Mario di Bernardo

Controllability of complex networks via pinning.

We study the problem of controlling a general complex network towards an assigned synchronous evolution, by means of a pinning control strategy. We define the pinning-controllability of the network in terms of the spectral properties of an extended network topology. The roles of the control and coupling gains as well as of the number of pinned nodes are also discussed.

Bifurcations in Nonsmooth Dynamical Systems

A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of “normal form” or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.

Criteria for global pinning-controllability of complex networks

In this paper, we study pinning-controllability of networks of coupled dynamical systems. In particular, we study the problem of asymptotically driving a network of coupled identical oscillators onto some desired common reference trajectory by actively controlling only a limited subset of the whole network. The reference trajectory is generated by an exogenous independent oscillator, and pinned nodes are coupled to it through a linear state feedback. We describe the time evolution of the complex dynamical system in terms of the error dynamics. Thereby, we reformulate the pinning-controllability problem as a global asymptotic stability problem. By using Lyapunov-stability theory and algebraic graph theory, we establish tractable sufficient conditions for global pinning-controllability in terms of the network topology, the oscillator dynamics, and the linear state feedback.

Synchronization of complex networks through local adaptive coupling

Two local adaptive strategies for the synchronization of complex networks are discussed in this paper. One, termed as vertex-based, uses local adaptive coupling gains at each node in the network. The other, named edge-based, associates to each edge in the network an adaptive coupling gain, determined solely on the basis of local information. The global asymptotic stability of the synchronous evolution is proven for both strategies using appropriate Lyapunov-based techniques. The effectiveness of the adaptive methodologies presented in the paper is shown via two representative examples: adaptive consensus and the adaptive synchronization of a network on N coupled Chuas circuits.

A comparative analysis of synthetic genetic oscillators

Synthetic biology is a rapidly expanding discipline at the interface between engineering and biology. Much research in this area has focused on gene regulatory networks that function as biological switches and oscillators. Here we review the state of the art in the design and construction of oscillators, comparing the features of each of the main networks published to date, the models used for in silico design and validation and, where available, relevant experimental data. Trends are apparent in the ways that network topology constrains oscillator characteristics and dynamics. Also, noise and time delay within the network can both have constructive and destructive roles in generating oscillations, and stochastic coherence is commonplace. This review can be used to inform future work to design and implement new types of synthetic oscillators or to incorporate existing oscillators into new designs.

Self-oscillations and sliding in relay feedback systems : Symmetry and bifurcations

This paper is concerned with the bifurcation analysis of linear dynamical systems with relay feedback. The emphasis is on the bifurcations of the system periodic solutions and their symmetry. It is shown that, despite what has been conjectured in the literature, a symmetric and unforced relay feedback system can exhibit asymmetric periodic solutions. Moreover, the occurrence of periodic solutions characterized by one or more sections lying within the system discontinuity set is outlined. The mechanisms underlying their formation are carefully studied and shown to be due to an interesting, novel class of local bifurcations.

Global Entrainment of Transcriptional Systems to Periodic Inputs

This paper addresses the problem of providing mathematical conditions that allow one to ensure that biological networks, such as transcriptional systems, can be globally entrained to external periodic inputs. Despite appearing obvious at first, this is by no means a generic property of nonlinear dynamical systems. Through the use of contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all their solutions converge to a fixed limit cycle. General results are proved, and the properties are verified in the specific cases of models of transcriptional systems as well as constructs of interest in synthetic biology. A self-contained exposition of all needed results is given in the paper.

Distributed Consensus Strategy for Platooning of Vehicles in the Presence of Time-Varying Heterogeneous Communication Delays

We analyze and solve the platooning problem by treating it as the problem of achieving consensus in a network of dynamical systems affected by time-varying heterogeneous delays due to wireless communication among vehicles. Specifically, a platoon is modeled as a dynamical network where: 1) each vehicle, with its own dynamics, is a node; 2) the presence of communication links between neighboring vehicles is represented by edges; and 3) the structure of the intervehicle communication is encoded in the network topology. A distributed control protocol, which acts on every vehicle in the platoon, is derived. It is composed of two terms: a local action depending on the state variables of the vehicle itself (measured onboard) and an action depending on the information received from neighboring vehicles through the communication network. The stability of the platoon is proven by using Lyapunov-Razumikhin theorem. Numerical results are included to confirm and illustrate the theoretical derivation.

On QUAD, Lipschitz, and Contracting Vector Fields for Consensus and Synchronization of Networks

In this paper, a relationship is discussed between three common assumptions made in the literature to prove local or global asymptotic stability of the synchronization manifold in networks of coupled nonlinear dynamical systems. In such networks, each node, when uncoupled, is described by a nonlinear ordinary differential equation of the form ẋ = <i>f</i> (<i>x</i>,<i>t</i>) . In this paper, we establish links between the QUAD condition on <i>f</i> (<i>x</i>, <i>t</i>), i.e.,(<i>x</i>-<i>y</i>)^T[<i>f</i>(<i>x</i>, <i>t</i>)-<i>f</i>(<i>y</i>, <i>t</i>)] - (<i>x</i>-<i>y</i>)^T Δ(<i>x</i>-<i>y</i>) ≤-ω(<i>x</i>-<i>y</i>)<i>T</i>(<i>x</i>-<i>y</i>) for some arbitrary Δ and ω, and contraction theory. We then investigate the relationship between the assumption of <i>f</i> being Lipschitz and the QUAD condition. We show the usefulness of the links highlighted in this paper to obtain proofs of asymptotic synchronization in networks of identical nonlinear oscillators and illustrate the results via numerical simulations on some representative examples.

Synchronization and control of complex networks via contraction, adaptation and evolution

Complex networked systems abound in Nature and Technology. They consist of a multitude of interacting agents communicating with each other over a web of complex interconnections. Flocks of birds, platoon of cooperating robots, swirling fishes in the Ocean are all examples whose intricate dynamics can be modeled in terms of three essential ingredients: (i) a mathematical description of the dynamical behavior of each of the agents in the network; (ii) an interaction (or coupling) protocol used by agents to communicate with each other and (iii) a graph describing the network of interconnections between neighboring agents. These three elements are actually mapped onto the mathematical model usually considered in the literature to describe a complex network which uses appropriate equations to describe the node dynamics, the coupling protocol and the network topology.