Self-oscillations of a third order PLL in periodic and chaotic mode and its tracking in a slave PLL

The dynamics of a third order phase locked loop (PLL) with a resonant low pass filter (LPF) has been studied numerically in the parameter space of the system. The range of stable synchronous operating zone of the PLL, expressed in terms of system and signal parameters, is estimated. The obtained results are in agreement with the analytically predicted results in the literature. The PLL dynamics in the unstable region is found to have a sequence of period doubling bifurcation and chaos. In the master–slave mode of operation of two 3rd order PLLs, the slave PLL can track the periodic as well as chaotic dynamics of the master PLL for a narrow range of effective frequency offset when other design parameters are within the stable zone as predicted for an isolated PLL. The synchronization of the master and slave PLLs in this condition is proved to be a generalized one using the auxiliary slave system approach. Experimental observations on prototype hardware circuits for an isolated PLL and for a master–slave PLL arrangement are also given.     

Modulation of synchronization dynamics in a network of self-sustained systems

This paper addresses the combined modulatory effects of non-nearest neighbor oscillators and local injection on synchronized states dynamics with their corresponding stability boundaries in a network of self-sustained systems. The Whittaker method and Floquet theory are used to predict analytically the stability of these states for identical and non-identical coupling parameters. Charts revealing the modulation of synchronized states and their stability boundaries at the second order of interaction in the cases of identical and non-identical coupling parameters are constructed with and without an external signal locally injected in the network. Numerical simulations validate and complement the results of analytical surveys. The limits of the stability regions are numerically explored when a small amount of Gaussian white noise is also injected in the network.     

Tracking and controlling unstable steady states of dynamical systems

An adaptive controller for stabilization of unknown unstable steady states (spirals, nodes and saddles) of nonlinear dynamical systems is considered and its robustness under the changes of the location of the fixed point in the phase space is demonstrated. An analog electronic controller, based on a low-pass filter technique, is described. It can be easily switched between a stable and an unstable mode of operation for stabilizing either spirals/nodes or saddles, respectively. Numerical and experimental results for two autonomous systems, the damped Duffing–Holmes oscillator and the chaotic Lorenz system, are presented.     

Minimum total coloring of planar graph

Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is familiar to us by the name of “total coloring”. Total coloring is a coloring of \(V\cup {E}\) such that no two adjacent or incident elements receive the same color. In other words, total chromatic number of \(G\) is the minimum number of disjoint vertex independent sets covering a total graph of \(G\) . Here, let \(G\) be a planar graph with \(\varDelta \ge 8\) . We proved that if for every vertex \(v\in V\) , there exists two integers \(i_{v},j_{v} \in \{3,4,5,6,7,8\}\) such that \(v\) is not incident with intersecting \(i_v\) -cycles and \(j_v\) -cycles, then the vertex chromatic number of total graph of \(G\) is \(\varDelta +1\) , i.e., the total chromatic number of \(G\) is \(\varDelta +1\) .     

Global exponential stability and existence of periodic solutions in BAM networks with delays and reaction–diffusion terms

Both exponential stability and periodic solutions are considered for a class of bi-directional associative memory (BAM) neural networks with delays and reaction–diffusion terms by constructing suitable Lyapunov functional and some analysis techniques. The general sufficient conditions are given ensuring the global exponential stability and existence of periodic solutions of BAM neural networks with delays and reaction–diffusion terms. These presented conditions are in terms of system parameters and have important leading significance in the design and applications of globally exponentially stable and periodic oscillatory neural circuits for BAM with delays and reaction–diffusion terms.     

