Physics
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Item Energy absorption and diffusion in chaotic systems under rapid periodic driving(2022) Hodson, Wade Daniel; Jarzynski, Christopher; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this thesis, we study energy absorption in classical chaotic, ergodic systems subject to rapid periodic driving, and in related systems. Under a rapid periodic drive, we find that the energy evolution of chaotic systems appears as a random walk in energy space, which can be described as a process of energy diffusion. We characterize this process, and show that it generally predicts three stages of energy evolution: Initial relaxation to a prethermal state, followed by slow evolution of the system’s energy probability distribution in accordance with a Fokker-Planck equation, followed by either unbounded energy absorption or relaxation to an infinite temperature state. We then study the energy diffusion model in detail in driven billiard systems specifically; in particular, we obtain numerical results which corroborate the energy diffusion description for a specific choice of billiard. This is followed by an analysis of energy diffusion in one-dimensional oscillator systems subject to weak, correlated noise. Finally, we begin to investigate energy absorption in periodically driven quantum chaotic systems, i.e., quantum systems with a classical chaotic analogue. We invoke tools from Floquet theory and random matrix theory to investigate whether the classical energy diffusion framework can be applied to quantum systems, and under what conditions. We conclude with a discussion of potential models for energy absorption in quantum chaotic systems, and with an overview of open questions and directions for future work.Item Extensions of the Kuramoto model: from spiking neurons to swarming drones(2020) Chandra, Sarthak; Girvan, Michelle; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The Kuramoto model (KM) was initially proposed by Yoshiki Kuramoto in 1975 to model the dynamics of large populations of weakly coupled phase oscillators. Since then, the KM has proved to be a paradigmatic model, demonstrating dynamics that are complex enough to model a wide variety of nontrivial phenomena while remaining simple enough for detailed mathematical analyses. However, as a result of the mathematical simplifications in the construction of the model, the utility of the KM is somewhat restricted in its usual form. In this thesis we discuss extensions of the KM that allow it to be utilized in a wide variety of physical and biological problems. First, we discuss an extension of the KM that describes the dynamics of theta neurons, i.e., quadratic-integrate-and-fire neurons. In particular, we study networks of such neurons and derive a mean-field description of the collective neuronal dynamics and the effects of different network topologies on these dynamics. This mean-field description is achieved via an analytic dimensionality reduction of the network dynamics that allows for an efficient characterization of the system attractors and their dependence not only on the degree distribution but also on the degree correlations. Then, motivated by applications of the KM to the alignment of members in a two-dimensional swarm, we construct a Generalized Kuramoto Model (GKM) that extends the KM to arbitrary dimensions. Like the KM, the GKM in even dimensions continues to demonstrate a transition to coherence at a positive critical coupling strength. However, in odd dimensions the transition to coherence occurs discontinuously as the coupling strength is increased through 0. In contrast to the unique stable incoherent equilibrium for the KM, we find that for even dimensions larger than 2 the GKM displays a continuum of different possible pretransition incoherent equilibria, each with distinct stability properties, leading to a novel phenomenon, which we call `Instability-Mediated Resetting.' To aid the analysis of such systems, we construct an exact dimensionality reduction technique with applicability to not only the GKM, but also other similar systems with high-dimensional agents beyond the GKM.Item DYNAMICS OF LARGE SYSTEMS OF NONLINEARLY EVOLVING UNITS(2017) Lu, Zhixin; Ott, Edward; Chemical Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The dynamics of large systems of many nonlinearly evolving units is a general research area that has great importance for many areas in science and technology, including biology, computation by artificial neural networks, statistical mechanics, flocking in animal groups, the dynamics of coupled neurons in the brain, and many others. While universal principles and techniques are largely lacking in this broad area of research, there is still one particular phenomenon that seems to be broadly applicable. In particular, this is the idea of emergence, by which is meant macroscopic behaviors that “emerge” from a large system of many “smaller or simpler entities such that ... large entities” [i.e., macroscopic behaviors] arise which “exhibit properties the smaller/simpler entities do not exhibit.” [1]. In this thesis we investigate mechanisms and manifestations of emergence in four dynamical systems consisting many nonlinearly evolving units. These four systems are as follows. (a) We first study the motion of a large ensemble of many noninteracting particles in a slowly changing Hamiltonian system that undergoes a separatrix crossing. In such systems, we find that separatrix-crossing induces a counterintuitive effect. Specifically, numerical simulation of two sets of densely sprinkled initial conditions on two energy curves appears to suggest that the two energy curves, one originally enclosing the other, seemingly interchange their positions. This, however, is topologically forbidden. We resolve this paradox by introducing a numerical simulation method we call “robust” and study its consequences. (b) We