arXiv subjects

Amenda Chow

Publications and source records attributed to Amenda Chow.

6 recordsLinked to original sources

arXiv2022

Hysteresis and Stabillity

Hysteresis can be defined from a dynamical systems perspective with respect to equilibrium points. Consequently, hysteresis naturally lends itself as a topic to illustrate and extend concepts in a dynamical systems course. A number of examples exhibiting hysteresis, most motivated by applications, are presented. Although the examples can be used to construct student exercises, specific questions are listed in an appendix. A brief extension on hysteresis in partial differential equations is also included.

math.DS↗

arXiv2017

Asymptotic Stability of the Landau-Lifshitz Equation

The Landau--Lifshitz equation describes the behaviour of magnetic domains in ferromagnetic structures. Recently such structures have been found to be favourable for storing digital data. Stability of magnetic domains is important for this. Consequently, asymptotic stability of the equilibrium points in the Landau--Lifshitz equation are established. A suitable Lyapunov function is presented.

math.AP↗

arXiv2016

Landau-Lifshitz Equation with Affine Control

The Landau-Lifshitz equation is a coupled set of nonlinear partial differential equations that describes the dynamics of magnetization in a ferromagnet. This equation has an infinite number of stable equilibria. Steering the system from one equilibrium to another is a problem of both theoretical and practical interest. Since the objective is to steer between equilibria, approaches based on linearization are not appropriate. It is proven that affine proportional control can be used to steer the system from an arbitrary initial state, including an equilibrium point, to a specified equilibrium point. The second point becomes a globally asymptotically stable equilibrium of the controlled system. The control also removes hysteresis from the Landau-Lifshitz equation. These results are illustrated with simulations.

math.AP↗

arXiv2015

Control of the Landau-Lifshitz Equation

The Landau--Lifshitz equation describes the dynamics of magnetization inside a ferromagnet. This equation is nonlinear and has an infinite number of stable equilibria. It is desirable to control the system from one equilibrium to another. A control that moves the system from an arbitrary initial state, including an equilibrium point, to a specified equilibrium is presented. It is proven that the second point is an asymptotically stable equilibrium of the controlled system. The results are illustrated with some simulations.

math.OC↗

arXiv2015

Hysteresis in the Linearized Landau-Lifshitz Equation

The Landau-Lifshitz equation describes the behaviour of magnetization inside a ferromagnetic object. It is known that the Landau-Lifshitz equation has an infinite number of stable equilibrium points. The existence of multiple stable equilibria is closely related to hysteresis. This is a phenomenon that is often characterized by a looping behaviour; however, the existence of a loop is not sufficient to identify hysteretic systems, but is defined more precisely as the presence of looping as the frequency of the input goes to zero. We describe these two approaches to identification of hysteresis and demonstrate that both the linear and nonlinear Landau-Lifshitz equations exhibit hysteresis. The presence of hysteresis in the linear Landau-Lifshitz equation, as well as in a simpler system also described here, indicates that nonlinearity is not necessary for hysteresis to exist.

math.DS↗

arXiv2015

Linearized stability analysis of nonlinear partial differential equations

Lyapunov's indirect method is an attractive method for analyzing stability of non-linear systems since only the stability of the corresponding linearized system needs to be determined. Unfortunately, the proof for finite-dimensional systems does not generalize to infinite-dimensions. In this paper a unified approach to Lyapunov's indirect method for infinite-dimensional system is described. It is shown how existing sufficient conditions fit this framework and a new sufficient condition is presented.

math.AP↗