Framework Of Dynamical Systems Research Articles

A Dynamic Graph-Based Systems Framework for Modeling, and Control of Cyber-Physical Systems Typified by Buildings

A Dynamic Graph-Based Systems Framework for Modeling, and Control of Cyber-Physical Systems Typified by Buildings

2θ-Burster for Rhythm-Generating Circuits

We propose a minimalistic model called the 2θ-burster due to two slow phase characteristics of endogenous bursters, which when coupled in 3-cell neural circuits generate a multiplicity of stable rhythmic outcomes. This model offers the benefits of simplicity for designing larger neural networks along with an acute reduction in the computation cost. We developed a dynamical system framework for explaining the existence and robustness of phase-locked states in activity patterns produced by small rhythmic neural circuits. Several 3-cell configurations, from multifunctional to monostable, are considered to demonstrate the versatility of the proposed approach, allowing the network dynamics to be reduced to the examination of 2D Poincaré return maps for the phase lags between three constituent 2θ-bursters.

A data-driven non-linear assimilation framework with neural networks

Complex dynamical systems are an integral part of predictive analysis that model diverse phenomena. As these models improve, they become more complex and depend on an increasing number of model or driver inputs. Uncertainty plagues these inputs (initial conditions, boundary conditions, key model parameters, signal noise, etc.), thereby introducing errors into the forecast of the model and significantly degrading its predictability. In this paper, we develop a new data-driven assimilation framework for non-linear dynamical systems. In particular, we develop assimilation methods by building powerful surrogates that emulate the evolution of the model observables of the dynamical system to efficiently perform assimilation on the reduced model. There are two distinct advantages of this approach: (1) we build a surrogate that captures the model uncertainty propagation, and (2) we use entirely data-driven techniques. We employ the Bayesian framework for data assimilation and use neural networks to learn the evolution operator of the observables. We demonstrate on a chaotic test case that (a) uncertainty in initial condition is accurately captured by the surrogate, and (b) the reduced-order model can be effectively used to get estimates of the posterior.

A framework for parametric reduction in large-scale nonlinear dynamical systems

This manuscript presents a general parametric model order reduction (pMOR) framework for nonlinear dynamical systems. At first, a family of local nonlinear reduced order models (ROMs) is generated for different parametric space vectors using the nonlinear moment matching (NLMM) scheme along-with the discrete empirical interpolation method (DEIM). Then, the nonlinear reduced order model for any new parameter is obtained by interpolating the neighbouring reduced order models after projecting them onto a universal subspace. The advantage of such a scheme is that the parametric dependency is maintained in the reduced nonlinear models. Finally, we substantiate our observations by a suite of numerical tests.

Analysis of synchronization in load ensembles

While coordinated control of a large population of electric loads can provide important services to the electric grid, situations have been observed where control of load ensembles may lead to highly nonlinear behavior such as synchronization, sustained oscillations and bifurcations. Synchronization of thermostatically controlled loads (TCLs) is undesirable since it can lead to increased short-cycling, sudden changes in power demand and network voltage issues. In this paper, we investigate the synchronizing tendency of TCLs under control strategies where updates are broadcast periodically for coordinating TCLs. We study the problem using a hybrid dynamical systems framework to model both the continuous and discrete dynamics of load ensembles. Analysis of eigenmodes of the underlying discrete-time system provide insights into synchronizing tendencies and rate of convergence to the synchronized state. Simulations are provided to illustrate the theory.

Seasonal food webs with migrations: multi-season models reveal indirect species interactions in the Canadian Arctic tundra.

