Dynamics and Synchronization of Weak Chimera States for a Coupled Oscillator System

This thesis is an investigation of chimera states in a network of identical coupled phase oscillators. Chimera states are intriguing phenomena that can occur in systems of coupled identical phase oscillators when synchronized and desynchronized oscillators coexist. We use the Kuramoto model and coupling function of Hansel for a specific system of six oscillators to prove the existence of chimera states. More precisely, we prove analytically there are chimera states in a small network of six phase oscillators previously investigated numerically by Ashwin and Burylko [8]. We can reduce to a two-dimensional system within an invariant subspace, in terms of phase differences. This system is found to have an integral of motion for a specific choice of parameters. Using this we prove there is a set of periodic orbits that is a weak chimera. Moreover, we are able to confirmthat there is an infinite number of chimera states at the special case of parameters, using the weak chimera definition of [8]. We approximate the Poincaré return map for these weak chimera solutions and demonstrate several results about their stability and bifurcation for nearby parameters. These agree with numerical path following of the solutions. We also consider another invariant subspace to reduce the Kuramoto model of six coupled phase oscillators to a first order differential equation. We analyse this equation numerically and find regions of attracting chimera states exist within this invariant subspace. By computing eigenvalues at a nonhyperbolic point for the system of phase differences, we numerically find there are chimera states in the invariant subspace that are attracting within full system.Republic of Iraq, Ministry of Higher Education and Scientific Research

Information-based Analysis and Control of Recurrent Linear Networks and Recurrent Networks with Sigmoidal Nonlinearities

Linear dynamical models have served as an analytically tractable approximation for a variety of natural and engineered systems. Recently, such models have been used to describe high-level diffusive interactions in the activation of complex networks, including those in the brain. In this regard, classical tools from control theory, including controllability analysis, have been used to assay the extent to which such networks might respond to their afferent inputs. However, for natural systems such as brain networks, it is not clear whether advantageous control properties necessarily correspond to useful functionality. That is, are systems that are highly controllable (according to certain metrics) also ones that are suited to computational goals such as representing, preserving and categorizing stimuli? This dissertation will introduce analysis methods that link the systems-theoretic properties of linear systems with informational measures that describe these functional characterizations. First, we assess sensitivity of a linear system to input orientation and novelty by deriving a measure of how networks translate input orientation differences into readable state trajectories. Next, we explore the implications of this novelty-sensitivity for endpoint-based input discrimination, wherein stimuli are decoded in terms of their induced representation in the state space. We develop a theoretical framework for the exploration of how networks utilize excess input energy to enhance orientation sensitivity (and thus enhanced discrimination ability). Next, we conduct a theoretical study to reveal how the background or default state of a network with linear dynamics allows it to best promote discrimination over a continuum of stimuli. Specifically, we derive a measure, based on the classical notion of a Fisher discriminant, quantifying the extent to which the state of a network encodes information about its afferent inputs. This measure provides an information value quantifying the knowablility of an input based on its projection onto the background state. We subsequently optimize this background state, and characterize both the optimal background and the inputs giving it rise. Finally, we extend this information-based network analysis to include networks with nonlinear dynamics--specifically, ones involving sigmoidal saturating functions. We employ a quasilinear approximation technique, novel here in terms of its multidimensionality and specific application, to approximate the nonlinear dynamics by scaling a corresponding linear system and biasing by an offset term. A Fisher information-based metric is derived for the quasilinear system, with analytical and numerical results showing that Fisher information is better for the quasilinear (hence sigmoidal) system than for an unconstrained linear system. Interestingly, this relation reverses when the noise is placed outside the sigmoid in the model, supporting conclusions extant in the literature that the relative alignment of the state and noise covariance is predictive of Fisher information. We show that there exists a clear trade-off between informational advantage, as conferred by the presence of sigmoidal nonlinearities, and speed of dynamics

