It is nearly 20 years since the concept of a small-world network was first quantitatively defined, by a combination of high clustering and short path length; and about 10 years since this metric of complex network topology began to be widely applied to analysis of neuroimaging and other neuroscience data as part of the rapid growth of the new field of connectomics. Here, we review briefly the foundational concepts of graph theoretical estimation and generation of small-world networks. We take stock of some of the key developments in the field in the past decade and we consider in some detail the implications of recent studies using high-resolution tract-tracing methods to map the anatomical networks of the macaque and the mouse. In doing so, we draw attention to the important methodological distinction between topological analysis of binary or unweighted graphs, which have provided a popular but simple approach to brain network analysis in the past, and the topology of weighted graphs, which retain more b...

https://journals.sagepub.com/doi/pdf/10.1177/1073858416667720

Small-World Brain Networks Revisited.

When nonlinear dynamical systems are coupled, depending on the intrinsic dynamics and the manner in which the coupling is organized, a host of novel phenomena can arise. In this context, an important emergent phenomenon is the complete suppression of oscillations, formally termed amplitude death (AD). Oscillations of the entire system cease as a consequence of the interaction, leading to stationary behavior. The fixed points that the coupling stabilizes can be the otherwise unstable fixed points of the uncoupled system or can correspond to novel stationary points. Such behaviour is of relevance in areas ranging from laser physics to the dynamics of biological systems. In this review we discuss the characteristics of the different coupling strategies and scenarios that lead to AD in a variety of different situations, and draw attention to several open issues and challenging problems for further study.

Amplitude Death: The emergence of stationarity in coupled nonlinear systems

Amplitude death: The emergence of stationarity in coupled nonlinear systems

A chaotic path planning generator for autonomous mobile robots

The penetration of dc distributed power systems is increasing rapidly in electric power grids and other isolated systems to cater demand for cheap, clean, high quality, and uninterrupted power demand of modern society. DC systems are more efficient and suite better to integrate some of the renewable energy sources, storage units, and dc loads. A dc distributed power system usually consists of large number of power electronic converters connected in cascad0ed configuration to satisfy the power quality and voltage magnitude requirements of the sources and loads. Tightly-regulated power converters in the aforementioned settings exhibit negative incremental impedance and behave as constant power loads (CPLs), and tend to destabilize their feeder systems and upstream converters. The presence of CPLs reduces effective damping of the system leading to instability of the whole system and present significant challenge in the system operation and control. In-depth knowledge of the instability effects of constant power loads (CPLs), available stabilizing techniques and stability analysis methods, is imperious to the young researchers, system designers, system integrators, and practicing engineers working in the field of dc power systems and emerging applications of dc power. This paper is intended to fill this gape by documenting present state of the art and research needs in one article. Modeling, behaviour and effects of typical CPL are discussed and a review of stability criteria used to study the stability of dc power systems are reviewed with their merits and limitations. Furthermore, available literature is reviewed to summarize the techniques to compensate the CPL effect. Finally, discussion and recent challenges in the dc distribution systems.

Constant power loads and their effects in DC distributed power systems: A review

The present study shows that diffusive connections with two long-time delays can induce the stabilization of a steady state in network oscillators. A linear stability analysis shows that, if the two delay times retain a proportional relation with a certain bias, the stabilization can be achieved independent of the delay times. Furthermore, a simple systematic procedure for designing the coupling strength and the delay times in the connections is proposed. The procedure has the following two advantages: one can employ time delays as long as one wants and the stabilization can be achieved independently of its network topology. Our analytical results are applied to the well-known double-scroll circuit model on a small-world network.

Stabilization of a steady state in network oscillators by using diffusive connections with two long time delays

This paper investigates the robust stability of a steady state in network systems with dynamic connections. A linear stability analysis reveals that the odd number property holds for any network topology. It is shown that the stability of the steady state in network systems with topological uncertainty is governed by the interval family of real characteristic polynomials. A parametric approach, which plays an important role in the field of robust control theory, allows us to analyze the robust stability against the network topological uncertainty. The stability of the steady state, which is valid for any arbitrary topology, can be evaluated by examining at most four characteristic polynomials. These analytical results are applied to coupled R\"ossler oscillators on a small-world network.

Topology-free stability of a steady state in network systems with dynamic connections.

This paper proposes a delayed feedback control (DFC) based on the act-and-wait concept, which reduces the dynamics of DFC systems to that of discrete-time systems. Based on this concept, a delayed feedback controller is designed for a prototype two-dimensional oscillator using a simple systematic procedure. This control has two advantages: the feedback delay time can be large and it can obtain deadbeat behavior. A numerical example using a double-scroll circuit model demonstrates these theoretical results.

Delayed feedback control based on the act-and-wait concept

Stability analysis and design of amplitude death induced by a time-varying delay connection

This study aims to show that a multiple delay feedback control method can stabilize unstable fixed points of time-delay nonlinear oscillators. The boundary curves of stability in a control parameter space are derived using linear stability analysis. A simple procedure for designing a feedback gain is provided. The main advantage of this procedure is that the designed controller can stabilize a system even if the controller delay times are long. These analytical results are experimentally verified using electronic circuits.

Naoyuki Hara

Papers

Stabilization of a steady state in network oscillators by using diffusive connections with two long time delays

Topology-free stability of a steady state in network systems with dynamic connections.

Delayed feedback control based on the act-and-wait concept

Stability analysis and design of amplitude death induced by a time-varying delay connection

Design and experimental verification of multiple delay feedback control for time-delay nonlinear oscillators