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VARIATIONAL EMBEDDED SOLITONS, AND TRAVELING WAVETRAINS GENERATED BY GENERALIZED HOPF BIFURCATIONS, IN SOME NLPDE SYSTEMS

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Date Issued:
2011
Abstract/Description:
In this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the family of the trial functions). Thus, the residual is calculated, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that only the parameter regimes for the existence of solitary waves had previously been analyzed for the microstructure PDE considered here, the results obtained here are both new and timely. Second, we consider generalized and degenerate Hopf bifurcations in three different models: i. a predator-prey model with general predator death rate and prey birth rate terms, ii. a laser-diode system, and iii. traveling-wave solutions of twospecies predator-prey/reaction-diusion equations with arbitrary nonlinear/reaction terms. For speci c choices of the nonlinear terms, the quasi-periodic orbit in the post-bifurcation regime is constructed for each system using the method of multiple scales, and its stability is analyzed via the corresponding normal form obtained by reducing the system down to the center manifold. The resulting predictions for the post-bifurcation dynamics provide an organizing framework for the variety of possible behaviors. These predictions are veri ed and supplemented by numerical simulations, including the computation of power spectra, autocorrelation functions, and fractal dimensions as appropriate for the periodic and quasiperiodic attractors, attractors at in nity, as well as bounded chaotic attractors obtained in various cases. The dynamics obtained in the three systems is contrasted and explained on the basis of the bifurcations occurring in each. For instance, while the two predator-prey models yield a variety of behaviors in the post-bifurcation regime, the laser-diode evinces extremely stable quasiperiodic solutions over a wide range of parameters, which is very desirable for robust operation of the system in oscillator mode.
Title: VARIATIONAL EMBEDDED SOLITONS, AND TRAVELING WAVETRAINS GENERATED BY GENERALIZED HOPF BIFURCATIONS, IN SOME NLPDE SYSTEMS.
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Name(s): Smith, Todd, Author
Choudhury, Roy, Committee Chair
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2011
Publisher: University of Central Florida
Language(s): English
Abstract/Description: In this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the family of the trial functions). Thus, the residual is calculated, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that only the parameter regimes for the existence of solitary waves had previously been analyzed for the microstructure PDE considered here, the results obtained here are both new and timely. Second, we consider generalized and degenerate Hopf bifurcations in three different models: i. a predator-prey model with general predator death rate and prey birth rate terms, ii. a laser-diode system, and iii. traveling-wave solutions of twospecies predator-prey/reaction-diusion equations with arbitrary nonlinear/reaction terms. For speci c choices of the nonlinear terms, the quasi-periodic orbit in the post-bifurcation regime is constructed for each system using the method of multiple scales, and its stability is analyzed via the corresponding normal form obtained by reducing the system down to the center manifold. The resulting predictions for the post-bifurcation dynamics provide an organizing framework for the variety of possible behaviors. These predictions are veri ed and supplemented by numerical simulations, including the computation of power spectra, autocorrelation functions, and fractal dimensions as appropriate for the periodic and quasiperiodic attractors, attractors at in nity, as well as bounded chaotic attractors obtained in various cases. The dynamics obtained in the three systems is contrasted and explained on the basis of the bifurcations occurring in each. For instance, while the two predator-prey models yield a variety of behaviors in the post-bifurcation regime, the laser-diode evinces extremely stable quasiperiodic solutions over a wide range of parameters, which is very desirable for robust operation of the system in oscillator mode.
Identifier: CFE0003634 (IID), ucf:48887 (fedora)
Note(s): 2011-05-01
Ph.D.
Sciences, Department of Mathematics
Masters
This record was generated from author submitted information.
Subject(s): Hopf bifurcations
nonlinear dynamics
variational solitons
multiple scale expansions
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0003634
Restrictions on Access: public
Host Institution: UCF

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