Towards understanding human locomotion

Die zentrale Frage, die in der vorliegenden Arbeit untersucht wurde, ist, wie man die komplizierte Dynamik des menschlichen Laufens besser verstehen kann. In der wissenschaftlichen Literatur werden zur Beschreibung von Laufbewegungen (Gehen und Rennen) oftmals minimalistische "Template"-Modelle verwendet. Diese sehr einfachen Modelle beschreiben nur einen ausgewählten Teil der Dynamik, meistens die Schwerpunktsbahn. In dieser Arbeit wird nun versucht, mittels Template-Modellen das Verständnis des Laufens voranzubringen. Die Analyse der Schwerpunktsbewegung durch Template-Modelle setzt eine präzise Bestimmung der Schwerpunktsbahn im Experiment voraus. Hierfür wird in Kapitel 2.3 eine neue Methode vorgestellt, welche besonders robust gegen die typischen Messfehler bei Laufexperimenten ist. Die am häfigsten verwendeten Template-Modelle sind das Masse-Feder-Modell und das inverse Pendel, welche zur Beschreibung der Körperschwerpunktsbewegung gedacht sind und das Drehmoment um den Schwerpunkt vernachlässigen. Zur Beschreibung der Stabilisierung der Körperhaltung (und damit der Drehimpulsbilanz) wird in Abschnitt 3.3 das Template-Modell "virtuelles Pendel" für das menschliche Gehen eingeführt und mit experimentellen Daten verglichen. Die Diskussion möglicher Realisierungsmechanismen legt dabei nahe, dass die Aufrichtung des menschlichen Gangs im Laufe der Evolution keine große mechanische Hürde war. In der Literatur wird oft versucht, Eigenschaften der Bewegung wie Stabilität durch Eigenschaften der Template-Modelle zu erklären. Dies wird in modifizierter Form auch in der vorliegen Arbeit getan. Hierzu wird zunächst eine experimentell bestimmte Schwerpunktsbewegung auf das Masse-Feder-Modell übertragen. Anschließend wird die Kontrollvorschrift der Schritt-zu-Schritt-Anpassung der Modellparameter identifiziert sowie eine geeignete Näherung angegeben, um die Stabilität des Modells, welches mit dieser Kontrollvorschrift ausgestattet wird, zu analysieren. Der Vergleich mit einer direkten Bestimmung der Stabilität aus einem Floquet-Modell zeigt qualitativ gute Übereinstimmung. Beide Ansätze führen auf das Ergebnis, dass beim langsamen menschlichen Rennen Störungen innerhalb von zwei Schritten weitgehend abgebaut werden. Zusammenfassend wurde gezeigt, wie Template-Modelle zum Verständnis der Laufbewegung beitragen können. Gerade die Identifikation der individuellen Kontrollvorschrift auf der Abstraktionsebene des Masse-Feder-Modells erlaubt zukünftig neue Wege, aktive Prothesen oder Orthesen in menschenähnlicher Weise zu steuern und ebnet den Weg, menschliches Rennen auf Roboter zu übertragen.Human locomotion is part of our everyday life, however the mechanisms are not well enough understood to be transferred into technical devices like orthoses, protheses or humanoid robots. In biomechanics often minimalistic so-called template models are used to describe locomotion. While these abstract models in principle offer a language to describe both human behavior and technical control input, the relation between human locomotion and locomotion of these templates was long unclear. This thesis focusses on how human locomotion can be described and analyzed using template models. Often, human running is described using the SLIP template. Here, it is shown that SLIP (possibly equipped with any controller) cannot show human-like disturbance reactions, and an appropriate extension of SLIP is proposed. Further, a new template to describe postural stabilization is proposed. Summarizing, this theses shows how simple template models can be used to enhance the understanding of human gait

Spectral, Combinatorial, and Probabilistic Methods in Analyzing and Visualizing Vector Fields and Their Associated Flows

