Articles via Databases
Articles via Journals
Online Catalog
Research & Information Literacy
Interlibrary loan
Theses & Dissertations
About / Contact Us
Littman Architecture Library

The New Jersey Institute of Technology's
Electronic Theses & Dissertations Project

Title: Pitchfork bifurcations of invariant manifolds
Author: Champanerkar, Jyoti
View Online: njit-etd2004-122
(ix, 60 pages ~ 3.7 MB pdf)
Department: Department of Mathematical Sciences
Degree: Doctor of Philosophy
Program: Mathematical Sciences
Document Type: Dissertation
Advisory Committee: Blackmore, Denis L. (Committee chair)
Miura, Robert M. (Committee member)
Mosher, Lee D. (Committee member)
Papageorgiou, Demetrius T. (Committee member)
Bose, Amitabha Koshal (Committee member)
Date: 2004-08
Keywords: Dynamical systems
Pitchfork bifurcation
Invariant manifolds
Availability: Unrestricted

In a parameter dependent, dynamical system, when the qualitative structure of the solutions changes due to a small change in the parameter, the system is said to have undergone a bifurcation. Bifurcations have been classified on the basis of the topological properties of fixed points and invariant manifolds of dynamical systems. A pitchfork bifurcation in R is said to have occurred when a stable fixed point becomes unstable and two new stable fixed points, separated by the unstable fixed point come into existence.

In this thesis, a pitchfork bifurcation of an (in- 1)-dimensional invariant submani-fold of a dynamical system in Rm is defined analogous to that in R. Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is shown under the stated hypotheses. The dynamical system is assumed to be a class C1 diffeomorphism or vector field in rtm. The existence of locally attracting invariant manifolds M+ and M- after the bifurcation has taken place, is proved by constructing a diffeomorphism of the unstable manifold M. Techniques used for proving the above mentioned result, involve differential topology and analysis and are adapted from Hartman [18] and Hirsch [19].

The main theorem of the thesis is illustrated by means of a canonical example and applied to a 2-dimensional discrete version of the Lotka-Volterra model, describing dynamics of a predator-prey population. The Lotka-Volterra model is slightly modified to depend on a continuously varying parameter. Significance of a pitchfork bifurcation in the Lotka-Volterra model is discussed with respect to population dynamics. Lastly, implications of the theorem are dicussed from a mathematical point of view.

If you have any questions please contact the ETD Team,

Browse ETDs by Adviser
Browse ETDs by Author
Browse ETDs by Program
Browse ETDs by Title
Browse ETDs by Year
ETD Policies & Procedures
ETD home

Request a Scan

NJIT's ETD project was given an ACRL/NJ Technology Innovation Honorable Mention Award in spring 2003