# Browsing by Subject "Dynamical systems"

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Item Cognition in dynamical systems(2017-12-07) Hall, John Wendell; Sentis, Luis; Fernandez, Benito R.; Djurdjanovic, Dragan; Stinchcombe, Maxwell; Miikkulainen, RistoShow more Cognition is the process of knowing. As carried out by a dynamical system, it is the process by which the system absorbs information into its state. A complex network of agents cognizes knowledge about its environment, internal dynamics and initial state by forming emergent, macro-level patterns. Such patterns require each agent to find its place while partially aware of the whole pattern. Such partial awareness can be achieved by separating the system dynamics into two parts by timescale: the propagation dynamics and the pattern dynamics. The fast propagation dynamics describe the spread of signals across the network. If they converge to a fixed point for any quasi-static state of the slow pattern dynamics, that fixed point represents an aggregate of macro-level information. On longer timescales, agents coordinate via positive feedback to form patterns, which are defined using closed walks in the graph of agents. Patterns can be coherent, in that every part of the pattern depends on every other part for context. Coherent patterns are acausal, in that a) they cannot be predicted and b) no part of the stored knowledge can be mapped to any part of the pattern, or vice versa. A cognitive network's knowledge is encoded or embodied by the selection of patterns which emerge. The theory of cognition summarized here can model autocatalytic reaction-diffusion systems, artificial neural networks, market economies and ant colony optimization, among many other real and virtual systems. This theory suggests a new understanding of complexity as a lattice of contexts rather than a single measure.Show more Item Gardening the landscape and bushwhacking through the swampland : exploring the consequences of quantum gravity for cosmic inflation(2022-02-07) Rosati, Robert James; Paban, Sonia; Distler, Jacques; Kilic, Can; Boylan-Kolchin, MichaelShow more This dissertation consists of five chapters. The first broadly and briefly orients the reader through an introduction to inflationary cosmology, and why we might expect multi-field inflation to take place. The next four chapters correspond to distinct lines of research conducted during my time as a graduate student. Chapter two is based on work conducted with my advisor Sonia Paban, studying the landscape of possible multi-field inflationary models through a random-matrix generated potential [1]. Chapter three is based on work with Diederik Roest and Perseas Christodoulidis, studying universality and prior dependence in multi-field inflation [2]. Chapters four and five are based on work with Sonia Paban and Vikas Aragam, studying inflation in potentials compatible with quantum gravity and rapidly turning trajectories [3, 4].Show more Item Iterative milestoning(2016-12) Bello Rivas, Juan Manuel; Elber, Ron; Engquist, Bjorn; Makarov, Dmitrii E; Rodin, Gregory J; Zariphopoulou, ThaleiaShow more Computer simulation of matter using Molecular Dynamics (MD) is a staple in the field of Molecular Biophysics. MD yields results suitable for comparison with laboratory experiments and, in addition, it serves as a computational microscope by providing insight into a variety of molecular mechanisms. However, some of the most interesting problems pertaining to the investigation of biomolecules remain outside of the scope of MD due to the long time scales at which they occur. Milestoning is a method that addresses the long time simulation of biomolecular systems without giving up the fully-atomistic spatial resolution necessary to understand biological processes such as signalling and biochemical reactions. The method works by partitioning the phase space of the system into regions whose boundaries are called milestones. The dynamics of the system restricted to the milestones defines a stochastic process whose transition probabilities and exit times can be efficiently computed by numerical simulation. By calculating the transition probabilities and exit times of this process, we can obtain global thermodynamic and kinetic properties of the original system such as its stationary probability, free energy, and reaction rates. The calculation of these properties would be unfeasible for many systems of interest if we were to approach the problem by plain MD simulation. The success of milestoning computations relies on certain modeling assumptions. In this dissertation we introduce an iterative variant of the Milestoning method that relaxes the assumptions required by the original method and can be applied in the non-equilibrium setting. The new method works by iteratively approximating the transition probabilities and exit times until convergence is attained. In addition to a detailed description of the method, we give various pedagogical examples, showcase its practical applications to molecular systems, and provide an alternative formulation of the method in terms of boundary value problems.Show more Item Learning for autonomy in the wild : theory, algorithms, and practice(2023-08) Djeumou, Franck; Topcu, Ufuk; Chinchali, Sandeep; Fridovich-Keil, David; Zhang, Amy; Putot, Sylvie; Lennon, CraigShow more How can autonomous systems learn to operate in the wild, i.e., complex, dynamic, and uncertain real-world environments? Despite recent and significant breakthroughs in artificial intelligence, there is still a tremendous gap between its current capabilities and what we need to do to develop systems that can autonomously operate in the wild. We aim to bridge this gap by addressing a few key challenges of learning in the wild. These challenges include learning with extremely scarce amounts of data, learning safely from a single and ongoing trial, learning to generalize to unseen situations, and learning with uncertainty-aware and explainability considerations for trustworthy human-robot interactions. We take an opinionated approach to address these challenges and argue that data are never the only source of knowledge available during training, and modern learning techniques should not treat them as such. Instead, we demonstrate that merging modern learning techniques' efficiency at extracting patterns from data with existing knowledge on how the world works is key for autonomous systems to achieve learning in the wild. This existing knowledge on how the world works may stem from structural knowledge such as fundamental principles of physics, qualitative expert knowledge such as design or mechanical constraints, or contextual