next study the collective dynamics of oscillatory pacemaker neurons in Suprachiasmatic Nucleus (SCN), which, through synchrony, govern the circadian rhythm of mammals. We start from a high-dimensional description of the many coupled oscillatory neuronal units within the SCN. This description is based on a forced Kuramoto model. We then reduce the system dimensionality by using the Ott Antonsen Ansatz and obtain a low-dimensional macroscopic description. Using this reduced macroscopic system, we explain the east-west asymmetry of jet-lag recovery and discus the consequences of our findings. (c) Thirdly, we study neuron firing in integrate-and-fire neural networks. We build a discrete-state/discrete-time model with both excitatory and inhibitory neurons and find a phase transition between avalanching dynamics and ceaseless firing dynamics. Power-law firing avalanche size/duration distributions are observed at critical parameter values. Furthermore, in this critical regime we find the same power law exponents as those observed from experiments and previous, more restricted, simulation studies. We also employ a mean-field method and show that inhibitory neurons in this system promote robustness of the criticality (i.e., an enhanced range of system parameter where power-law avalanche statistics applies). (d) Lastly, we study the dynamics of “reservoir computing networks” (RCN’s), which is a recurrent neural network (RNN) scheme for machine learning. The ad- vantage of RCN’s over traditional RNN’s is that the training is done only on the output layer, usually via a simple least-square method. We show that RCN’s are very effective for inferring unmeasured state variables of dynamical systems whose system state is only partially measured. Using the examples of the Lorenz system and the Rossler system we demonstrate the potential of an RCN to perform as an universal model-free “observer”.Item Complex dynamics of a microwave time-delayed feedback loop(2013) Dao, Hien Thi Le; Murphy, Thomas E; Rodgers, John C; Chemical Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The subject of this thesis is deterministic behaviors generated from a microwave time-delayed feedback loop. Time-delayed feedback systems are especially interesting because of the rich variety of dynamical behaviors that they can support. While ordinary differential equations must be of at least third-order to produce chaos, even a simple first-order nonlinear delay differential equation can produce higher-dimensional chaotic dynamics. The system reported in the thesis is governed by a very simple nonlinear delay differential equation. The experimental implementation uses both microwave and digital components to achieve the nonlinearity and time-delayed feedback, respectively. When a sinusoidal nonlinearity is incorporated, the dynamical behaviors range from fixed-point to periodic to chaotic depending on the feedback strength. The microwave frequency modulated chaotic signal generated by this system offers advantages in range and velocity sensing applications. When the sinusoidal nonlinearity is replaced by a binary nonlinearity, the system exhibits a complex periodic attractor with no fixed-point solution. Although there are many classic electronic circuits that produce chaotic behavior, microwave sources of chaos are especially relevant in communication and sensing applications where the signal must be transmitted between locations. The system also can exhibit random walk behavior when being operated in a higher feedback strength regime. Depending on the feedback strength values, the random behaviors can have properties of a regular or fractional Brownian motion. By unidirectional coupling two systems in the baseband, envelope synchronization between two deterministic Brownian motions can be achieved.Item Synchronization of Chaotic Optoelectronic Oscillators: Adaptive Techniques and the Design of Optimal Networks(2011) Ravoori, Bhargava; Roy, Rajarshi; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Synchronization in networks of chaotic systems is an interesting phenomenon with potential applications to sensing, parameter estimation and communications. Synchronization of chaos, in addition to being influenced by the dynamical nature of the constituent network units, is critically dependent upon the maintenance of a proper coupling between the systems. In practical situations, however, synchronization in chaotic networks is negatively affected by perturbations in the coupling channels. Here, using a fiber-optic network of chaotic optoelectronic oscillators, we experimentally demonstrate an adaptive algorithm that maintains global network synchrony even when the coupling strengths are unknown and time-varying. Our adaptive algorithm operates by generating real-time estimates of the coupling perturbations which are subsequently used to suitably adjust internal node parameters in order to compensate for external disturbances. In our work, we also examine the influence of network configuration on synchronization. Through measurements of the convergence rate to synchronization in networks of optoelectronic systems, we show that having more network links does not necessarily imply faster or better synchronization as is generally thought. We find that the convergence rate is maximized for certain network configurations, called optimal networks, which are identified based on the eigenvalues of the coupling matrix. Further, based on an analysis of the eigenvectors of the coupling matrix, we introduce a classification system that categorizes networks according to their sensitivity to coupling perturbations as sensitive and nonsensitive configurations. Though our experiments are performed on networks consisting of specific nonlinear optoelectronic oscillators, the theoretical basis of our studies is general and consequently many of our results are applicable to networks of arbitrary dynamical oscillators.