Slim Fractals: The Geometry of Doubly Transient Chaos

Traditional studies of chaos in conservative and driven dissipative systems have established a correspondence between sensitive dependence on initial conditions and fractal basin boundaries, but much less is known about the relation between geometry and dynamics in undriven dissipative systems. These systems can exhibit a prevalent form of complex dynamics, dubbed doubly transient chaos because not only typical trajectories but also the (otherwise invariant) chaotic saddles are transient. This property, along with a manifest lack of scale invariance, has hindered the study of the geometric properties of basin boundaries in these systems--most remarkably, the very question of whether they are fractal across all scales has yet to be answered. Here we derive a general dynamical condition that answers this question, which we use to demonstrate that the basin boundaries can indeed form a true fractal; in fact, they do so generically in a broad class of transiently chaotic undriven dissipative systems. Using physical examples, we demonstrate that the boundaries typically form a slim fractal, which we define as a set whose dimension at a given resolution decreases when the resolution is increased. To properly characterize such sets, we introduce the notion of equivalent dimension for quantifying their relation with sensitive dependence on initial conditions at all scales. We show that slim fractal boundaries can exhibit complex geometry even when they do not form a true fractal and fractal scaling is observed only above a certain length scale at each boundary point. Thus, our results reveal slim fractals as a geometrical hallmark of transient chaos in undriven dissipative systems.Comment: 13 pages, 9 figures, proof corrections implemente

Ballistic Orbits and Front Speed Enhancement for ABC Flows

We study the two main types of trajectories of the ABC flow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the non-integrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also find and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of flows, in particular, the question of linear (i.e., maximal possible) front speed enhancement rate for ABC flows.Comment: 39 pages, 26 figure

Rotation Vectors for Torus Maps by the Weighted Birkhoff Average

In this paper, we focus on distinguishing between the types of dynamical behavior that occur for typical one- and two-dimensional torus maps, in particular without the assumption of invertibility. We use three fast and accurate numerical methods: weighted Birkhoff averages, Farey trees, and resonance orders. The first of these allows us to distinguish between chaotic and regular orbits, as well as to calculate the frequency vectors for the regular case to high precision. The second method allows us to distinguish between the periodic and quasiperiodic orbits, and the third allows us to distinguish among the quasiperiodic orbits to determine the dimension the resulting attracting tori. We first consider the well-studied Arnold circle map, comparing our results to the universal power law of Jensen and Ecke. We next consider quasiperiodically forced circle maps, inspired by models introduced by Ding, Grebogi, and Ott. We use the Birkhoff average to distinguish between "strong" chaos (positive Lyapunov exponents) and "weak" chaos (strange nonchaotic attractors). Finally, we apply our methods to 2D torus maps, building on the work of Grebogi, Ott, and Yorke. We distinguish incommensurate, resonant, periodic, and chaotic orbits and accurately compute the proportions of each as the strength of the nonlinearity grows. We compute generalizations of Arnold tongues corresponding to resonances and to periodic orbits, and we show that chaos typically begins before the map becomes noninvertible. We show that the proportion of nonresonant orbits does not obey a universal power law like that seen in the 1D case.Comment: Keywords: Circle maps, Quasiperiodic forcing, Arnold tongues, Resonance, Birkhoff averages, Strange nonchaotic attractor

Methods and Measures for Analyzing Complex Street Networks and Urban Form

Complex systems have been widely studied by social and natural scientists in terms of their dynamics and their structure. Scholars of cities and urban planning have incorporated complexity theories from qualitative and quantitative perspectives. From a structural standpoint, the urban form may be characterized by the morphological complexity of its circulation networks - particularly their density, resilience, centrality, and connectedness. This dissertation unpacks theories of nonlinearity and complex systems, then develops a framework for assessing the complexity of urban form and street networks. It introduces a new tool, OSMnx, to collect street network and other urban form data for anywhere in the world, then analyze and visualize them. Finally, it presents a large empirical study of 27,000 street networks, examining their metric and topological complexity relevant to urban design, transportation research, and the human experience of the built environment.Comment: PhD thesis (2017), City and Regional Planning, UC Berkele