International Journal of Bifurcation and Chaos

This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulliσ_τ-shift rules and eight hyper Bernoulliσ_τ-shift rules, the latter including such famous rules and . All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞.

Basin tree diagrams of all ten complex Bernoulli σ_τ-shift rules are exhibited for lengths L = 3, 4, …, 8. Superficial as it may seem, these basin tree diagrams suggest general qualitative properties which have since been proved to be true in general. Two such properties form the main results of this paper; namely,

Explicit global state transition formulas are given for local rules , and . Such formulas led to the rigorous proof of several surprising periodicity constraints for rule , and to the discovery of a new global, quasi-equivalence class, defined via an alternating transformation. In particular, local rules and are globally quasi-equivalent where corresponding space-time patterns can be derived from each other by simply complementing every other row.

Another important result of this paper is the discovery of a scale-free phenomenon exhibited by the local rules , and . In particular, the period "T" of all attractors of rules , and , as well as of all isles of Eden of rules and , increases linearly with unit slope, in logarithmic scale, with the length L.

The parameter dependence of the various attractive solutions of the three variable nonlinear Lorenz equations is studied as a function of r, the normalized Rayleigh number, and of σ, the Prandtl number. Previous work, either for fixed σ and all r or along σ ∝ r and , is extended to the entire (r, σ) parameter plane. An onion-like periodic pattern is found which is due to the alternating stability of symmetric and nonsymmetric periodic orbits. This periodic pattern is explained by considering non-trivial limits of large r and σ and thus interpolating between the above mentioned cases. The mathematical analysis uses Airy functions as introduced in previous work, but instead of concentrating on the Lorenz map we analyze the trajectories in full phase space. The periodicity of the Airy function allows to calculate analytically the periodic onion structure in the (r, σ)-plane. Previous observations about sequences of bifurcations are confirmed, and more details regarding their symmetry are reported.

In this paper, an entire organic Chua's circuit is presented. The adopted technology is one of the most advanced in the post-silicon era and extends the applications of classical silicon devices. The electronic circuit design is based on Organic Thin-Film Transistors (OTFTs). New electronic blocks based on OTFT are designed to be suitable with the organic technology features. Typical dynamics of the Chua's circuit have been reproduced.

In this paper, smooth Chua's equation is generalized to a higher order system from a special viewpoint of interconnected systems. Simple conditions for Lagrange stability are established. And a detailed Lagrange stable region analysis is given for the canonical Chua's oscillator. In addition, a new nonlinearly coupled Chua's circuit that appeared in the recent literature is also discussed and a Lagrange stability condition is presented. Several examples are presented to illustrate the results.

In this work we continue the study of the centers which are limits of linear type centers. It is proved that if a degenerate center has an inverse integrating factor V(x, y) with V(0, 0) ≠ 0, then this degenerate center is also the limit of linear type centers. Moreover, we show that the degenerate centers with characteristic directions that are the limits of degenerate centers without characteristic directions are also detectable, from a theoretical point of view, with the Bautin method.

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

Chaotic neurons are interconnected with each other through a matrix of dynamic synaptic weights to constitute an associative network. First, the dynamics of the two chaotic neurons connected with dynamic synapses is analyzed. A quick memory switching phenomenon in the Associative Network with Chaotic Neurons and Dynamic Synapses (ANCNDS) is then demonstrated. It is further discovered that the memory retrieval frequency of the network among stored patterns distributes more uniformly, and is much higher than that of a chaotic network without dynamic synapses. Moreover, the network acts as a peculiar pattern recognition system when applied with external stimulations. If the external stimulation is close to only one stored pattern, the network recalls the stored pattern exactly. On the other hand, if the external stimulation shares similarity with two or several stored patterns, the network transits among these related stored patterns. When the external stimulation does not resemble any stored patterns, the network wanders among all the stored patterns.

Exponential stability of reaction–diffusion fuzzy recurrent neural networks (RDFRNNs) with time-varying delays are considered. By using the method of variational parameters, M-matrix properties and inequality technique, some delay-independent or delay-dependent sufficient conditions for guaranteeing the exponential stability of an equilibrium solution are obtained. One example is given to demonstrate the theoretical results.

