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Local Orthogonal Rectification: A New Tool for Geometric Phase Space Analysis

Letson, Benjamin (2020) Local Orthogonal Rectification: A New Tool for Geometric Phase Space Analysis. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Local orthogonal rectification (LOR) provides a natural and useful geometric frame for analyzing dynamics relative to manifolds embedded in flows. LOR can be applied to any embedded base manifold in a system of ODEs of arbitrary dimension to establish a corresponding system of LOR equations for analyzing dynamics within the LOR frame. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. Additionally, we illustrate the utility of LOR by showing a wide range of application domains.
In the plane, we use the LOR approach to derive a novel definition for rivers, long-recognized but poorly understood trajectories that locally attract other orbits yet need not be related to invariant manifolds or other familiar phase space structures, and to identify rivers within several example systems.
In higher dimensions, we apply LOR to identify periodic orbits and study the transient dynamics nearby. In the LOR method, %in $\R^n$ for any $n$,
the standard approach of finding periodic orbits by solving for fixed points of a Poincar\'{e} return map is replaced by the solution of a boundary value problem with fixed endpoints, and the computation provides information about the stability of the identified orbit. We detail the general method and derive theory to show that once a periodic orbit has been identified with LOR, the LOR coordinate system allows us to characterize the stability of the periodic orbit, to continue the orbit with respect to system parameters, to identify invariant manifolds attendant to the periodic orbit, and to compute the asymptotic phase associated with points in a neighborhood of the periodic orbit in a novel way.
Finally, we generalize the definition of rivers beyond planar systems, and demonstrate a fundamental connection between canard solutions in two-timescale systems and generalized rivers. We will again use a blow-up transformation on the LOR equations, which provides a useful decomposition for studying trajectories' behavior relative to the embedded base curve.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Letson, Benjaminbgl14@pitt.edubgl140000-0001-9343-3312
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairRubin, Jonathanjonrubin@pitt.edujonrubin0000-0002-1513-1551
Committee MemberErmentrout, Bardbard@pitt.edubard0000-0002-5854-0654
Committee MemberManfredi, Juanmanfredi@pitt.edumanfredi
Committee MemberDeBlois, Jasonjdeblois@pitt.edujdeblois
Committee MemberVo,
Date: 16 January 2020
Date Type: Publication
Defense Date: 18 November 2019
Approval Date: 16 January 2020
Submission Date: 5 December 2019
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 172
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Dynamical systems, differential geometry, coordinate transformation, approximation.
Date Deposited: 16 Jan 2020 18:56
Last Modified: 16 Jan 2020 18:56

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