type of coupling. In this paper we address pattern formation in a chain of bistable elements so coupled [Kuznetsov & Shalfeev, 2000].

In this paper we investigate the onset of instabilities in a model describing the propagation of the steady planar premixed combustion wave. In particular, we are interested in determining the Bogdanov–Takens bifurcation condition, which is investigated semi-analytically. We derive an analytic condition for the existence of this type of bifurcation and based on this criterion we numerically determine the parameter values for which the Bogdanov–Takens bifurcation occurs. This numerical method is found to be more efficient than the previous methods.

In this paper, based on a generalized Lyapunov function, a simple proof is given to improve the estimation of globally attractive and positive invariant set of the Lorenz system. In particular, a new estimation is derived for the variable x. On the globally attractive set, the Lorenz system satisfies Lipschitz condition, which is very useful in the study of chaos control and chaos synchronization. Applications are presented for globally, exponentially tracking periodic solutions, stabilizing equilibrium points and synchronizing two Lorenz systems.

In this paper, we consider the planar differential system associated with the potential Hamiltonian H(x,y) = (1/2)y²+V(x) where V(x) = (1/2)x²+(a/4)x⁴+(b/6)x⁶ with b ≠ 0. This family of differential systems always has a center at the origin and, eventually, three other period annuli. We prove that the corresponding period functions can have at most one critical period altogether. More precisely, we show that this critical period is simple and that it corresponds to the period function associated to the center at the origin. To prove the result we use that the period function verifies a Picard–Fuchs equation. Finally we describe the bifurcation diagram of the period function of the center at the origin.