Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators

In recent years there has been an increasing interest in studying time-delayed coupled networks of oscillators since these occur in many real life applications. In many cases symmetry patterns can emerge in these networks, as a consequence a part of the system might repeat itself, and properties of this subsystem are representative of the dynamics on the whole phase space. In this paper an analysis of the second order N-node time-delay fully connected network is presented which is based on previous work by Correa and Piqueira \cite{Correa2013} for a 2-node network. This study is carried out using symmetry groups. We show the existence of multiple eigenvalues forced by symmetry, as well as the existence of Hopf bifurcations. Three different models are used to analyze the network dynamics, namely, the full-phase, the phase, and the phase-difference model. We determine a finite set of frequencies $\omega$, that might correspond to Hopf bifurcations in each case for critical values of the delay. The $S_n$ map is used to actually find Hopf bifurcations along with numerical calculations using the Lambert W function. Numerical simulations are used in order to confirm the analytical results. Although we restrict attention to second order nodes, the results could be extended to higher order networks provided the time-delay in the connections between nodes remains equal.Comment: 41 pages, 18 figure

A mathematical view of biological complexity

This essay is a trial on giving some mathematical ideas about the concept of biological complexity, trying to explore four different attributes considered to be essential to characterize a complex system in a biological context: decomposition, heterogeneous assembly, self-organization, and adequacy. It is a theoretical and speculative approach, opening some possibilities to further numerical and experimental work, illustrated by references to several researches that applied the concepts presented here. (C) 2008 Elsevier B.V. All rights reserved.CNPq (Brazilian Research Council)FAPESP (SaoPaulo Research Agency

Using bifurcations in the determination of lock-in ranges for third-order phase-locked loops

Transmission and switching in digital telecommunication networks require distribution of precise time signals among the nodes. Commercial systems usually adopt a master-slave (MS) clock distribution strategy building slave nodes with phase-locked loop (PLL) circuits. PLLs are responsible for synchronizing their local oscillations with signals from master nodes, providing reliable clocks in all nodes. The dynamics of a PLL is described by an ordinary nonlinear differential equation, with order one plus the order of its internal linear low-pass filter. Second-order loops are commonly used because their synchronous state is asymptotically stable and the lock-in range and design parameters are expressed by a linear equivalent system [Gardner FM. Phaselock techniques. New York: John Wiley & Sons: 1979]. In spite of being simple and robust, second-order PLLs frequently present double-frequency terms in PD output and it is very difficult to adapt a first-order filter in order to cut off these components [Piqueira JRC, Monteiro LHA. Considering second-harmonic terms in the operation of the phase detector for second order phase-locked loop. IEEE Trans Circuits Syst [2003;50(6):805-9; Piqueira JRC, Monteiro LHA. All-pole phase-locked loops: calculating lock-in range by using Evan`s root-locus. Int J Control 2006;79(7):822-9]. Consequently, higher-order filters are used, resulting in nonlinear loops with order greater than 2. Such systems, due to high order and nonlinear terms, depending on parameters combinations, can present some undesirable behaviors, resulting from bifurcations, as error oscillation and chaos, decreasing synchronization ranges. In this work, we consider a second-order Sallen-Key loop filter [van Valkenburg ME. Analog filter design. New York: Holt, Rinehart & Winston; 1982] implying a third order PLL The resulting lock-in range of the third-order PLL is determined by two bifurcation conditions: a saddle-node and a Hopf. (C) 2008 Elsevier B.V. All rights reserved

Phase-Locked loops lock-in range in Frequency Modulated-Atomic Force Microscope nonlinear control system

Since the mid 1980s the Atomic Force Microscope is one the most powerful tools to perform surface investigation, and since 1995 Non-Contact AFM achieved true atomic resolution. The Frequency-Modulated Atomic Force Microscope (FM-AFM) operates in the dynamic mode, which means that the control system of the FM-AFM must force the micro-cantilever to oscillate with constant amplitude and frequency. However, tip-sample interaction forces cause modulations in the microcantilever motion. A Phase-Locked loop (PLL) is used to demodulate the tip-sample interaction forces from the microcantilever motion. The demodulated signal is used as the feedback signal to the control system, and to generate both topographic and dissipation images. As a consequence, a proper design of the PLL is vital to the FM-AFM performance. In this work, using bifurcation analysis, the lock-in range of the PLL is determined as a function of the frequency shift (Q) of the microcantilever and of the other design parameters, providing a technique to properly design the PLL in the FM-AFM system. (C) 2011 Elsevier B.V. All rights reserved