In this thesis, we introduce several tools, each coming from a different branch of mathematics, for analyzing real vector fields and their associated flows. Beginning with a discussion about generalized vector field decompositions, that mainly have been derived from the classical Helmholtz-Hodge-decomposition, we decompose a field into a kernel and a rest respectively to an arbitrary vector-valued linear differential operator that allows us to construct decompositions of either toroidal flows or flows obeying differential equations of second (or even fractional) order and a rest. The algorithm is based on the fast Fourier transform and guarantees a rapid processing and an implementation that can be directly derived from the spectral simplifications concerning differentiation used in mathematics. Moreover, we present two combinatorial methods to process 3D steady vector fields, which both use graph algorithms to extract features from the underlying vector field. Combinatorial approaches are known to be less sensitive to noise than extracting individual trajectories. Both of the methods are extensions of an existing 2D technique to 3D fields. We observed that the first technique can generate overly coarse results and therefore we present a second method that works using the same concepts but produces more detailed results. Finally, we discuss several possibilities for categorizing the invariant sets with respect to the flow. Existing methods for analyzing separation of streamlines are often restricted to a finite time or a local area. In the frame of this work, we introduce a new method that complements them by allowing an infinite-time-evaluation of steady planar vector fields. Our algorithm unifies combinatorial and probabilistic methods and introduces the concept of separation in time-discrete Markov chains. We compute particle distributions instead of the streamlines of single particles. We encode the flow into a map and then into a transition matrix for each time direction. Finally, we compare the results of our grid-independent algorithm to the popular Finite-Time-Lyapunov-Exponents and discuss the discrepancies. Gauss\'' theorem, which relates the flow through a surface to the vector field inside the surface, is an important tool in flow visualization. We are exploiting the fact that the theorem can be further refined on polygonal cells and construct a process that encodes the particle movement through the boundary facets of these cells using transition matrices. By pure power iteration of transition matrices, various topological features, such as separation and invariant sets, can be extracted without having to rely on the classical techniques, e.g., interpolation, differentiation and numerical streamline integration

Bio-inspired robotic control in underactuation: principles for energy efficacy, dynamic compliance interactions and adaptability.

Biological systems achieve energy efficient and adaptive behaviours through extensive autologous and exogenous compliant interactions. Active dynamic compliances are created and enhanced from musculoskeletal system (joint-space) to external environment (task-space) amongst the underactuated motions. Underactuated systems with viscoelastic property are similar to these biological systems, in that their self-organisation and overall tasks must be achieved by coordinating the subsystems and dynamically interacting with the environment. One important question to raise is: How can we design control systems to achieve efficient locomotion, while adapt to dynamic conditions as the living systems do? In this thesis, a trajectory planning algorithm is developed for underactuated microrobotic systems with bio-inspired self-propulsion and viscoelastic property to achieve synchronized motion in an energy efficient, adaptive and analysable manner. The geometry of the state space of the systems is explicitly utilized, such that a synchronization of the generalized coordinates is achieved in terms of geometric relations along the desired motion trajectory. As a result, the internal dynamics complexity is sufficiently reduced, the dynamic couplings are explicitly characterised, and then the underactuated dynamics are projected onto a hyper-manifold. Following such a reduction and characterization, we arrive at mappings of system compliance and integrable second-order dynamics with the passive degrees of freedom. As such, the issue of trajectory planning is converted into convenient nonlinear geometric analysis and optimal trajectory parameterization. Solutions of the reduced dynamics and the geometric relations can be obtained through an optimal motion trajectory generator. Theoretical background of the proposed approach is presented with rigorous analysis and developed in detail for a particular example. Experimental studies are conducted to verify the effectiveness of the proposed method. Towards compliance interactions with the environment, accurate modelling or prediction of nonlinear friction forces is a nontrivial whilst challenging task. Frictional instabilities are typically required to be eliminated or compensated through efficiently designed controllers. In this work, a prediction and analysis framework is designed for the self-propelled vibro-driven system, whose locomotion greatly relies on the dynamic interactions with the nonlinear frictions. This thesis proposes a combined physics-based and analytical-based approach, in a manner that non-reversible characteristic for static friction, presliding as well as pure sliding regimes are revealed, and the frictional limit boundaries are identified. Nonlinear dynamic analysis and simulation results demonstrate good captions of experimentally observed frictional characteristics, quenching of friction-induced vibrations and satisfaction of energy requirements. The thesis also performs elaborative studies on trajectory tracking. Control schemes are designed and extended for a class of underactuated systems with concrete considerations on uncertainties and disturbances. They include a collocated partial feedback control scheme, and an adaptive variable structure control scheme with an elaborately designed auxiliary control variable. Generically, adaptive control schemes using neural networks are designed to ensure trajectory tracking. Theoretical background of these methods is presented with rigorous analysis and developed in detail for particular examples. The schemes promote the utilization of linear filters in the control input to improve the system robustness. Asymptotic stability and convergence of time-varying reference trajectories for the system dynamics are shown by means of Lyapunov synthesis