We investigate the behavior of a neural network model consisting of three neurons with delayed self and nearest-neighbor connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D₃-equivariant, Hopf and pitchfork bifurcation points. We use numerical simulation and numerical bifurcation analysis to investigate the dynamics near the pitchfork–Hopf interaction points. Our numerical investigations reveal that multiple secondary Hopf bifurcations and pitchfork bifurcations of limit cycles may emanate from the pitchfork–Hopf points. Further, these secondary bifurcations give rise to ten different types of periodic solutions. In addition, the secondary bifurcations can lead to multistability between equilibrium points and periodic solutions in some regions of parameter space. We conclude by generalizing our results into conjectures about the secondary bifurcations that emanate from codimension two pitchfork–Hopf bifurcation points in systems with D_n symmetry.

In a series of papers published in the seventies, Grossberg had developed a geometric approach for analyzing the global dynamical behavior and convergence properties of a class of competitive dynamical systems. The approach is based on the property that it is possible to associate a decision scheme with each competitive system in that class, and that global consistency of the decision scheme implies convergence of each solution toward some stationary state. In this paper, the Grossberg approach is extended to the class of competitive standard Cellular Neural Networks (CNNs), and it is used to investigate convergence under the hypothesis that the competitive CNN has a globally consistent decision scheme. The extension is nonobvious and requires to deal with the set-valued vector field describing the dynamics of the CNN output solutions. It is also stressed that the extended approach does not require the existence of a Lyapunov function, hence it is applicable to address convergence in the general case where the CNN neuron interconnections are not necessarily symmetric. By means of the extended approach, a number of classes of third-order nonsymmetric competitive CNNs are discovered, which have a globally consistent decision scheme and are convergent. Moreover, global consistency and convergence hold for interconnection parameters belonging to sets with non-empty interior, and thus they represent physically robust properties. The paper also shows that when the dimension is higher than three, there are fundamental differences between the convergence properties of competitive CNNs implied by a globally consistent decision scheme, and those of the class of competitive dynamical systems considered by Grossberg. These differences lead to the need to introduce a stronger notion of global consistency of decisions, with respect to that proposed by Grossberg, in order to guarantee convergence of competitive CNNs with more than three neurons.

In this paper, a general problem of the robust template decomposition with restricted weights for cellular neural networks (CNNs) implementing an arbitrary Boolean function is investigated. First, the geometric margin of a linear classifier with respect to a training data set is used to define the robustness of an uncoupled CNN implementing a linearly separable Boolean function. Second, maximal margin classifiers, i.e. robust CNNs, for such Boolean functions can be designed via support vector machines (SVMs). Third, some general properties of robust CNNs with or without restricted weights are discussed. Moreover, all robust CNNs with restricted weights are characterized. Finally, for an arbitrarily given Boolean function, we propose an algorithm, which is the generalized version of the well-known CFC algorithm, to find a sequence of robust uncoupled CNNs implementing the given Boolean function. Several illustrative examples demonstrate the efficiency of the proposed method. It will be seen that, in general, the tradeoff between the complexity regarding the number of terms in the decomposition and the guaranteed robustness regarding the geometric margins of the resulting CNNs must be made in the robust template decomposition with restricted weights.

The following paper explores possibilities of an application of Cellular Neural Networks for spontaneous generation of complex stochastic textures. The considered class of textures includes images, which are combinations of large-scale and fine-scale stochastic patterns. Accurate modeling of properties of such complex stochastic fields by means of a single 1-neighborhood CNN poses great difficulties. Therefore, we propose a method that exploits texture decomposition into two elements and involves an application of two networks, designated for performing distinct partial tasks. The first step of the proposed CNN-based texture generation procedure focuses on modeling of a coarse, large-scale textural pattern. The second step refines this pattern into a final image that is expected to closely imitate a target texture.

In this paper, we are concerned with the existence of invariant manifolds and complete foliations for a class of PDEs with boundary coupling. Some new form of gap relative coupling and inequality conditions are obtained. Further, we prove the topological equivalence of the flows in the respective attractors between the system and its spatial discretization system (an ODE system). Finally, the chaotification of the system is discussed through an example and simulation is generated to illustrate the theoretical results.