Models incorporating seasonality are necessary to fully assess the impact of global warming on Arctic communities. Seasonal migrations are a key component of Arctic food webs that still elude current theories predicting a single community equilibrium. We develop a multi-season model of predator-prey dynamics using a hybrid dynamical systems framework applied to a simplified tundra food web (lemming-fox-goose-owl). Hybrid systems models can accommodate multiple equilibria, which is a basic requirement for modelling food webs whose topology changes with season. We demonstrate that our model can generate multi-annual cycling in lemming dynamics, solely from a combined effect of seasonality and state-dependent behaviour. We compare our multi-season model to a static model of the predator-prey community dynamics and study the interactions between species. Interestingly, including seasonality reveals indirect interactions between migrants and residents not captured by the static model. Further, we find that the direction and magnitude of interactions between two species are not necessarily accurate using only summer time-series. Our study demonstrates the need for the development of multi-season models and provides the tools to analyse them. Integrating seasonality in food web modelling is a vital step to improve predictions about the impacts of climate change on ecosystem functioning. This article is part of the theme issue 'The changing Arctic Ocean: consequences for biological communities, biogeochemical processes and ecosystem functioning'.

Learning stabilizable nonlinear dynamics with contraction-based regularization

We propose a novel framework for learning stabilizable nonlinear dynamical systems for continuous control tasks in robotics. The key contribution is a control-theoretic regularizer for dynamics fitting rooted in the notion of stabilizability, a constraint which guarantees the existence of robust tracking controllers for arbitrary open-loop trajectories generated with the learned system. Leveraging tools from contraction theory and statistical learning in reproducing kernel Hilbert spaces, we formulate stabilizable dynamics learning as a functional optimization with a convex objective and bi-convex functional constraints. Under a mild structural assumption and relaxation of the functional constraints to sampling-based constraints, we derive the optimal solution with a modified representer theorem. Finally, we utilize random matrix feature approximations to reduce the dimensionality of the search parameters and formulate an iterative convex optimization algorithm that jointly fits the dynamics functions and searches for a certificate of stabilizability. We validate the proposed algorithm in simulation for a planar quadrotor, and on a quadrotor hardware testbed emulating planar dynamics. We verify, both in simulation and on hardware, significantly improved trajectory generation and tracking performance with the control-theoretic regularized model over models learned using traditional regression techniques, especially when learning from small supervised datasets. The results support the conjecture that the use of stabilizability constraints as a form of regularization can help prune the hypothesis space in a manner that is tailored to the downstream task of trajectory generation and feedback control. This produces models that are not only dramatically better conditioned, but also data efficient.

COVID-19 ORDER PARAMETERS AND ORDER PARAMETER TIME CONSTANTS OF ITALY AND CHINA: A MODELING APPROACH BASED ON SYNERGETICS

From a dynamical systems perspective, COVID-19 infectious disease emerges via an instability in human populations. Accordingly, the human population free of COVID-19 infected individuals is an unstable state and the dynamics away from that unstable state is a bifurcation. Recent research has determined COVID-19 relevant bifurcation parameters for various countries in terms of basic reproduction ratios. However, little is known about the relevant order parameters (synergetics) of COVID-19 bifurcations and the corresponding time constants. Those order parameters describe directions in compartment model spaces in which infection dynamics initially evolves. The corresponding time constants describe the speed of the dynamics along those directions. COVID-19 order parameters and their time constants are derived within a standard SEIR dynamical systems framework and determined explicitly for two published studies on COVID-19 trajectories in Italy and China. The results suggest the existence of certain relationships between order parameters, time constants, and reproduction ratios. However, the examples from Italy and China also suggest that COVID-19 order parameters and time constants in general depend on regional differences and the stage of the local COVID-19 epidemic under consideration. These findings may help to improve the forecasting of COVID-19 outbreaks in new hotspots around the world.

Hybrid Observer Concept for Sensor Fusion of Sporadic Measurements for Underwater Navigation

Accurate underwater navigation systems are required for closed-loop guidance and control of unmanned underwater vehicles (UUV). This paper proposes a sensor-based hybrid translational observer concept for underwater navigation using the hybrid dynamical systems framework, accounting for noisy, asynchronous and sporadic sensor measurements. Sensor measurements from an acoustic positioning system, a Doppler Velocity Log (DVL), an Inertial Measurement Unit (IMU) and a pressure gauge are used in the proposed observer. A method for filtering high-frequency noise is proposed, where the estimated states are obtained by taking a weighted discounted average of a finite number of previous measurements predicted forwards to the current time. The attitude of the vehicle is assumed known, and the acceleration measurements are assumed to be continuously available. Measurements of position, depth and linear velocity are assumed to be asynchronous and sporadically available, that is, they do not arrive at the same time, and their sampling rates are not constant. Uniform global asymptotic stability (UGAS) is established using Lyapunov theory for hybrid systems. Results from simulations are presented in order to demonstrate the performance of the proposed method.

Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems

Data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a data-based, probabilistic perspective that enables the quantification of predictive uncertainties. One of the outstanding problems has been the introduction of physical constraints in the probabilistic machine learning objectives. The primary utility of such constraints stems from the undisputed physical laws such as conservation of mass, energy etc. that they represent. Furthermore and apart from leading to physically realistic predictions, they can significantly reduce the requisite amount of training data which for high-dimensional, multiscale systems are expensive to obtain (Small Data regime). We formulate the coarse-graining process by employing a probabilistic state-space model and account for the aforementioned equality constraints as virtual observables in the associated densities. We demonstrate how deep neural nets in combination with probabilistic inference tools can be employed to identify the coarse-grained variables and their evolution model without ever needing to define a fine-to-coarse (restriction) projection and without needing time-derivatives of state variables.We advocate a sparse Bayesian learning perspective which avoids overfitting and reveals the most salient features in the CG evolution law. The formulation adopted enables the quantification of a crucial, and often neglected, component in the CG process, i.e. the predictive uncertainty due to information loss. Furthermore, it is capable of reconstructing the evolution of the full, fine-scale system and therefore the observables of interest need not be selected a priori. We demonstrate the efficacy of the proposed framework by applying it to systems of interacting particles and a series of images of a nonlinear pendulum. In both cases we identify the underlying coarse dynamics and can generate extrapolative predictions including the forming and propagation of a shock for the particle systems and a stable trajectory in the phase space for the pendulum.

A neuro-inspired general framework for the evolution of stochastic dynamical systems: Cellular automata, random Boolean networks and echo state networks towards criticality

Although deep learning has recently increased in popularity, it suffers from various problems including high computational complexity, energy greedy computation, and lack of scalability, to mention a few. In this paper, we investigate an alternative brain-inspired method for data analysis that circumvents the deep learning drawbacks by taking the actual dynamical behavior of biological neural networks into account. For this purpose, we develop a general framework for dynamical systems that can evolve and model a variety of substrates that possess computational capacity. Therefore, dynamical systems can be exploited in the reservoir computing paradigm, i.e., an untrained recurrent nonlinear network with a trained linear readout layer. Moreover, our general framework, called EvoDynamic, is based on an optimized deep neural network library. Hence, generalization and performance can be balanced. The EvoDynamic framework contains three kinds of dynamical systems already implemented, namely cellular automata, random Boolean networks, and echo state networks. The evolution of such systems towards a dynamical behavior, called criticality, is investigated because systems with such behavior may be better suited to do useful computation. The implemented dynamical systems are stochastic and their evolution with genetic algorithm mutates their update rules or network initialization. The obtained results are promising and demonstrate that criticality is achieved. In addition to the presented results, our framework can also be utilized to evolve the dynamical systems connectivity, update and learning rules to improve the quality of the reservoir used for solving computational tasks and physical substrate modeling.

Global attractor for damped forced nonlinear logarithmic Schrödinger equations

We consider here a damped forced nonlinear logarithmic Schrodinger equation in \begin{document}$ \mathbb{R}^N $\end{document} . We prove the existence of a global attractor in a suitable energy space. We complete this article with some open issues for nonlinear logarithmic Schrodinger equations in the framework of infinite-dimensional dynamical systems.