knowledge such as formal specifications on the underlying task. Thus, by leveraging prior knowledge into learning through formal techniques, we propose data-driven modeling and control approaches that enable autonomous systems to operate even under severely limited amounts of data, such as streaming data from a single and ongoing trial. We additionally demonstrate that the data-driven approaches generalize beyond the training regime, improve explainability over traditional black-box models, and exhibit principled uncertainty awareness. Specifically, we focus on theoretical analyses that quantify the benefits of exploiting prior knowledge as inductive bias in terms of data efficiency, safety, computational requirements, and optimality of learning. We derive these theoretical analyses through novel ideas at the intersection of control, learning, and formal methods. Based on the theoretical insights, we develop practical and computationally efficient algorithms, some of which have provable performance, real-time, and safety guarantees. To validate the effectiveness of our algorithms, we conduct experiments in high-fidelity robotics and flight simulators, as well as on real-world hardware such as a Toyota Supra car and a custom-built hexacopter. Remarkably, when applied in real-world settings, our algorithms provide high performance for control tasks that push the system beyond the limits of the prior knowledge and data coverage, despite being trained on only a handful of system trajectories or a few minutes worth of data.Show more Item Non-unique topological sofic entropy and a von Neumann algebra multiplicative ergodic theorem(2020-05-07) Lin, Yuqing, Ph. D.; Bowen, Lewis, 1972-; Kerr, David; Neeman, Joseph; Sadun, LorenzoShow more A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. All previously known examples showing this dependence rely on degenerate behavior. In joint work with D. Airey and L.Bowen an explicit example is exhibited of a mixing subshift of finite type with two different positive sofic entropies. The example is inspired by statistical physics literature on 2-coloringsof random hyper-graphs. Also, in joint work with L. Bowen and B. Hayes, the classical Multiplicative Ergodic Theorem (MET) of Oseledets is generalized to cocycles taking values in a type II von Neumann algebra. This appears to be the first MET involving operators with continuous spectrum.Show more Item On the role of invariant objects in applications of dynamical systems(2012-05) Blazevski, Daniel, 1984-; Llave, Rafael de la; Chen, Thomas (Ph. D. in mechanical engineering and Ph. D. in mathematical physics); Koch, Hans; Morrison, Phil; Ocampo, Cesar; Pavlovic, Natasa; Vasseur, AlexisShow more In this dissertation, we demonstrate the importance of invariant objects in many areas of applied research. The areas of application we consider are chemistry, celestial mechanics and aerospace engineering, plasma physics, and coupled map lattices. In the context of chemical reactions, stable and unstable manifolds of fixed points separate regions of phase space that lead to a certain outcome of the reaction. We study how these regions change under the influence of exposing the molecules to a laser. In celestial mechanics and aerospace engineering, we compute periodic orbits and their stable and unstable manifolds for a object of negligible mass (e.g. a satellite or spacecraft) under the presence of Jupiter and two of its moons, Europa and Ganymede. The periodic orbits serve as convenient spot to place a satellite for observation purposes, and computing their stable and unstable manifolds have been used in constructing low-energy transfers between the two moons. In plasma physics, an important and practical problem is to study barriers for heat transport in magnetically confined plasma undergoing fusion. We compute barriers for which heat cannot pass through. However, such barriers break down and lead to robust partial barriers. In this latter case, heat can flow across the barrier, but at a very slow rate. Finally, infinite dimensional coupled map lattice systems are considered in a wide variety of areas, most notably in statistical mechanics, neuroscience, and in the discretization of PDEs. We assume that the interaction amont the lattice sites decays with the distance of the sites, and assume the existence of an invariant whiskered torus that is localized near a collection of lattice sites. We prove that the torus has invariant stable and unstable manifolds that are also localized near the torus. This is an important step in understanding the global dynamics of such systems and opens the door to new possible results, most notably studying the problem of energy transfer between the sites.Show more Item Toward seamless multiscale computations(2013-05) Lee, Yoonsang, active 2013; Engquist, Björn, 1945-Show more Efficient and robust numerical simulation of multiscale problems encountered in science and engineering is a formidable challenge. Full resolution of multiscale problems using direct numerical simulations requires enormous amounts of computational time and resources. This thesis develops seamless multiscale methods for ordinary and partial differential equations under the framework of the heterogeneous multiscale method (HMM). The first part of the thesis is devoted to the development of seamless multiscale integrators for ordinary differential equations. The first method, which we call backward-forward HMM (BFHMM), uses splitting and on-the-fly filtering techniques to capture slow variables of highly oscillatory problems without any a priori information. The second method, denoted by variable step size HMM (VSHMM), as the name implies, uses variable mesoscopic step sizes for the unperturbed equation, which gives computational efficiency and higher accuracy. VSHMM can be applied to dissipative problems as well as highly oscillatory problems, while BFHMM has some difficulties when applied to the dissipative case. The effect of variable time stepping is analyzed and the two methods are tested numerically. Multi-spatial problems and numerical methods are discussed in the second part. Seamless heterogeneous multiscale methods (SHMM) for partial differential equations, especially the parabolic case without scale separation are proposed. SHMM is developed first for the multiscale heat equation with a continuum of scales in the diffusion coefficient. This seamless method uses a hierarchy of local grids to capture effects from each scale and uses filtering in Fourier space to impose an artificial scale gap. SHMM is then applied to advection enhanced diffusion problems under incompressible turbulent velocity fields.Show more