Renormalized oscillation theory for regular linear non-Hamiltonian systems

Peter Howard

doi:10.3934/cpaa.2022145

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2022, Volume 21, Issue 12: 4311-4345. Doi: 10.3934/cpaa.2022145

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Renormalized oscillation theory for regular linear non-Hamiltonian systems

Peter Howard

Department of Mathematics, Texas A&M University, College Station, TX 77843 USA

Received: May 2022

Early access: September 2022

Published: December 2022

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Abstract

In recent work, Baird et al. have generalized the definition of the Maslov index to paths of Grassmannian subspaces that are not necessarily contained in the Lagrangian Grassmannian [2]. Such an extension opens up the possibility of applications to non-Hamiltonian systems of ODE, and Baird and his collaborators have taken advantage of this observation to establish oscillation-type results for obtaining lower bounds on eigenvalue counts in this generalized setting. In the current analysis, the author shows that renormalized oscillation theory, appropriately defined in this generalized setting, can be applied in a natural way, and that it has the advantage, as in the traditional setting of linear Hamiltonian systems, of ensuring monotonicity of crossing points as the independent variable increases for a wide range of system/boundary-condition combinations. This seems to mark the first effort to extend the renormalized oscillation approach to the non-Hamiltonian setting.

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Mathematics Subject Classification: Primary: 34C10, 34L15; Secondary: 34L16.

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Figure 1. Local contributions to $ \mathfrak{m} $

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Figure 2. The Maslov Box

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Figure 3. Full Maslov Box and spectral curve for (6.15)

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Figure 4. Full Maslov Box and spectral curves for (6.18)–(6.30)

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Figure 5. Spectral curves for the system (6.18)–(6.31)

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