# A geometric approach to eigenanalysis: With applications to structural stability: Dissertation

Research output: ThesisDissertationMonograph

### Abstract

This thesis clarifies certain aspects of non-linear eigenanalysis with the help of differential geometry tools. Non-linear eigenproblems are generalizations of the linear eigenproblem that comprise higher order terms with respect to the bifurcation parameter. They arise from singularity investigations of dynamical system equilibrium sets: for a vast majority of physical problems, the eigenspace associated to the eigenproblem gives, precisely, the bifurcating direction at the singular point. The research question consists in assessing the sensitivity of the eigenspace with respect to a prescribed variation of internal parameters of the defining dynamical system. A possible source of variation comes from using the linear predictor instead of the original non-linear eigenproblem. One of the questions an answer shall be given to, consists in assessing the difference between two eigenspaces: one given by the linear predictor and the other given by the original non-linear eigenproblem. The tools used are taken from differential geometry. The parameter dependent Jacobian matrix evaluated at the primary equilibrium branch defines the non-linear eigenproblem. Geometrically, it can be interpreted as a smooth curve, called the Jacobian curve, that evolutes in the ambient matrix space. At a bifurcation point, the Jacobian matrix is rank deficient, by definition. Hence, the geometric interpretation of a nonlinear eigenproblem is the intersection of the Jacobian curve with the fixed rank matrix submanifold embedded in the ambient matrix space. The intersection point constitutes, therefore, a reference point on the fixed rank matrix manifold, around which one can draw a sphere of given radius computed with respect to the Riemannian distance function. The geometric interpretation of the eigenproblem enables the eigenspace associated with the eigenproblem to be considered as a smooth map, called the eigenspace map. It is defined on the fixed rank matrix manifold and has values in the projective space. Sensitivity analysis of the eigenspace map consists, then, in computing the ratio of two distances. The first distance, which is located in the denominator of the ratio, is the Riemannian distance on the fixed rank matrix manifold between the centre point and a point on the sphere. It is, by definition, equal to the radius of the sphere. The second distance, located in the numerator of the ratio, is the distance, in the projective space, between the corresponding two image points by the eigenspace map. Numerical examples are given at the end of the work to illustrate the use of eigenspace sensitivity analysis in the context of structural stability.
Original language English Doctor Degree Aalto University Paavola, Juha, Supervisor, External personReijo, Kouhia, Advisor, External person 22 Feb 2016 Espoo Aalto University 978-952-60-6629-5 978-952-60-6630-1 Published - 2016 G4 Doctoral dissertation (monograph)

### Fingerprint

Eigenproblem
Structural Stability
Geometric Approach
Eigenspace
Jacobian matrix
Differential Geometry
Projective Space
Curve
Sensitivity Analysis
Predictors
Dynamical system
Intersection
Numerator
Reference Point
Bifurcation Point
Denominator
Distance Function
Singular Point
Submanifolds

### Keywords

• dynamical systems
• equilibrium sets
• singular points
• tangent cones
• bifurcating directions
• non-linear eigenproblem
• eigenspace map
• fixed rank matrix manifold
• spheres on fixed rank matrix manifold
• local sensitivity analysis

### Cite this

@phdthesis{2194bc923a624af0a2eb614be6d82670,
title = "A geometric approach to eigenanalysis: With applications to structural stability: Dissertation",
abstract = "This thesis clarifies certain aspects of non-linear eigenanalysis with the help of differential geometry tools. Non-linear eigenproblems are generalizations of the linear eigenproblem that comprise higher order terms with respect to the bifurcation parameter. They arise from singularity investigations of dynamical system equilibrium sets: for a vast majority of physical problems, the eigenspace associated to the eigenproblem gives, precisely, the bifurcating direction at the singular point. The research question consists in assessing the sensitivity of the eigenspace with respect to a prescribed variation of internal parameters of the defining dynamical system. A possible source of variation comes from using the linear predictor instead of the original non-linear eigenproblem. One of the questions an answer shall be given to, consists in assessing the difference between two eigenspaces: one given by the linear predictor and the other given by the original non-linear eigenproblem. The tools used are taken from differential geometry. The parameter dependent Jacobian matrix evaluated at the primary equilibrium branch defines the non-linear eigenproblem. Geometrically, it can be interpreted as a smooth curve, called the Jacobian curve, that evolutes in the ambient matrix space. At a bifurcation point, the Jacobian matrix is rank deficient, by definition. Hence, the geometric interpretation of a nonlinear eigenproblem is the intersection of the Jacobian curve with the fixed rank matrix submanifold embedded in the ambient matrix space. The intersection point constitutes, therefore, a reference point on the fixed rank matrix manifold, around which one can draw a sphere of given radius computed with respect to the Riemannian distance function. The geometric interpretation of the eigenproblem enables the eigenspace associated with the eigenproblem to be considered as a smooth map, called the eigenspace map. It is defined on the fixed rank matrix manifold and has values in the projective space. Sensitivity analysis of the eigenspace map consists, then, in computing the ratio of two distances. The first distance, which is located in the denominator of the ratio, is the Riemannian distance on the fixed rank matrix manifold between the centre point and a point on the sphere. It is, by definition, equal to the radius of the sphere. The second distance, located in the numerator of the ratio, is the distance, in the projective space, between the corresponding two image points by the eigenspace map. Numerical examples are given at the end of the work to illustrate the use of eigenspace sensitivity analysis in the context of structural stability.",
keywords = "dynamical systems, equilibrium sets, singular points, tangent cones, bifurcating directions, non-linear eigenproblem, eigenspace map, fixed rank matrix manifold, spheres on fixed rank matrix manifold, local sensitivity analysis",
author = "Alexis Fedoroff",
note = "BA2125",
year = "2016",
language = "English",
isbn = "978-952-60-6629-5",
series = "Aalto University Publication Series: Doctoral Dissertations",
publisher = "Aalto University",
number = "14",
school = "Aalto University",

}

Espoo : Aalto University, 2016. 216 p.