Prediction and control of nonlinear dynamical systems using machine learning

Künstliche Intelligenz und Machine Learning erfreuen sich in Folge der rapide gestiegenen Rechenleistung immer größerer Popularität. Sei es autonomes Fahren, Gesichtserkennung, bildgebende Diagnostik in der Medizin oder Robotik – die Anwendungsvielfalt scheint keine Grenzen zu kennen. Um jedoch systematischen Bias und irreführende Ergebnisse zu vermeiden, ist ein tiefes Verständnis der Methoden und ihrer Sensitivitäten von Nöten. Anhand der Vorhersage chaotischer Systeme mit Reservoir Computing – einem künstlichen rekurrenten neuronalem Netzwerk – wird im Rahmen dieser Dissertation beleuchtet, wie sich verschiedene Eigenschaften des Netzwerks auf die Vorhersagekraft und Robustheit auswirken. Es wird gezeigt, wie sich die Variabilität der Vorhersagen – sowohl was die exakte zukünftige Trajektorie betrifft als auch das statistische Langzeitverhalten (das "Klima") des Systems – mit geeigneter Parameterwahl signifikant reduzieren lässt. Die Nichtlinearität der Aktivierungsfunktion spielt hierbei eine besondere Rolle, weshalb ein Skalierungsparameter eingeführt wird, um diese zu kontrollieren. Des Weiteren werden differenzielle Eigenschaften des Netzwerkes untersucht und gezeigt, wie ein kontrolliertes Entfernen der "richtigen" Knoten im Netzwerk zu besseren Vorhersagen führt und die Größe des Netzwerkes stark reduziert werden kann bei gleichzeitig nur moderater Verschlechterung der Ergebnisse. Dies ist für Hardware Realisierungen von Reservoir Computing wie zum Beispiel Neuromorphic Computing relevant, wo möglichst kleine Netzwerke von Vorteil sind. Zusätzlich werden unterschiedliche Netzwerktopologien wie Small World Netzwerke und skalenfreie Netzwerke beleuchtet. Mit den daraus gewonnenen Erkenntnissen für bessere Vorhersagen von nichtlinearen dynamischen Systemen wird eine neue Kontrollmethode entworfen, die es ermöglicht, dynamische Systeme flexibel in verschiedenste Zielzustände zu lenken. Hierfür wird – anders als bei vielen bisherigen Ansätzen – keine Kenntnis der zugrundeliegenden Gleichungen des Systems erfordert. Ebenso wird nur eine begrenzte Datenmenge verlangt, um Reservoir Computing hinreichend zu trainieren. Zudem ist es nicht nur möglich, chaotisches Verhalten in einen periodischen Zustand zu zwingen, sondern auch eine Kontrolle auf komplexere Zielzustände wie intermittentes Verhalten oder ein spezifischer anderer chaotischer Zustand. Dies ermöglicht eine Vielzahl neuer potenzieller realer Anwendungen, von personalisierten Herzschrittmachern bis hin zu Kontrollvorrichtungen für Raketentriebwerke zur Unterbindung von kritischen Verbrennungsinstabilitäten. Als Schritt zur Weiterentwicklung von Reservoir Computing zu einem verbesserten hybriden System, das nicht nur rein datenbasiert arbeitet, sondern auch physikalische Zusammenhänge berücksichtigt, wird ein Ansatz vorgestellt, um lineare und nichtlinearen Kausalitätsstrukturen zu separieren. Dies kann verwendet werden, um Systemgleichungen oder Restriktionen für ein hybrides System zur Vorhersage oder Kontrolle abzuleiten.Artificial intelligence and machine learning are becoming increasingly popular as a result of the rapid increase in computing power. Be it autonomous driving, facial recognition, medical imaging diagnostics or robotics – the variety of applications seems to have no limits. However, to avoid systematic bias and misleading results, a deep understanding of the methods and their sensitivities is needed. Based on the prediction of chaotic systems with reservoir computing – an artificial recurrent neural network – this dissertation sheds light on how different properties of the network affect the predictive power and robustness. It is shown how the variability of the predictions – both in terms of the exact short-term predictions and the long-term statistical behaviour (the "climate") of the system – can be significantly reduced with appropriate parameter choices. The nonlinearity of the activation function plays a special role here, thus a scaling parameter is introduced to control it. Furthermore, differential properties of the network are investigated and it is shown how a controlled removal of the right nodes in the network leads to better predictions, whereas the size of the network can be greatly reduced while only moderately degrading the results. This is relevant for hardware realizations of reservoir computing such as neuromorphic computing, where networks that are as small as possible are advantageous. Additionally, different network topologies such as small world networks and scale-free networks are investigated. With the insights gained for better predictions of nonlinear dynamical systems, a new control method is designed that allows dynamical systems to be flexibly forced into a wide variety of dynamical target states. For this – unlike many previous approaches – no knowledge of the underlying equations of the system is required. Further, only a limited amount of data is needed to sufficiently train reservoir computing. Moreover, it is possible not only to force chaotic behavior to a periodic state, but also to control for more complex target states such as intermittent behavior or a specific different chaotic state. This enables a variety of new potential real-world applications, from personalized cardiac pacemakers to control devices for rocket engines to suppress critical combustion instabilities. As a step toward advancing reservoir computing to an improved hybrid system that is not only purely data-based but also takes into account physical relationships, an approach is presented to separate linear and nonlinear causality structures. This can be used to derive system equations or constraints for a hybrid prediction or control system

