Floquet Theory on Banach Space

<p>In this thesis we study Floquet theory on a Banach space. We are concerned about the linear differential equation of the form: <em>y'</em>(<em>t</em>) = <em>A</em>(<em>t</em>)<em>y</em>(<em>t</em>), where t ∈ R, <em>y</em>(<em>t</em>) is a function with values in a Banach space <em>X</em>, and <em>A</em>(<em>t</em>) are linear, bounded operators on <em>X</em>. If the system is periodic, meaning <em>A</em>(<em>t</em>+<em>ω</em>) = <em>A</em>(<em>t</em>) for some period <em>ω</em>, then it is called a Floquet system. We will investigate the existence and uniqueness of the periodic solution, as well as the stability of a Floquet system. This thesis will be presented in five main chapters. In the first chapter, we review the system of linear differential equations on Rⁿ: <em>y'</em>= <em>A</em>(<em>t</em>)<em>y</em>(<em>t</em>) + <em>f</em>(<em>t</em>), where <em>A</em>(<em>t</em>) is an <em>n</em> x<em> n</em> matrix-valued function, <em>y</em>(<em>t</em>) are vectors and<em> f</em>(<em>t</em>) are functions with values in Rⁿ. We establish the general form of the all solutions by using the fundamental matrix, consisting of <em>n</em> independent solutions. Also, we can find the stability of solutions directly by using the eigenvalues of A. In the second chapter, we study the Floquet system on Rⁿ, where <em>A</em>(<em>t</em>+<em>ω</em>) = <em>A</em>(<em>t</em>). We establish the Floquet theorem, in which we show that the Floquet system is closely related to a linear system with constant coefficients, so the properties of those systems can be applied. In particular, we can answer the questions about the stability of the Floquet system. Then we generalize the Floquet theory to a linear system on Banach spaces. So we introduce to the readers Banach spaces in chapter three and the linear operators on Banach spaces in chapter four. In the fifth chapter we study the asymptotic properties of solutions of the system: <em>y'</em>(<em>t</em>) = <em>A</em>(<em>t</em>)<em>y</em>(<em>t</em>), where <em>y</em>(<em>t)</em> is a function with values in a Banach space <em>X</em> and<em> A</em>(<em>t</em>) are linear, bounded operators on<em> X</em> with <em>A</em> (<em>t</em>+<em>ω</em>) =<em> A</em>(<em>t</em>). For that system, we can show the evolution family <em>U</em>(<em>t</em>,<em>s</em>) representing the solutions is periodic, i.e. <em>U</em>(<em>t</em>+<em>ω</em>, <em>s</em>+<em>ω</em>) = <em>U</em>(<em>t</em>,<em>s</em>). Next we study the monodromy of the system <em>V</em> := <em>U</em>(<em>ω</em>,0). We point out that the spectrum set of <em>V</em> actually determines the stability of the Floquet system. Moreover, we show that the existence and uniqueness of the periodic solution of the nonhomogeneous equation in a Floquet system is equivalent to the fact that 1 belongs to the resolvent set of <em>V</em>.</p>

Name thesis:degree_name

Master of Science

Discipline thesis:degree_discipline

Department of Mathematics

Year

2013

Author dc:creator

• Albasrawi, Fatimah Hassan

Contributors dc:contributor

• Lan Nguyen (Director), John Spraker, Nezam Iraniparast

• Floquet Theory
• Banach Spaces
• Linear Operators
• Differential Equations
• Applied Mathematics
• Mathematics
• Physical Sciences and Mathematics

Repository record dc:identifier

https://digitalcommons.wku.edu/theses/1234

OAI identifier oai:identifier

oai:digitalcommons.wku.edu:theses-2237