Browsing by Author "Grover, Piyush"

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    Finding and exploiting structure in complex systems via geometric and statistical methods
    Grover, Piyush (Virginia Tech, 2010-06-01)
    The dynamics of a complex system can be understood by analyzing the phase space structure of that system. We apply geometric and statistical techniques to two Hamiltonian systems to find and exploit structure in the phase space that helps us get qualitative and quantitative results about the phase space transport. While the structure can be revealed by the study of invariant manifolds of fixed points and periodic orbits in the first system, there do not exist any fixed points (and hence invariant manifolds) in the second system. The use of statistical (or measure theoretic) and topological methods reveals the phase space structure even in the absence of fixed points or stable and unstable invariant manifolds. The first problem we study is the four-body problem in the context of a spacecraft in the presence of a planet and two of its moons, where we exploit the phase space structure of the problem to devise an intelligent control strategy to achieve mission objectives. We use a family of analytically derived controlled Keplerian Maps in the Patched-Three-Body framework to design fuel efficient trajectories with realistic flight times. These maps approximate the dynamics of the Planar Circular Restricted Three Body Problem (PCR3BP) and we patch solutions in two different PCR3BPs to form the desired trajectories in the four body system. The second problem we study concerns phase space mixing in a two-dimensional time dependent Stokes flow system. Topological analysis of the braiding of periodic points has been recently used to find lower bounds on the complexity of the flow via the Thurston-Nielsen classification theorem (TNCT). We extend this framework by demonstrating that in a perturbed system with no apparent periodic points, the almost-invariant sets computed using a transfer operator approach are the natural objects on which to pin the TNCT.
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    Topological Chaos and Periodic Braiding of Almost-Cyclic Sets
    Stremler, Mark A.; Ross, Shane D.; Grover, Piyush; Kumar, Pankaj (American Physical Society, 2011-03-18)
    In certain (2 + 1)-dimensional dynamical systems, the braiding of periodic orbits provides a framework for analyzing chaos in the system through application of the Thurston-Nielsen classification theorem. Periodic orbits generated by the dynamics can behave as physical obstructions that "stir" the surrounding domain and serve as the basis for this topological analysis. We provide evidence that, even in the absence of periodic orbits, almost-cyclic regions identified using a transfer operator approach can reveal an underlying structure that enables topological analysis of chaos in the domain.
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    Topological chaos, braiding and bifurcation of almost-cyclic sets
    Grover, Piyush; Ross, Shane D.; Stremler, Mark A.; Kumar, Pankaj (American Institute of Physics, 2012-12-01)
    In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or " ghost rods" around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.