High performance hybrid reinforcement of nitrile rubber using short pineapple leaf fiber and carbon black

Rubber composites with very high moduli at low elongation, high elongation at break and high ultimate breaking strength have been developed. The matrix was acrylonitrile butadiene rubber (NBR) and the hybrid (fibrous and particulate) reinforcements were short, fine pineapple leaf fiber (PALF) and carbon black. The amount of PALF was fixed at 10 parts (by weight) per hundred of rubber (phr) while that of carbon black was varied from 0 to 30 phr. Uniaxial NBR composites were prepared. Tensile strength, elongation at break, modulus and tear strength of the hybrid composites were characterized in both longitudinal (parallel to the fiber axis) and transverse (perpendicular to the fiber axis) directions. The addition of carbon black causes the slope of the early part of the stress–strain curve to increase and also extends breaking to greater strains. At carbon black contents of 20 phr and above, the stress–strain relation displays an upturn at high elongations, providing greater ultimate strength. Comparison with the usual carbon black filled rubber shows that the composite behavior at low strains is determined by the PALF, and at high strains by the carbon black. This high performance PALF-carbon black reinforced NBR shows great promise for engineering applications.     

Stability of the split-step backward Euler scheme for stochastic delay integro-differential equations with Markovian switching

In this paper, we concentrate on the numerical approximation of solutions of stochastic delay integro-differential equations with Markovian switching (SDIDEsMS). We establish the split-step backward Euler (SSBE) scheme for solving linear SDIDEsMS and discuss its convergence and stability. Moreover, the SSBE method is convergent with strong order γ = 1/2 in the mean-square sense. The conditions under which the SSBE method is mean-square stable and general mean-square stable are obtained. Some illustrative numerical examples are presented to demonstrate the stability of the numerical method and show that SSBE method is superior to Euler method.     

Microeconomics-based resource allocation in overlay networks by using non-strategic behavior modeling

Behavior modeling has recently been investigated for designing self-organizing mechanisms in the context of communication networks in order to exploit the natural selfishness of the users with the goal of maximizing the overall utility. In strategic behavior modeling, the users of the network are assumed to be game players who seek to maximize their utility with taking into account the decisions that the other players might make. The essential difference between the aforementioned researches and this work is that it incorporates the non-strategic decisions in order to design the mechanism for the overlay network. In this solution concept, the decisions that a peer might make does not affect the actions of the other peers at all. The theory of consumer-firm developed in microeconomics is a model of the non-strategic behavior that we have adopted in our research. Based on it, we have presented distributed algorithms for peers’ “joining” and “leaving” operations. We have modeled the overlay network as a competitive economy in which the content provided by an origin server can be viewed as commodity and the origin server and the peers who multicast the content to their downside are considered as the firms. On the other hand, due to the dual role of the peers in the overlay network, they can be considered as the consumers as well. On joining to the overlay economy, each peer is provided with an income and tries to get hold of the service regardless to the behavior of the other peers. We have designed the scalable algorithms in such a way that the existence of equilibrium price (known as Walrasian equilibrium price) is guaranteed.     

Controllability and stabilization of periodic switched Boolean control networks with application to asynchronous updating

This paper investigates the periodic switching point controllability and stabilization of periodic switched Boolean control networks (PSBCNs), and applies the obtained results to the stabilization of deterministic asynchronous Boolean control networks (DABCNs). Firstly, using the algebraic state space representation of PSBCNs, a kind of periodic switching point controllability matrix is constructed, based on which, a necessary and sufficient condition is presented for the periodic switching point reachability and controllability of PSBCNs. Secondly, using the reachable set of PSBCNs, a constructive procedure is proposed to design time-variant state feedback controllers for the periodic switching point stabilization of PSBCNs. Finally, by converting the dynamics of DABCNs into the form of PSBCNs, the time-variant state feedback stabilization problem of DABCNs is solved.     

A dual neural network for convex quadratic programming subject to linear equality and inequality constraints

A recurrent neural network called the dual neural network is proposed in this Letter for solving the strictly convex quadratic programming problems. Compared to other recurrent neural networks, the proposed dual network with fewer neurons can solve quadratic programming problems subject to equality, inequality, and bound constraints. The dual neural network is shown to be globally exponentially convergent to optimal solutions of quadratic programming problems. In addition, compared to neural networks containing high-order nonlinear terms, the dynamic equation of the proposed dual neural network is piecewise linear, and the network architecture is thus much simpler. The global convergence behavior of the dual neural network is demonstrated by an illustrative numerical example.