Pitchfork bifurcations of invariant manifolds

Author

Jyoti Champanerkar, New Jersey Institute of Technology

Document Type

Dissertation

Date of Award

Summer 8-31-2004

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Denis L. Blackmore

Second Advisor

Robert M. Miura

Third Advisor

Lee D. Mosher

Fourth Advisor

Demetrius T. Papageorgiou

Fifth Advisor

Amitabha Koshal Bose

Abstract

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 R^m 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.

Recommended Citation

Champanerkar, Jyoti, "Pitchfork bifurcations of invariant manifolds" (2004). Dissertations. 654.
https://digitalcommons.njit.edu/dissertations/654