Continuation with Non-invasive Control Schemes: Revealing Unstable States in a Pedestrian Evacuation Scenario

This paper presents a framework to perform bifurcation analysis in laboratory experiments or simulations. We employ control-based continuation to study the dynamics of a macroscopic variable of a microscopically defined model, exploring the potential viability of the underlying feedback control techniques in an experiment. In contrast to previous experimental studies that used iterative root-finding methods on the feedback control targets, we propose a feedback control law that is inherently non-invasive. That is, the control discovers the location of equilibria and stabilizes them simultaneously. We call the proposed control zero-in-equilibrium feedback control and we prove that it is able to stabilize branches of equilibria, except at singularities of codimension n+1, where n is the number of state space dimensions the feedback can depend on. We apply the method to a simulated evacuation scenario were pedestrians have to reach an exit after maneuvering left or right around an obstacle. The scenario shows a hysteresis phenomenon with bistability and tipping between two possible steady pedestrian flows in microscopic simulations. We demonstrate for the evacuation scenario that the proposed control law is able to uniformly discover and stabilize steady flows along the entire branch, including points where other non-invasive approaches to feedback control become singular.Comment: submitted (34 pages 12 figures

Classification and detection of Critical Transitions: from theory to data

From population collapses to cell-fate decision, critical phenomena are abundant in complex real-world systems. Among modelling theories to address them, the critical transitions framework gained traction for its purpose of determining classes of critical mechanisms and identifying generic indicators to detect and alert them (“early warning signals”). This thesis contributes to such research field by elucidating its relevance within the systems biology landscape, by providing a systematic classification of leading mechanisms for critical transitions, and by assessing the theoretical and empirical performance of early warning signals. The thesis thus bridges general results concerning the critical transitions field – possibly applicable to multidisciplinary contexts – and specific applications in biology and epidemiology, towards the development of sound risk monitoring system

Invariant manifold theory for impulsive functional differential equations with applications

The primary contribution of this thesis is a development of invariant manifold theory for impulsive functional differential equations. We begin with an in-depth analysis of linear systems, immersed in a nonautonomous dynamical systems framework. We prove a variation-of-constants formula, introduce appropriate generalizations of stable, centre and unstable subspaces, and develop a Floquet theory for periodic systems. Using the Lyapunov-Perron method, we prove the existence of local centre manifolds at a nonhyperbolic equilibrium of nonlinear impulsive functional differential equations. Using a formal differentiation procedure in conjunction with machinery from functional analysis -- specifically, contraction mappings on scales of Banach spaces -- we prove that the centre manifold is smooth in the state space. By introducing a coordinate system, we are able to prove that the coefficients of any Taylor expansion of the local centre manifold are unique and sufficiently regular in the time and lag arguments that they can be computed by solving an impulsive boundary-value problem. After proving a reduction principle, this leads naturally to explorations into bifurcation theory, where we establish generalizations of the classical fold and Hopf bifurcations for impulsive delay differential equations. Aside from the centre manifold, we demonstrate the existence and smoothness of stable and unstable manifolds and prove a linearized stability theorem. One of the applications of the theory above is an analysis of a SIR model with pulsed vaccination and finite temporary immunity modeled by a discrete delay. We determine an analytical stability criteria for the disease-free equilibrium and prove the existence of a transcritical bifurcation of periodic solutions at some critical vaccination coverage level for generic system parameters. Then, using numerical continuation and a monodromy operator discretization scheme, we track the bifurcating endemic periodic solution until a Hopf point is identifed. A cylinder bifurcation is observed; the periodic orbit expands into a cylinder in the extended phase space before eventually contracting onto a periodic orbit as the vaccination coverage vanishes. The other application is an impulsive stabilization method based on centre manifold reduction and optimization principles. Assuming a cost structure on the impulsive controller and a desired convergence rate target, we prove that under certain conditions there is always an impulsive controller that can stabilize a nonhyperbolic equilibrium with a trivial unstable subspace, robustly with respect to parameter perturbation, while guaranteeing a minimal cost. We then exploit the low-dimensionality of the centre manifold to develop a two-stage program that can be implemented to compute the optimal controller. To demonstrate the effectiveness of the two-stage program, which we call the centre probe method, we use the method to stabilize a complex network of 100 diffusively coupled nodes at a Hopf point. The cost structure is one that assigns higher cost to controlling of nodes that have more neighbours, while the jump functionals are required to be diagonal -- that is, they do not introduce further coupling. We also introduce a secondary goal, which is that the number of nodes that are controlled is minimized

On fast-slow consensus networks with a dynamic weight

We study dynamic networks under an undirected consensus communication protocol and with one state-dependent weighted edge. We assume that the aforementioned dynamic edge can take values over the whole real numbers, and that its behaviour depends on the nodes it connects and on an extrinsic slow variable. We show that, under mild conditions on the weight, there exists a reduction such that the dynamics of the network are organized by a transcritical singularity. As such, we detail a slow passage through a transcritical singularity for a simple network, and we observe that an exchange between consensus and clustering of the nodes is possible. In contrast to the classical planar fast-slow transcritical singularity, the network structure of the system under consideration induces the presence of a maximal canard. Our main tool of analysis is the blow-up method. Thus, we also focus on tracking the effects of the blow-up transformation on the network's structure. We show that on each blow-up chart one recovers a particular dynamic network related to the original one. We further indicate a numerical issue produced by the slow passage through the transcritical singularity

