Data based identification and prediction of nonlinear and complex dynamical systems

We thank Dr. R. Yang (formerly at ASU), Dr. R.-Q. Su (formerly at ASU), and Mr. Zhesi Shen for their contributions to a number of original papers on which this Review is partly based. This work was supported by ARO under Grant No. W911NF-14-1-0504. W.-X. Wang was also supported by NSFC under Grants No. 61573064 and No. 61074116, as well as by the Fundamental Research Funds for the Central Universities, Beijing Nova Programme.Peer reviewedPostprin

Stability and Control in Complex Networks of Dynamical Systems

Stability analysis of networked dynamical systems has been of interest in many disciplines such as biology and physics and chemistry with applications such as LASER cooling and plasma stability. These large networks are often modeled to have a completely random (Erdös-Rényi) or semi-random (Small-World) topologies. The former model is often used due to mathematical tractability while the latter has been shown to be a better model for most real life networks. The recent emergence of cyber physical systems, and in particular the smart grid, has given rise to a number of engineering questions regarding the control and optimization of such networks. Some of the these questions are: How can the stability of a random network be characterized in probabilistic terms? Can the effects of network topology and system dynamics be separated? What does it take to control a large random network? Can decentralized (pinning) control be effective? If not, how large does the control network needs to be? How can decentralized or distributed controllers be designed? How the size of control network would scale with the size of networked system? Motivated by these questions, we began by studying the probability of stability of synchronization in random networks of oscillators. We developed a stability condition separating the effects of topology and node dynamics and evaluated bounds on the probability of stability for both Erdös-Rényi (ER) and Small-World (SW) network topology models. We then turned our attention to the more realistic scenario where the dynamics of the nodes and couplings are mismatched. Utilizing the concept of ε-synchronization, we have studied the probability of synchronization and showed that the synchronization error, ε, can be arbitrarily reduced using linear controllers. We have also considered the decentralized approach of pinning control to ensure stability in such complex networks. In the pinning method, decentralized controllers are used to control a fraction of the nodes in the network. This is different from traditional decentralized approaches where all the nodes have their own controllers. While the problem of selecting the minimum number of pinning nodes is known to be NP-hard and grows exponentially with the number of nodes in the network we have devised a suboptimal algorithm to select the pinning nodes which converges linearly with network size. We have also analyzed the effectiveness of the pinning approach for the synchronization of oscillators in the networks with fast switching, where the network links disconnect and reconnect quickly relative to the node dynamics. To address the scaling problem in the design of distributed control networks, we have employed a random control network to stabilize a random plant network. Our results show that for an ER plant network, the control network needs to grow linearly with the size of the plant network

Informational cascade as a pinning control problem

Informational cascades are imitation phenomena which can emerge in financial markets. When an informational cascade occurs, the agents completely disregard their own information and blindly follow the behavior of the other traders. The models in the literature, although capable of replicating this phenomenon, do not take into account the possibility of reproducing cascades of different intensities, displayed by empirical evidences. To overcome this limitation, we introduce a new model of opinion dynamics capable of replicating informational cascades of different magnitudes. This is accomplished by viewing informational cascades as a diffusion of a certain opinion in a network of financial agents, whose trading strategies dynamically depend on that of their neighbors according to a nonlinear law. Following the logic of pinning control, we model the generic exogenous information triggering informational cascades as a control signal fed by an external entity, the pinner, to a subset of agents. By virtue of the received information, they take the trading action that will be imitated by the non informed traders. In this framework, we can exploit some results of the so called “partial pinning control” to assess the number of non informed agents which reach consensus on the pinner’s opinion, and thus are involved in the informational cascade. This assessment is based on the topological structure connecting the agents: different topologies generate informational cascades of different magnitudes. We test our model of opinion dynamics in an agent-based artificial financial market, showing its capability of replicating informational cascades of different and predictable intensities

Connectivity Influences on Nonlinear Dynamics in Weakly-Synchronized Networks: Insights from Rössler Systems, Electronic Chaotic Oscillators, Model and Biological Neurons

Natural and engineered networks, such as interconnected neurons, ecological and social networks, coupled oscillators, wireless terminals and power loads, are characterized by an appreciable heterogeneity in the local connectivity around each node. For instance, in both elementary structures such as stars and complex graphs having scale-free topology, a minority of elements are linked to the rest of the network disproportionately strongly. While the effect of the arrangement of structural connections on the emergent synchronization pattern has been studied extensively, considerably less is known about its influence on the temporal dynamics unfolding within each node. Here, we present a comprehensive investigation across diverse simulated and experimental systems, encompassing star and complex networks of Rössler systems, coupled hysteresis-based electronic oscillators, microcircuits of leaky integrate-and-fire model neurons, and finally recordings from in-vitro cultures of spontaneously-growing neuronal networks. We systematically consider a range of dynamical measures, including the correlation dimension, nonlinear prediction error, permutation entropy, and other information-theoretical indices. The empirical evidence gathered reveals that under situations of weak synchronization, wherein rather than a collective behavior one observes significantly differentiated dynamics, denser connectivity tends to locally promote the emergence of stronger signatures of nonlinear dynamics. In deterministic systems, transition to chaos and generation of higher-dimensional signals were observed; however, when the coupling is stronger, this relationship may be lost or even inverted. In systems with a strong stochastic component, the generation of more temporally-organized activity could be induced. These observations have many potential implications across diverse fields of basic and applied science, for example, in the design of distributed sensing systems based on wireless coupled oscillators, in network identification and control, as well as in the interpretation of neuroscientific and other dynamical data