Phase-Locked Loop design applied to frequency-modulated atomic force microscope

The atomic force microscope (AFM) introduced the surface investigation with true atomic resolution. In the frequency modulation technique (FM-AFM) both the amplitude and the frequency of oscillation of the micro-cantilever must be kept constant even in the presence of tip-surface interaction forces. For that reason, the proper design of the Phase-Locked Loop (PLL) used in FM-AFM is vital to system performance. Here, the mathematical model of the FM-AFM control system is derived considering high order PLL In addition a method to design stable third-order Phase-Locked Loops is presented. (C) 2010 Elsevier B.V. All rights reserved.FAPESPCNP

Architectures, stability and optimization for clock distribution networks

Synchronous telecommunication networks, distributed control systems and integrated circuits have its accuracy of operation dependent on the existence of a reliable time basis signal extracted from the line data stream and acquirable to each node. In this sense, the existence of a sub-network (inside the main network) dedicated to the distribution of the clock signals is crucially important. There are different solutions for the architecture of the time distribution sub-network and choosing one of them depends on cost, precision, reliability and operational security. In this work we expose: (i) the possible time distribution networks and their usual topologies and arrangements. (ii) How parameters of the network nodes can affect the reachability and stability of the synchronous state of a network. (iii) Optimizations methods for synchronous networks which can provide low cost architectures with operational precision, reliability and security. (C) 2011 Elsevier B. V. All rights reserved

COMPLEXITY MEASURE: A QUANTUM INFORMATION APPROACH

In the past decades, all of the efforts at quantifying systems complexity with a general tool has usually relied on using Shannon's classical information framework to address the disorder of the system through the Boltzmann-Gibbs-Shannon entropy, or one of its extensions. However, in recent years, there were some attempts to tackle the quantification of algorithmic complexities in quantum systems based on the Kolmogorov algorithmic complexity, obtaining some discrepant results against the classical approach. Therefore, an approach to the complexity measure is proposed here, using the quantum information formalism, taking advantage of the generality of the classical-based complexities, and being capable of expressing these systems' complexity on other framework than its algorithmic counterparts. To do so, the Shiner-Davison-Landsberg (SDL) complexity framework is considered jointly with linear entropy for the density operators representing the analyzed systems formalism along with the tangle for the entanglement measure. The proposed measure is then applied in a family of maximally entangled mixed state.Brazilian National Council for Scientific and Technological Development (Conselho Nacional de Desenvolvimento Cientiico e Tecnologico) - CNPqBrazilian National Council for Scientific and Technological Development (Conselho Nacional de Desenvolvimento Cientiico e Tecnologico) CNP

Epidemic Processes on Complex Networks

As a fascinating branch of the emerging network science, the mission of the epidemic dynamics on networks is to understand how objects spread in networks and thereby to work out cost-effective strategy for restraining undesirable objects or promoting desirable objects. This special issue contains ten excellent papers about this subject.Network Architectures and Service

Modeling and Filtering Double-Frequency Jitter in One-Way Master-Slave Chain Networks

One-way master-slave (OWMS) chain networks are widely used in clock distribution systems due to their reliability and low cost. As the network nodes are phase-locked loops (PLLs), double-frequency jitter (DFJ) caused by their phase detectors appears as an impairment to the performance of the clock recovering process found in communication systems and instrumentation applications. A nonlinear model for OWMS chain networks with P + 1 order PLLs as slave nodes is presented, considering the DFJ. Since higher order filters are more effective in filtering DFJ, the synchronous state stability conditions for an OWMS chain network with third-order nodes are derived, relating the loop gain and the filter coefficients. By using these conditions, design examples are discussed.FAPESPCNP