Finite-Time Transport in Aperiodic Dynamical Systems

In this dissertation, we consider the problem of quantifying finite-time transport in aperiodic dynamical systems. To address this problem, we concentrate primarily on a class of aperiodic systems that we call "transitory." These systems exhibit time-dependent behavior only on a compact time interval, and we present a new method for quantifying transport between isolated coherent structures of such systems, in the globally Liouville case. Both 2D and 3D examples are given. Moreover, our treatment of the 3D case represents the first quantitative analysis of transport between Lagrangian coherent structures in fully 3D aperiodic flows. We also present a numerical method that facilitates the application of our transport formulas to systems defined by discrete velocity data. In each case we consider, transport is quantified by computing the areas (or volumes in dimensions greater than two) of lobes bounded by codimension-one objects that are past or future invariant, and our method is Lagrangian, in the sense that it relies only on knowing certain key trajectories. These trajectories form a codimension-two set at the intersections of lobe boundary components. Thus, our transport computations require little Lagrangian information relative to various other methods involving finite-time Lyapunov exponents (FTLEs) or distinguished hyperbolic trajectories. To show this, we compare our method to one that identifies coherent structures as regions bounded by ridges of the FTLE field, and additionally present a new computational method for efficiently extracting such ridges

Modeling diversity by strange attractors with application to temporal pattern recognition

This thesis belongs to the general discipline of establishing black-box models from real-word data, more precisely, from measured time-series. This is an old subject and a large amount of papers and books has been written about it. The main difficulty is to express the diversity of data that has essentially the same origin without creating confusion with data that has a different origin. Normally, the diversity of time-series is modeled by a stochastic process, such as filtered white noise. Often, it is reasonable to assume that the time series is generated by a deterministic dynamical system rather than a stochastic process. In this case, the diversity of the data is expressed by the variability of the parameters of the dynamical system. The parameter variability itself is then, once again, modeled by a stochastic process. In both cases the diversity is generated by some form of exogenous noise. In this thesis a further step has been taken. A single chaotic dynamical system is used to model the data and their diversity. Indeed, a chaotic system produces a whole family of trajectories that are different but nonetheless very similar. It is believed that chaotic dynamics not only are a convenient means to represent diversity but that in many cases the origin of diversity stems actually from chaotic dynamic. Since the approach of this thesis explores completely new grounds the most suitable kind of data is considered, namely approximately periodic signals. In nature such time-series are rather common, in particular the physiological signal of living beings, such as the electrocardiograms (ECG), parts of speech signals, electroencephalograms (EEG), etc. Since there are strong arguments in favor of the chaotic nature of these signals, they appear to be the best candidates for modeling diversity by chaos. It should be stressed however, that the modeling approach pursued in this thesis is thought to be quite general and not limited to signals produced by chaotic dynamics in nature. The intended application of the modeling effort in this thesis is temporal signal classification. The reason for this is twofold. Firstly, classification is one of the basic building block of any cognitive system. Secondly, the recently studied phenomenon of synchronization of chaotic systems suggests a way to test a signal against its chaotic model. The essential content of this work can now be formulated as follows. Thesis: The diversity of approximately periodic signals found in nature can be modeled by means of chaotic dynamics. This kind of modeling technique, together with selective properties of the synchronization of chaotic systems, can be exploited for pattern recognition purposes. This Thesis is advocated by means of the following five points. Models of randomness (Chapter 2) It is argued that the randomness observed in nature is not necessarily the result of exogenous noise, but it could be endogenally generated by deterministic chaotic dynamics. The diversity of real signals is compared with signals produced by the most common chaotic systems. Qualitative resonance (Chapter 3) The behavior of chaotic systems forced by periodic or approximately periodic input signals is studied theoretically and by numerical simulation. It is observed that the chaotic system "locks" approximately to an input signal that is related to its internal chaotic dynamic. In contrast to this, its chaotic behavior is reinforced when the input signal has nothing to do with its internal dynamics. This new phenomenon is called "qualitative resonance". Modeling and recognizing (Chapter 4) In this chapter qualitative resonance is used for pattern recognition. The core of the method is a chaotic dynamical system that is able to reproduce the class of time-series that is to be recognized. This model is excited in a suitable way by an input signal such that qualitative resonance is realized. This means that if the input signal belongs to the modeled class of time-series, the system approximately "locks" into it. If not, the trajectory of the system and the input signal remain unrelated. Automated design of the recognizer (Chapters 5 and 6) For the kind of signals considered in this thesis a systematic design method of the recognizer is presented. The model used is a system of Lur'e type, i.e. a model where the linear dynamic and nonlinear static part are separated. The identification of the model parameters from the given data proceed iteratively, adapting in turn the linear and the nonlinear part. Thus, the difficult nonlinear dynamical system identification task is decomposed into the easier problems of linear dynamical and nonlinear static system identification. The way to apply the approximately periodic input signal in order to realize qualitative resonance is chosen with the help of periodic control theory. Validation (Chapter 7) The pattern recognition method has been validated on the following examples — A synthetic example — Laboratory measurement from Colpitts oscillator — ECG — EEG — Vowels of a speech signals In the first four cases a binary classification and in the last example a classification with five classes was performed. To the best of the knowledge of the author the recognition method is original. Chaotic systems have been already used to produce pseudo-noise and to model signal diversity. Also, parameter identification of chaotic systems has been already carried out. However, the direct establishment of the model from the given data and its subsequent use for classification based on the phenomenon of qualitative resonance is entirely new

Analytical solutions of orientation aggregation models, multiple solutions and path following with the Adomian decomposition method

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this work we apply the Adomian decomposition method to an orientation aggregation problem modelling the time distribution of filaments. We find analytical solutions under certain specific criteria and programmatically implement the Adomian method to two variants of the orientation aggregation model. We extend the utility of the Adomian decomposition method beyond its original capability to enable it to converge to more than one solution of a nonlinear problem and further to be used as a corrector in path following bifurcation problems