Browsing by Subject "dynamical systems"

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    Applications Of Functional-Analytic Methods In Nonlocal And Dynamical System Problems
    (2020-05) Tao, Tianyu
    This paper is concerned with functional analytic techniques in problems related to dynamical systems and contains two parts. In the first part we show bifurcation of localized spike solutions from spatially constant states in systems of nonlocally coupled equations in the whole space. The main assumptions are a generic bifurcation of saddle-node or transcritical type for spatially constant profiles, and a symmetry and second moment condition on the convolution kernel. Rather than relying on center manifolds methods, we pursue a more direct approach, deriving leading order asymptotics and Newton corrections for error terms. We are able to extend well-known results for spots, spikes, and fronts in locally coupled systems on the real line, and for radially symmetric profiles in higher space dimensions. In the second part, we revisit the classical problem of determining the asymptotic expansion of the solution near the passage of a fold point in a singularly perturbed system, where the theory of normally hyperbolic invariant manifold cannot be directly applied. While the standard remedy is the blow-up method first demonstrated by Krupa and Szmolyan, we will show how one can use a functional analytic approach to achieve the same goal.
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    Attractors and Attracting Neighborhoods for Multiflows
    (2019-05) Negaard-Paper, Shannon
    We already know a great deal about dynamical systems with uniqueness in forward time. Indeed, flows, semiflows, and maps (both invertible and not) have been studied at length. A view that has proven particularly fruitful is topological: consider invariant sets (attractors, repellers, periodic orbits, etc.) as topological objects, and the connecting sets between them form gradient like flows. In the case of systems with uniqueness in forward time, an attractor in one system is related to nearby attractors in a family of other, "close enough" systems. One way of seeing that connection is through the Conley decomposition (and the Conley index) [2], [13]. This approach requires focusing on isolated invariant sets - that is, invariant sets with isolating neighborhoods. If there is an invariant set I, which has an isolating neighborhood N, we say that I is the invariant set associated to N, and N is an isolating neighborhood associated to I. When the invariant set in question is an attractor or a repeller, then the isolating neighborhood is called an attracting neighborhood or a repelling neighborhood, respectively. A more specialized case may be called an attractor block or a repeller block. This approach was expanded to discrete time systems which lack uniqueness in forward time, using relations, in [7] and [11]. Relations do not rely on uniqueness in forward time, but the graph of any map is a relation; thus they serve to generalize maps. Some of this is reviewed in the next few sections. In addition, I expanded on work done in [7] to show that in compact metric spaces, attractors for closed relations continue (see Section 6.1). On the continuous time side, more work needs to be done. This paper is a step toward a more systematic approach for continuous time systems which lack uniqueness in forward time. This work applies to Filippov systems [4] and in control theory [12]. In the following pages, we establish a tool (multiflows) for discussing the continuous time case and develop a framework for understanding attractors (and therefore stability) in these systems. A crucial part of this work was establishing attractor / attracting neighborhood pairs, which happens in Section 5.5.
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    Relative Equilibria of Dumbbells Orbiting in a Planar Newtonian Gravitational System
    (2022-04) Morey, Jodin
    In the cosmos, any two bodies share a gravitational attraction. When in proximity to one another in empty space, their motions can be modeled by Newtonian gravity. Newton found their orbits when the two bodies are infinitely small, the so-called two-body problem. The general situation in which the bodies have varying shapes and sizes, called the full two-body problem, remains open. We find relative equilibria (RE) and their stability for an approximation of the full two-body problem, where each body is restricted to a plane and consists of two point masses connected by a massless rod, a dumbbell. In particular, we find symmetric RE in which the bodies are arranged colinearly, perpendicularly, or trapezoidally. When the masses of the dumbbells are pairwise equal, we find asymmetric RE bifurcating from the symmetric RE. And while we find that only the colinear RE have nonlinear/energetic stability (for sufficiently large radii), we also find that the perpendicular and trapezoid configurations have radial intervals of linear stability. We also provide a geometric restriction on the location of RE for a dumbbell body and any number of planar rigid bodies in planar orbit (an extension of the Conley Perpendicular Bisector Theorem).