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue

Actuation-Aware Simplified Dynamic Models for Robotic Legged Locomotion

In recent years, we witnessed an ever increasing number of successful hardware implementations of motion planners for legged robots. If one common property is to be identified among these real-world applications, that is the ability of online planning. Online planning is forgiving, in the sense that it allows to relentlessly compensate for external disturbances of whatever form they might be, ranging from unmodeled dynamics to external pushes or unexpected obstacles and, at the same time, follow user commands. Initially replanning was restricted only to heuristic-based planners that exploit the low computational effort of simplified dynamic models. Such models deliberately only capture the main dynamics of the system, thus leaving to the controllers the issue of anchoring the desired trajectory to the whole body model of the robot. In recent years, however, we have seen a number of new approaches attempting to increase the accuracy of the dynamic formulation without trading-off the computational efficiency of simplified models. In this dissertation, as an example of successful hardware implementation of heuristics and simplified model-based locomotion, I describe the framework that I developed for the generation of an omni-directional bounding gait for the HyQ quadruped robot. By analyzing the stable limit cycles for the sagittal dynamics and the Center of Pressure (CoP) for the lateral stabilization, the described locomotion framework is able to achieve a stable bounding while adapting to terrains of mild roughness and to sudden changes of the user desired linear and angular velocities. The next topic reported and second contribution of this dissertation is my effort to formulate more descriptive simplified dynamic models, without trading off their computational efficiency, in order to extend the navigation capabilities of legged robots to complex geometry environments. With this in mind, I investigated the possibility of incorporating feasibility constraints in these template models and, in particular, I focused on the joint torques limits which are usually neglected at the planning stage. In this direction, the third contribution discussed in this thesis is the formulation of the so called actuation wrench polytope (AWP), defined as the set of feasible wrenches that an articulated robot can perform given its actuation limits. Interesected with the contact wrench cone (CWC), this yields a new 6D polytope that we name feasible wrench polytope (FWP), defined as the set of all wrenches that a legged robot can realize given its actuation capabilities and the friction constraints. Results are reported where, thanks to efficient computational geometry algorithms and to appropriate approximations, the FWP is employed for a one-step receding horizon optimization of center of mass trajectory and phase durations given a predefined step sequence on rough terrains. For the sake of reachable workspace augmentation, I then decided to trade off the generality of the FWP formulation for a suboptimal scenario in which a quasi-static motion is assumed. This led to the definition of the, so called, local/instantaneous actuation region and of the global actuation/feasible region. They both can be seen as different variants of 2D linear subspaces orthogonal to gravity where the robot is guaranteed to place its own center of mass while being able to carry its own body weight given its actuation capabilities. These areas can be intersected with the well known frictional support region, resulting in a 2D linear feasible region, thus providing an intuitive tool that enables the concurrent online optimization of actuation consistent CoM trajectories and target foothold locations on rough terrains