This paper investigates the synchronization of coupled chaotic systems using techniques from the theory of robust control based on Linear Matrix Inequalities. To deal with the synchronization of a class of Lur'e discrete time systems, a project methodology is proposed. A laboratory setup based on Chua's oscillator circuit is used to demonstrate the main ideas of the paper in the context of the problem of information transmission.

We obtain numerically a horseshoe in a Poincaré map derived from a cellular neural network described by four-dimensional autonomous ordinary differential equations. Contrary to the horseshoe numerically found in the Hodgkin–Huxley model, which showed evidence that the Poincaré map derived from the Hodgkin–Huxley model has just one expanding direction on some invariant subset, the horseshoe obtained in this paper proves that the Poincaré map derived from the neural network have two expanding directions on some invariant subset.

Stochastic effects on the stability property of reaction–diffusion generalized Cohen–Grossberg neural networks (GDCGNNs) with time-varying delay are considered. By skillfully constructing suitable Lyapunov functionals and employing the method of variational parameters, inequality technique and stochastic analysis, the delay independent and easily verifiable sufficient conditions to guarantee the mean-value exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained. One example is given to illustrate the theoretical results.

Cardiac ablation is increasingly used to interdict the complex propagation of excitatory waves during atrial fibrillation. While such procedures are useful, they can often fail. Here we use fluorescence imaging to observe the electrical activity of an ablated porcine heart. We find that, while ablation lines do attenuate cardiac waves, the attenuation is incomplete. A remnant of the incident wave survives passage through the ablation barrier, albeit as a subthreshold signal. More importantly, we have found that these subthreshold signals may add constructively and thereby dynamically reduce the effective attenuation, an effect which we call "dynamic transmurality". We suspect this as a factor in the persistence of arrhythmia after ablative.

The chaotic behavior of the Rabinovich–Fabrikant system, a model with multiple topologically different chaotic attractors, is analyzed. Because of the complexity of this system, analytical and numerical studies of the system are very difficult tasks. Following the investigation of this system carried out in [Danca & Chen, 2004], this paper verifies the presence of multiple chaotic attractors in the system. Moreover, the Monte Carlo hypothesis test (or, equivalently, surrogate data test) is applied to the system for the detection of chaos.

This letter proposes a procedure for generation and control of chaotic beats in a dynamical system that is initially in the periodic state. The dynamical system describes a simple nonlinear optical process — second-harmonic generation of light. The periodic states of the system are found to be in analytical forms. We also investigate some aspects of synchronization of chaotic beats in two systems, detuned in the pump fields.

This work studies transitions to chaos and adaptive synchronization of a nonlinear emitter–receiver system in a drive–response framework. Safe bifurcations and explosive bifurcations are observed. A robust adaptive observer-based response system is designed to synchronize the emitter–receiver system with unknown parameters and external disturbances. Lyapunov stability ensures global synchronization between the drive and response systems even if Lipschitz constants on functions matrices and bound on uncertainties are unknown. Computer simulations are provided to illustrate the designed adaptive synchronization scheme.

The so-called renormalization group equation or Cvitanovic–Feigenbaum (CF) equation arises in the universal scaling theory of iterated maps. In this Letter, a set of formal analytic solutions of this equation is obtained by assuming that the solutions have a particular functional form.

Addressing the weakened Hilbert's 16th problem or the Hilbert–Arnold problem, this paper gives an upper bound B(n) ≤ 7n + 5 for the number of zeros of the Abelian integrals for a class of Liénard systems. We proved the main result using the Picard–Fuchs equations and the algebraic structure of the integrals.

The well–known Rossler system can be written as ẋ = -(z + y), ẏ = x + ay and ż = b + xz - cz. Here, we study the first integrals of the Rossler system that can be described by formal power series.

In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in ℝ³. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a "generalized heteroclinic loop" formed by two singular points e⁺ and e^- at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of dimension 1 formed by an orbit going from e^- to e⁺, Γ is contained in ℝ³ and is transversal to Σ. Λ is an invariant manifold of dimension 2 at infinity. In fact, Λ is the two-dimensional sphere at infinity in the Poincaré compactification minus the singular points e⁺ and e^-. The main tool for proving the existence of such periodic orbits is the construction of a Poincaré map along the generalized heteroclinic loop together with the symmetry with respect to Σ.