{"id":{"repo_id":"njit","oai_identifier":"oai:digitalcommons.njit.edu:dissertations-1709"},"canonical_url":"https://search.dev.ndltd.org/etd/njit/oai:digitalcommons.njit.edu:dissertations-1709","repository":{"repo_id":"njit","name":"NJIT","base_url":"https://digitalcommons.njit.edu/do/oai/"},"display":{"title":"Pitchfork bifurcations of invariant manifolds","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 C^1 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.","abstract_html":"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 C^1 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.","abstract_has_math":false,"creators":["Champanerkar, Jyoti"],"institution":null,"degree_name":"Doctor of Philosophy in Mathematical Sciences - (Ph.D.)","degree_level":null,"degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":["Denis L. Blackmore","Robert M. Miura","Lee D. Mosher"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2004,"date_issued":"2004-08-31T07:00:00Z","date_published":"2004-08-31T07:00:00Z","updated_at":"2026-07-24T03:22:58Z","subjects":["Dynamical systems","Pitchfork bifurcation","Bifurcation","Invariant manifolds","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.njit.edu/dissertations/654","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Denis L. Blackmore","Robert M. Miura","Lee D. 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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 C^1 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."]},{"key":"dc:title","label":"Title","values":["Pitchfork bifurcations of invariant manifolds"]}]}],"canonical_facts":{"dc:contributor":["Denis L. Blackmore","Robert M. Miura","Lee D. Mosher"],"dc:creator":["Champanerkar, Jyoti"],"dc:description.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 C^1 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."],"dc:identifier":["https://digitalcommons.njit.edu/dissertations/654"],"dc:subject":["Dynamical systems","Pitchfork bifurcation","Bifurcation","Invariant manifolds","Mathematics"],"dc:title":["Pitchfork bifurcations of invariant manifolds"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_name":["Doctor of Philosophy in Mathematical Sciences - (Ph.D.)"]},"updated_at":"2026-07-24T03:22:58Z"}