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Title: Cascading mode interactions in discrete dynamical systems
Author: Mir, Himat
ISNI:       0000 0001 3410 7672
Awarding Body: University of Surrey
Current Institution: University of Surrey
Date of Award: 2007
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Mode interactions arise when two types of bifurcation come closer and meet at a codimension 2 point and cross each other on the branch. It happens when multiple eigenvalues involving more than one parameter pass through their critical values together. A cascade happens when these mode interactions go through, in this case, successive period doubling. In this thesis we study cascades of codimension 2 mode interactions in discrete dynamical systems. The underlying focus is on understanding mode interaction cascades in iterated maps when their solutions go through successive period doubling. We consider a large system which has a mode interaction involving a period doubling bifurcation and a symmetry breaking bifurcation. If the system is more then two dimensional we perform a Centre Manifold reduction to get a two dimensional reduced form. We then reformulate the period doubling bifurcation as a symmetry breaking bifurcation and use Liapunov-Schmidt reduction to get two bifurcation equations which describe the solutions in a neighbourhood of the mode interaction. We give a detailed description of solutions of the bifurcation equations. We calculate the values of the parameters for primary and secondary bifurcations from the trivial solution and primary branches respectively and derive conditions for their sub(super) criticality. We also consider whether there is any tertiary Hopf bifurcation on the mixed mode solutions and derive conditions for it to exist. We study the conditions for a mode interaction cascade for a class of iterated maps which satisfy a few basic conditions. We list all possible mode interactions and carefully study which of these mode interactions go through a cascade and categorise them. At higher periods we study the relationship among the bifurcation equations of different points in the cycle and derive equations to find all the sets of bifurcation equations from one. We apply these techniques to different examples. We study the limiting behaviour of the cascade using renormalisation theory first studying a sequence of one dimensional functions and then a two dimensional problem which includes the one dimensional functions. We derive the parameter scaling for each function and then for the two dimensional problem. We used a second parameter mu in the mode interaction cascade. We calculate the limiting values muinfinity and universal constant for mu similar to Feigenbaum point and Feigenbaum Number.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available