Multilayer Networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure

Optimal Control Strategies for Complex Biological Systems

To better understand and to improve therapies for complex diseases such as cancer or diabetes, it is not sufficient to identify and characterize the interactions between molecules and pathways in complex biological systems, such as cells, tissues, and the human body. It also is necessary to characterize the response of a biological system to externally supplied agents (e.g., drugs, insulin), including a proper scheduling of these drugs, and drug combinations in multi drugs therapies. This obviously becomes important in applications which involve control of physiological processes, such as controlling the number of autophagosome vesicles in a cell, or regulating the blood glucose level in patients affected by diabetes. A critical consideration when controlling physiological processes in biological systems is to reduce the amount of drugs used, as in some therapies drugs may become toxic when they are overused. All of the above aspects can be addressed by using tools provided by the theory of optimal control, where the externally supplied drugs or hormones are the inputs to the system. Another important aspect of using optimal control theory in biological systems is to identify the drug or the combination of drugs that are effective in regulating a given therapeutic target, i.e., a biological target of the externally supplied stimuli. The dynamics of the key features of a biological system can be modeled and described as a set of nonlinear differential equations. For the implementation of optimal control theory in complex biological systems, in what follows we extract \textit{a network} from the dynamics. Namely, to each state variable $x_i$ we will assign a network node $v_i$ ($i=1,...,N$) and a network directed edge from node $v_i$ to another node $v_j$ will be assigned every time $x_j$ is present in the time derivative of $x_i$. The node which directly receives an external stimulus is called a \emph{driver nodes} in a network. The node which directly connected to an output sensor is called a \emph{target node}. %, and it has a prescribed final state that we wish to achieve in finite time. From the control point of view, the idea of controllability of a system describes the ability to steer the system in a certain time interval towards thea desired state with a suitable choice of control inputs. However, defining controllability of large complex networks is quite challenging, primarily because of the large size of the network, its complex structure, and poor knowledge of the precise network dynamics. A network can be controllable in theory but not in practice when a very large control effort is required to steer the system in the desired direction. This thesis considers several approaches to address some of these challenges. Our first approach is to reduce the control effort is to reduce the number of target nodes. We see that by controlling the states of a subset of the network nodes, rather than the state of every node, while holding the number of control signals constant, the required energy to control a portion of the network can be reduced substantially. The energy requirements exponentially decay with the number of target nodes, suggesting that large networks can be controlled by a relatively small number of inputs as long as the target set is appropriately sized. We call this strategy \emph{target control}. As our second approach is based on reducing the control efforts by allowing the prescribed final states are satisfied approximately rather than strictly. We introduce a new control strategy called \textit{balanced control} for which we set our objective function as a convex combination of two competitive terms: (i) the distance between the output final states at a given final time and given prescribed states and (ii) the total control efforts expenditure over the given time period. Based on the above two approaches, we propose an algorithm which provides a locally optimal control technique for a network with nonlinear dynamics. We also apply pseudo-spectral optimal control, together with the target and balance control strategies previously described, to complex networks with nonlinear dynamics. These optimal control techniques empower us to implement the theoretical control techniques to biological systems evolving with very large, complex and nonlinear dynamics. We use these techniques to derive the optimal amounts of several drugs in a combination and their optimal dosages. First, we provide a prediction of optimal drug schedules and combined drug therapies for controlling the cell signaling network that regulates autophagy in a cell. Second, we compute an optimal dual drug therapy based on administration of both insulin and glucagon to control the blood glucose level in type I diabetes. Finally, we also implement the combined control strategies to investigate the emergence of cascading failures in the power grid networks

International Symposium on Magnetic Suspension Technology, Part 1

The goal of the symposium was to examine the state of technology of all areas of magnetic suspension and to review related recent developments in sensors and controls approaches, superconducting magnet technology, and design/implementation practices. The symposium included 17 technical sessions in which 55 papers were presented. The technical session covered the areas of bearings, sensors and controls, microgravity and vibration isolation, superconductivity, manufacturing applications, wind tunnel magnetic suspension systems, magnetically levitated trains (MAGLEV), space applications, and large gap magnetic suspension systems