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East Tennessee State University

1.
Frazier, William.
Application of *Symplectic* *Integration* on a Dynamical System.

Degree: MS, Mathematical Sciences, 2017, East Tennessee State University

URL: https://dc.etsu.edu/etd/3213

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic integrators, and often, these techniques are developed for well-understood Hamiltonian systems such as Hill’s lunar equation. In this presentation, we explore how well symplectic techniques developed for well-understood systems (specifically, Hill’s Lunar equation) address discretization errors in MD systems which fail for one or more reasons.

Subjects/Keywords: Lie algebra; Lie group; symplectic integration; molecular dynamics; Algebra; Dynamic Systems; Non-linear Dynamics; Numerical Analysis and Computation; Ordinary Differential Equations and Applied Dynamics

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Frazier, W. (2017). Application of Symplectic Integration on a Dynamical System. (Masters Thesis). East Tennessee State University. Retrieved from https://dc.etsu.edu/etd/3213

Chicago Manual of Style (16^{th} Edition):

Frazier, William. “Application of Symplectic Integration on a Dynamical System.” 2017. Masters Thesis, East Tennessee State University. Accessed October 31, 2020. https://dc.etsu.edu/etd/3213.

MLA Handbook (7^{th} Edition):

Frazier, William. “Application of Symplectic Integration on a Dynamical System.” 2017. Web. 31 Oct 2020.

Vancouver:

Frazier W. Application of Symplectic Integration on a Dynamical System. [Internet] [Masters thesis]. East Tennessee State University; 2017. [cited 2020 Oct 31]. Available from: https://dc.etsu.edu/etd/3213.

Council of Science Editors:

Frazier W. Application of Symplectic Integration on a Dynamical System. [Masters Thesis]. East Tennessee State University; 2017. Available from: https://dc.etsu.edu/etd/3213

2. Tuwankotta, J.M. Higher-Order Resonances in Dynamical Systems.

Degree: 2002, University Utrecht

URL: https://dspace.library.uu.nl/handle/1874/877 ; URN:NBN:NL:UI:10-1874-877 ; URN:NBN:NL:UI:10-1874-877 ; https://dspace.library.uu.nl/handle/1874/877

This thesis is a collection of studies on higher-order resonances in an important class of dynamical systems called coupled oscillators systems. After giving an overview of the mathematical background, we start in Chapter 1 by presenting a study on resonances in two degrees of freedom, autonomous, Hamiltonian systems. Due to the presence of a symmetry condition on one of the degrees of freedom, we show that some of the resonances vanish as lower order resonances. After giving a sharp estimate of the resonance domain, we investigate this order change of resonance in a rather general potential problem with discrete symmetry and consider as an example the H´enon-Heiles family of Hamiltonians. We also study a classical example of a mechanical system with symmetry, the elastic pendulum, which leads to a natural hierarchy of resonances with the 4 : 1-resonance as the most prominent after the 2 : 1-resonance and which explains why the 3 : 1- resonance is neglected.
In chapter 2 of this thesis we study the performance of a symplectic numerical integrator based on the splitting method. This method is applied to a subtle problem i.e. higher order resonance of the elastic pendulum. In order to numerically study the phase space of the elastic pendulum at higher order resonance, a numerical integrator which preserves qualitative features after long integration times is needed. We show by means of an example that our symplectic method offers a relatively cheap and accurate numerical integrator.
In chapter 3 we study two degree of freedom Hamiltonian systems and applications to nonlinear wave equations. Near the origin, we assume that near the linearized system has purely imaginary eigenvalues: ±i[omega]1 and ±i[omega]2 with 0 <[omega]2/[omega]1«1 or w2/w1»1,
which is interpreted as a perturbation of a problem with double zero eigenvalues. Using the averaging method, we compute the normal form and show that the dynamics differs from the usual one for Hamiltonian systems at higher order resonances. Under certain conditions, the normal form is degenerate which forces us to normalize to higher degree. The asymptotic character of the normal form and the corresponding invariant tori is validated using KAM theorem. This analysis is then applied to widely separated mode-interaction in a family of nonlinear wave equations containing various degeneracies.
In chapter 4 we present an analysis of a system of coupled-oscillators. We make two assumptions for our system. The first assumption is that the frequencies of the characteristic oscillations are widely separated, and the second is that the nonlinear part of the vector field preserves the distance to the origin. Using the first assumption, we prove that the reduced normal form of our system, exhibits an invariant manifold which, exists for all values of the parameters and cannot be perturbed away by including higher order terms in the normal form. Using the second assumption, we view the normal form as an energy-preserving three-dimensional system which is linearly perturbed.…

Subjects/Keywords: Hamiltonian mechanics; higher-order resonance; normal forms; symmetry; elastic pendulum; symplectic numerical integration; widely separated frequencies; singular perturbation; bifurcation; coupled oscillators

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Tuwankotta, J. M. (2002). Higher-Order Resonances in Dynamical Systems. (Doctoral Dissertation). University Utrecht. Retrieved from https://dspace.library.uu.nl/handle/1874/877 ; URN:NBN:NL:UI:10-1874-877 ; URN:NBN:NL:UI:10-1874-877 ; https://dspace.library.uu.nl/handle/1874/877

Chicago Manual of Style (16^{th} Edition):

Tuwankotta, J M. “Higher-Order Resonances in Dynamical Systems.” 2002. Doctoral Dissertation, University Utrecht. Accessed October 31, 2020. https://dspace.library.uu.nl/handle/1874/877 ; URN:NBN:NL:UI:10-1874-877 ; URN:NBN:NL:UI:10-1874-877 ; https://dspace.library.uu.nl/handle/1874/877.