Stochastic Stability of the Classical Lorenz Flow Under Impulsive Type Forcing

We introduce a novel type of random perturbation for the classical Lorenz flow in order to better model phenomena slowly varying in time such as anthropogenic forcing in climatology and prove stochastic stability for the unperturbed flow. The perturbation acts on the system in an impulsive way, hence is not of diffusive type as those already discussed in Keller (Attractors and bifurcations of the stochastic Lorenz system Report 389, Institut für Dynamische Systeme, Universität Bremen, 1996), Kifer (Random Perturbations of Dynamical Systems. Birkhäuser, Basel, 1988), and Metzger (Commun. Math. Phys. 212, 277–296, 2000). Namely, given a cross-section \(\mathcal {M}\) for the unperturbed flow, each time the trajectory of the system crosses \(\mathcal {M}\) the phase velocity field is changed with a new one sampled at random from a suitable neighborhood of the unperturbed one. The resulting random evolution is therefore described by a piecewise deterministic Markov process. The proof of the stochastic stability for the umperturbed flow is then carried on working either in the framework of the Random Dynamical Systems or in that of semi-Markov processes.

Design and analysis of control strategies for pedestrian flows

Exploiting the full potential of pedestrian infrastructure is becoming critical in many environments which cannot be easily expanded to cope with the increasing pedestrian demand. This is particularly true for train stations as in many dense cities space is limited and expansion is difficult and very costly. In this paper, we investigate how to improve the level-of-service experienced by pedestrians by regulating and controlling their movements with a dynamic traffic management system. Although dynamic traffic management systems have been widely investigated in the last two decades to mitigate vehicular traffic congestion, little attention has been given in the literature to dynamic traffic management systems for pedestrian flows. The objective of this paper is to develop the concept of a dynamic traffic management system for pedestrian flows by building on the experience acquired from vehicular traffic management systems. We first propose a general framework for dynamic traffic management systems which takes into account the specificities of pedestrian traffic. The specificities of pedestrian traffic are discussed and emphasized. Then we illustrate the framework by using a control strategy designed for pedestrian flows that mitigates the issues induced by bidirectional flows. We show the effectiveness of this strategy by simulating a subpart of the train station in Lausanne (Switzerland). The results show a substantial improvement despite the relative simplicity of the method. These results emphasize the under-explored potential of pedestrian control and guidance when integrated into a dynamic pedestrian management system.

A numerical approach to hybrid nonlinear optimal control

This paper proposes a novel optimal control design framework for hybrid nonlinear dynamical systems involving an interacting combination of continuous-time and discrete-time dynamics. Two numerical algorithms are proposed to approximate the continuous-time and discrete-time portions of the hybrid Hamilton-Jacobi-Bellman (HJB) equation. Galerkin’s spectral method is utilised to approximate the value function involved in the continuous-time HJB equation, thereby computing the optimal control gains between impulsive events. Employing the spectral collocation method, the discrete-time HJB equation is then approximated to find the optimal control gain vector at impulsive instants. These two algorithms are ultimately combined to obtain the desired hybrid nonlinear optimal control law. Describing practical considerations for implementing the algorithms, some illustrative examples are presented to evaluate the functionality of the proposed hybrid nonlinear optimal controller.

Long-time behavior of the Cahn–Hilliard equation with dynamic boundary condition

We study the long-time behavior, within the framework of infinite dimensional dynamical systems, of the Cahn–Hilliard equation endowed with a new class of dynamic boundary conditions. The system under investigation was recently derived by Liu–Wu (Arch Ration Mech Anal 233:167–247, 2019) via an energetic variational approach such that it naturally fulfills physical properties like mass conservation, energy dissipation and force balance. For the system with regular potentials, we prove the existence of exponential attractors, which also yields the existence of a global attractor with finite fractal dimension. For the system with singular potentials, we obtain the existence of a global attractor in a suitable complete metric space.