Research output: ThesisDissertationMonograph

TY - THES

T1 - A geometric approach to eigenanalysis

T2 - With applications to structural stability: Dissertation

AU - Fedoroff, Alexis

N1 - BA2125

PY - 2016

Y1 - 2016

N2 - This thesis clarifies certain aspects of non-linear eigenanalysis with the help of differential geometry tools. Non-linear eigenproblems are generalizations of the linear eigenproblem that comprise higher order terms with respect to the bifurcation parameter. They arise from singularity investigations of dynamical system equilibrium sets: for a vast majority of physical problems, the eigenspace associated to the eigenproblem gives, precisely, the bifurcating direction at the singular point. The research question consists in assessing the sensitivity of the eigenspace with respect to a prescribed variation of internal parameters of the defining dynamical system. A possible source of variation comes from using the linear predictor instead of the original non-linear eigenproblem. One of the questions an answer shall be given to, consists in assessing the difference between two eigenspaces: one given by the linear predictor and the other given by the original non-linear eigenproblem. The tools used are taken from differential geometry. The parameter dependent Jacobian matrix evaluated at the primary equilibrium branch defines the non-linear eigenproblem. Geometrically, it can be interpreted as a smooth curve, called the Jacobian curve, that evolutes in the ambient matrix space. At a bifurcation point, the Jacobian matrix is rank deficient, by definition. Hence, the geometric interpretation of a nonlinear eigenproblem is the intersection of the Jacobian curve with the fixed rank matrix submanifold embedded in the ambient matrix space. The intersection point constitutes, therefore, a reference point on the fixed rank matrix manifold, around which one can draw a sphere of given radius computed with respect to the Riemannian distance function. The geometric interpretation of the eigenproblem enables the eigenspace associated with the eigenproblem to be considered as a smooth map, called the eigenspace map. It is defined on the fixed rank matrix manifold and has values in the projective space. Sensitivity analysis of the eigenspace map consists, then, in computing the ratio of two distances. The first distance, which is located in the denominator of the ratio, is the Riemannian distance on the fixed rank matrix manifold between the centre point and a point on the sphere. It is, by definition, equal to the radius of the sphere. The second distance, located in the numerator of the ratio, is the distance, in the projective space, between the corresponding two image points by the eigenspace map. Numerical examples are given at the end of the work to illustrate the use of eigenspace sensitivity analysis in the context of structural stability.

AB - This thesis clarifies certain aspects of non-linear eigenanalysis with the help of differential geometry tools. Non-linear eigenproblems are generalizations of the linear eigenproblem that comprise higher order terms with respect to the bifurcation parameter. They arise from singularity investigations of dynamical system equilibrium sets: for a vast majority of physical problems, the eigenspace associated to the eigenproblem gives, precisely, the bifurcating direction at the singular point. The research question consists in assessing the sensitivity of the eigenspace with respect to a prescribed variation of internal parameters of the defining dynamical system. A possible source of variation comes from using the linear predictor instead of the original non-linear eigenproblem. One of the questions an answer shall be given to, consists in assessing the difference between two eigenspaces: one given by the linear predictor and the other given by the original non-linear eigenproblem. The tools used are taken from differential geometry. The parameter dependent Jacobian matrix evaluated at the primary equilibrium branch defines the non-linear eigenproblem. Geometrically, it can be interpreted as a smooth curve, called the Jacobian curve, that evolutes in the ambient matrix space. At a bifurcation point, the Jacobian matrix is rank deficient, by definition. Hence, the geometric interpretation of a nonlinear eigenproblem is the intersection of the Jacobian curve with the fixed rank matrix submanifold embedded in the ambient matrix space. The intersection point constitutes, therefore, a reference point on the fixed rank matrix manifold, around which one can draw a sphere of given radius computed with respect to the Riemannian distance function. The geometric interpretation of the eigenproblem enables the eigenspace associated with the eigenproblem to be considered as a smooth map, called the eigenspace map. It is defined on the fixed rank matrix manifold and has values in the projective space. Sensitivity analysis of the eigenspace map consists, then, in computing the ratio of two distances. The first distance, which is located in the denominator of the ratio, is the Riemannian distance on the fixed rank matrix manifold between the centre point and a point on the sphere. It is, by definition, equal to the radius of the sphere. The second distance, located in the numerator of the ratio, is the distance, in the projective space, between the corresponding two image points by the eigenspace map. Numerical examples are given at the end of the work to illustrate the use of eigenspace sensitivity analysis in the context of structural stability.

KW - dynamical systems

KW - equilibrium sets

KW - singular points

KW - tangent cones

KW - bifurcating directions

KW - non-linear eigenproblem

KW - eigenspace map

KW - fixed rank matrix manifold

KW - spheres on fixed rank matrix manifold

KW - local sensitivity analysis

M3 - Dissertation

SN - 978-952-60-6629-5

T3 - Aalto University Publication Series: Doctoral Dissertations

PB - Aalto University

CY - Espoo

ER -