MLA Handbook (7^{th} Edition):

Tuwankotta, J M. “Higher-Order Resonances in Dynamical Systems.” 2002. Web. 31 Oct 2020.

Vancouver:

Tuwankotta JM. Higher-Order Resonances in Dynamical Systems. [Internet] [Doctoral dissertation]. University Utrecht; 2002. [cited 2020 Oct 31]. Available from: https://dspace.library.uu.nl/handle/1874/877 ; URN:NBN:NL:UI:10-1874-877 ; URN:NBN:NL:UI:10-1874-877 ; https://dspace.library.uu.nl/handle/1874/877.

Council of Science Editors:

Tuwankotta JM. Higher-Order Resonances in Dynamical Systems. [Doctoral Dissertation]. University Utrecht; 2002. Available from: https://dspace.library.uu.nl/handle/1874/877 ; URN:NBN:NL:UI:10-1874-877 ; URN:NBN:NL:UI:10-1874-877 ; https://dspace.library.uu.nl/handle/1874/877

Universiteit Utrecht

3. Tuwankotta, J.M. Higher-Order Resonances in Dynamical Systems.

Degree: 2002, Universiteit Utrecht

URL: http://dspace.library.uu.nl:8080/handle/1874/877

This thesis is a collection of studies on higher-order resonances in an important class of dynamical systems called coupled oscillators systems. After giving an overview of the mathematical background, we start in Chapter 1 by presenting a study on resonances in two degrees of freedom, autonomous, Hamiltonian systems. Due to the presence of a symmetry condition on one of the degrees of freedom, we show that some of the resonances vanish as lower order resonances. After giving a sharp estimate of the resonance domain, we investigate this order change of resonance in a rather general potential problem with discrete symmetry and consider as an example the H´enon-Heiles family of Hamiltonians. We also study a classical example of a mechanical system with symmetry, the elastic pendulum, which leads to a natural hierarchy of resonances with the 4 : 1-resonance as the most prominent after the 2 : 1-resonance and which explains why the 3 : 1- resonance is neglected.
In chapter 2 of this thesis we study the performance of a symplectic numerical integrator based on the splitting method. This method is applied to a subtle problem i.e. higher order resonance of the elastic pendulum. In order to numerically study the phase space of the elastic pendulum at higher order resonance, a numerical integrator which preserves qualitative features after long integration times is needed. We show by means of an example that our symplectic method offers a relatively cheap and accurate numerical integrator.
In chapter 3 we study two degree of freedom Hamiltonian systems and applications to nonlinear wave equations. Near the origin, we assume that near the linearized system has purely imaginary eigenvalues: ±i[omega]1 and ±i[omega]2 with 0 <[omega]2/[omega]1«1 or w2/w1»1,
which is interpreted as a perturbation of a problem with double zero eigenvalues. Using the averaging method, we compute the normal form and show that the dynamics differs from the usual one for Hamiltonian systems at higher order resonances. Under certain conditions, the normal form is degenerate which forces us to normalize to higher degree. The asymptotic character of the normal form and the corresponding invariant tori is validated using KAM theorem. This analysis is then applied to widely separated mode-interaction in a family of nonlinear wave equations containing various degeneracies.
In chapter 4 we present an analysis of a system of coupled-oscillators. We make two assumptions for our system. The first assumption is that the frequencies of the characteristic oscillations are widely separated, and the second is that the nonlinear part of the vector field preserves the distance to the origin. Using the first assumption, we prove that the reduced normal form of our system, exhibits an invariant manifold which, exists for all values of the parameters and cannot be perturbed away by including higher order terms in the normal form. Using the second assumption, we view the normal form as an energy-preserving three-dimensional system which is linearly perturbed.…

Subjects/Keywords: Wiskunde en Informatica; Hamiltonian mechanics; higher-order resonance; normal forms; symmetry; elastic pendulum; symplectic numerical integration; widely separated frequencies; singular perturbation; bifurcation; coupled oscillators

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Tuwankotta, J. M. (2002). Higher-Order Resonances in Dynamical Systems. (Doctoral Dissertation). Universiteit Utrecht. Retrieved from http://dspace.library.uu.nl:8080/handle/1874/877

Chicago Manual of Style (16^{th} Edition):

Tuwankotta, J M. “Higher-Order Resonances in Dynamical Systems.” 2002. Doctoral Dissertation, Universiteit Utrecht. Accessed October 31, 2020. http://dspace.library.uu.nl:8080/handle/1874/877.

MLA Handbook (7^{th} Edition):

Tuwankotta, J M. “Higher-Order Resonances in Dynamical Systems.” 2002. Web. 31 Oct 2020.

Vancouver:

Tuwankotta JM. Higher-Order Resonances in Dynamical Systems. [Internet] [Doctoral dissertation]. Universiteit Utrecht; 2002. [cited 2020 Oct 31]. Available from: http://dspace.library.uu.nl:8080/handle/1874/877.

Council of Science Editors:

Tuwankotta JM. Higher-Order Resonances in Dynamical Systems. [Doctoral Dissertation]. Universiteit Utrecht; 2002. Available from: http://dspace.library.uu.nl:8080/handle/1874/877