Distributed algorithm for resource allocation problems under persistent attacks

This paper explores the ability of a distributed resource allocation algorithm under attacks to identify the optimal solution in the framework of hybrid dynamical systems. Since the attacks can result in time-varying topologies, differential inclusions are used to model attack modes and an average dwell-time automaton and time-ratio monitor are employed to constrain attacks. To address the effect induced by such attacks, a switched algorithm is considered. Then, to analyze stability, the switched algorithm is first modeled as a hybrid system and then a Lyapunov function is constructed to prove that the distributed algorithm under attacks is able to converge to the optimal solution of the resource allocation problem. Moreover, based on automaton and monitor constraints, an inequality condition is provided to guarantee exponential stability of the optimal solution. Finally, an example is presented to verify the theoretical result.

Further exploring the dynamicity, situatedness, and emergence of the self: The key role of context

Drawing on theoretical insights from a complex dynamic systems framework, this work explores the ways that learner selves, as they relate to learning and using languages, manifest across different contexts and timescales and emerge in interaction with various factors. First, a broad overview of dynamically-oriented L2 motivation research is provided before critically considering the need for research that aligns with conceptual advances made under the dynamic turn in SLA. In particular, this critical overview highlights a crucial need for more research employing dynamic methods capable of revealing how learner perceptions of self emerge in relation to their interlocutors and in interaction with external factors, including language ideologies that may uniquely characterize sociocultural contexts where target languages other than English are learned. The chapter concludes by discussing ways to implement dynamically oriented methodology that can provide much needed insights into the inherent dynamic, emergent, and contextually and socially embedded nature of learner selves.

Measuring Information Coupling between the Solar Wind and the Magnetosphere–Ionosphere System

The interaction between the solar wind and the Earth’s magnetosphere–ionosphere system is very complex, being essentially the result of the interplay between an external driver, the solar wind, and internal processes to the magnetosphere–ionosphere system. In this framework, modelling the Earth’s magnetosphere–ionosphere response to the changes of the solar wind conditions requires a correct identification of the causality relations between the different parameters/quantities used to monitor this coupling. Nowadays, in the framework of complex dynamical systems, both linear statistical tools and Granger causality models drastically fail to detect causal relationships between time series. Conversely, information theory-based concepts can provide powerful model-free statistical quantities capable of disentangling the complex nature of the causal relationships. In this work, we discuss how to deal with the problem of measuring causal information in the solar wind–magnetosphere–ionosphere system. We show that a time delay of about 30–60 min is found between solar wind and magnetospheric and ionospheric overall dynamics as monitored by geomagnetic indices, with a great information transfer observed between the z component of the interplanetary magnetic field and geomagnetic indices, while a lower transfer is found when other solar wind parameters are considered. This suggests that the best candidate for modelling the geomagnetic response to solar wind changes is the interplanetary magnetic field component . A discussion of the relevance of our results in the framework of Space Weather is also provided.

A data-driven phase and isostable reduced modeling framework for oscillatory dynamical systems.

Phase-amplitude reduction is of growing interest as a strategy for the reduction and analysis of oscillatory dynamical systems. Augmentation of the widely studied phase reduction with amplitude coordinates can be used to characterize transient behavior in directions transverse to a limit cycle to give a richer description of the dynamical behavior. Various definitions for amplitude coordinates have been suggested, but none are particularly well suited for implementation in experimental systems where output recordings are readily available but the underlying equations are typically unknown. In this work, a reduction framework is developed for inferring a phase-amplitude reduced model using only the observed model output from an arbitrarily high-dimensional system. This framework employs a proper orthogonal reduction strategy to identify important features of the transient decay of solutions to the limit cycle. These features are explicitly related to previously developed phase and isostable coordinates and used to define so-called data-driven phase and isostable coordinates that are valid in the entire basin of attraction of a limit cycle. The utility of this reduction strategy is illustrated in examples related to neural physiology and is used to implement an optimal control strategy that would otherwise be computationally intractable. The proposed data-driven phase and isostable coordinate system and associated reduced modeling framework represent a useful tool for the study of nonlinear dynamical systems in situations where the underlying dynamical equations are unknown and in particularly high-dimensional or complicated numerical systems for which standard phase-amplitude reduction techniques are not computationally feasible.