Narrow Results

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, we present theorems which give sufficient conditions for the uniform convergence of measure differential inclusions with certain maximal monotonicity properties. The framework of measure differential inclusions allows us to describe systems with state discontinuities. Moreover, we illustrate how these convergence results for measure differential inclusions can be exploited to solve tracking problems for certain classes of nonsmooth mechanical systems with friction and one-way clutches. Illustrative examples of convergent mechanical systems are discussed in detail.

In this paper, the convergence of C₀ complexity is proved, some examples and applications are given to verify the results. From these examples, we can see that C₀ complexity can also obtain robust results with short signals, and consequently it can be applied to the nonstationary signals, such as EEG signals.

The paper analyzes some fundamental properties of the solution semiflow of nonsymmetric cooperative standard (S) cellular neural networks (CNNs) with a typical three-segment piecewise-linear (pwl) neuron activation. Two relevant subclasses of SCNNs, corresponding to one-dimensional circular SCNNs with two-sided or single-sided positive interconnections between nearest neighboring neurons only, are considered. For these subclasses it is shown that the associated solution semiflow satisfies the fundamental properties of the CONVERGENCE CRITERION, the NONORDERING OF LIMIT SETS and the LIMIT SET DICHOTOMY, and that this is true although the semiflow is not eventually strongly monotone. As a consequence such CNNs are almost convergent, i.e., almost all solutions converge toward an equilibrium point as time tends to infinity. To the authors' knowledge the paper is the first rigorous investigation on the geometry of limit sets and convergence properties of cooperative SCNNs with a pwl neuron activation. All available convergence results in the literature indeed concern a modified cooperative CNN model where the original pwl activation of the SCNN model is replaced by a continuously differentiable strictly increasing sigmoid function. The main results in the paper are established by conducting a deep analysis of the properties of the omega-limit sets of the solution semiflow defined by the considered subclasses of SCNNs. In doing so the paper exploits and extends some mathematical tools for monotone systems in order that they can be applied to pwl vector fields that govern the dynamics of SCNNs. By using some transformations and referring to specific examples it is also shown that the treatment in the paper can be extended to other subclasses of SCNNs.

Synchronization of growth rates are an important feature of international business cycles, particularly in relation to regional integration projects such as the single currency in Europe. Synchronization of growth rates clearly enhances the effectiveness of European Central Bank monetary policy, ensuring that policy changes are attuned to the dynamics of growth and business cycles in the majority of member states. In this paper, a dissimilarity metric is constructed by measuring the topological differences between the GDP growth patterns in recurrence plots for individual countries. The results show that synchronization of growth rates were higher among the euro area member states during the second half of the 1980s and from 1997 to roughly 2002. Apart from these two time periods, euro area member states do not appear to be more synchronized than a group of major international countries, suggesting that apart from specific times when European integration initiatives were being implemented, globalization was likely the dominant factor behind international business cycle synchronization.

Strange attractors are fascinating not only for the fractal structure most of them display, but also for the freakish chaotic and multiscroll trajectories they portray. In this paper, Haar wavelet numerical method is used to address the solvability of the 3D Lu–Chen model combined with the classical Caputo fractional differentiation. The full model, named Caputo–Lu–Chen model is solved analytically and a complete error analysis is performed to test the convergence of the method. It reveals that the error made using such a technique is negligible. Numerical simulations, performed in pure fractional and standard integer cases, show that the Caputo–Lu–Chen system maintains its status of strange, chaotic and multiscroll attractor. The graphics in both cases present similar features characterized by attractors with many scrolls. Hence, the Caputo derivative order when introduced as parameter in the Lu–Chen system does not remove the multiscroll property of generated attractors; it even becomes an important control parameter able to change the dynamics of the whole system. This result can be very important in control theory.

Improving running-in quality is an important project of Green Tribology. In actual environment, the running-in quality of mechanical equipment is greatly affected by system parameters. In view of this, running-in experiments were performed at different rotating speeds on a pin-on-disc tribometer. Chaos theory was used to investigate the effect of rotating speeds on the running-in quality. The experimental results showed that the running-in quality could be better evaluated by using the chaotic characteristic parameter “Convergence (CON)”. The greater the CON, the better the running-in quality. With an increase of the rotating speed, CON first gradually increased and then decreased to a certain level, whilst the running-in quality first gradually became better and then became worse. A moderate rotating speed is advantageous to obtain the optimum running-in quality. This contributes to the service life extension and the efficiency improvement, so as to reduce energy consumption.

The paper considers a class of discrete-time cellular neural networks (DT-CNNs) obtained by applying Euler’s discretization scheme to standard CNNs. Let $T$ be the DT-CNN interconnection matrix which is defined by the feedback cloning template. The paper shows that a DT-CNN is convergent, i.e. each solution tends to an equilibrium point, when $T$ is symmetric and, in the case where $T+E_n$ is not positive-semidefinite, the step size of Euler’s discretization scheme does not exceed a given bound ($E_n$ is the $n\times n$ unit matrix). It is shown that two relevant properties hold as a consequence of the local and space-invariant interconnecting structure of a DT-CNN, namely: (1) the bound on the step size can be easily estimated via the elements of the DT-CNN feedback cloning template only; (2) the bound is independent of the DT-CNN dimension. These two properties make DT-CNNs very effective in view of computer simulations and for the practical applications to high-dimensional processing tasks. The obtained results are proved via Lyapunov approach and LaSalle’s Invariance Principle in combination with some fundamental inequalities enjoyed by the projection operator on a convex set. The results are compared with previous ones in the literature on the convergence of DT-CNNs and also with those obtained for different neural network models as the Brain-State-in-a-Box model. Finally, the results on convergence are illustrated via the application to some relevant 2D and 1D DT-CNNs for image processing tasks.