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 Title
 HOPF BIFURCATION ANALYSIS OF CHAOTIC CHEMICAL REACTOR MODEL.
 Creator

Mandragona, Daniel, Choudhury, Roy, University of Central Florida
 Abstract / Description

Bifurcations in Huang's chaotic chemical reactor system leading from simple dynamics into chaotic regimes are considered. Following the linear stability analysis, the periodic orbit resulting from a Hopf bifurcation of any of the six fixed points is constructed analytically by the method of multiple scales across successively slower time scales, and its stability is then determined by the resulting final secularity condition. Furthermore, we run numerical simulations of our chemical reactor...
Show moreBifurcations in Huang's chaotic chemical reactor system leading from simple dynamics into chaotic regimes are considered. Following the linear stability analysis, the periodic orbit resulting from a Hopf bifurcation of any of the six fixed points is constructed analytically by the method of multiple scales across successively slower time scales, and its stability is then determined by the resulting final secularity condition. Furthermore, we run numerical simulations of our chemical reactor at a particular fixed point of interest, alongside a set of parameter values that forces our system to undergo Hopf bifurcation. These numerical simulations then verify our analysis of the normal form.
Show less  Date Issued
 2018
 Identifier
 CFH2000342, ucf:45831
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFH2000342
 Title
 SOLITARY WAVE FAMILIES IN TWO NONINTEGRABLE MODELS USING REVERSIBLE SYSTEMS THEORY.
 Creator

Leto, Jonathan, Choudhury, S. Roy, University of Central Florida
 Abstract / Description

In this thesis, we apply a recently developed technique to comprehensively categorize all possible families of solitary wave solutions in two models of topical interest. The models considered are: a) the Generalized PochhammerChree Equations, which govern the propagation of longitudinal waves in elastic rods, and b) a generalized microstructure PDE. Limited analytic results exist for the occurrence of one family of solitary wave solutions for each of these equations. Since, as mentioned...
Show moreIn this thesis, we apply a recently developed technique to comprehensively categorize all possible families of solitary wave solutions in two models of topical interest. The models considered are: a) the Generalized PochhammerChree Equations, which govern the propagation of longitudinal waves in elastic rods, and b) a generalized microstructure PDE. Limited analytic results exist for the occurrence of one family of solitary wave solutions for each of these equations. Since, as mentioned above, solitary wave solutions often play a central role in the longtime evolution of an initial disturbance, we consider such solutions of both models here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves for each model, we find a continuum of delocalized solitary waves (or homoclinics to smallamplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. For the microstructure equation, the new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work, including the dynamics of each family of solitary waves using exponential asymptotics techniques, are also mentioned.
Show less  Date Issued
 2008
 Identifier
 CFE0002151, ucf:47930
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0002151
 Title
 ALGEBRAIC ASPECTS OF (BIO) NANOCHEMICAL REACTION NETWORKS AND BIFURCATIONS IN VARIOUS DYNAMICAL SYSTEMS.
 Creator

Chen, Teng, Brennan, Joseph, University of Central Florida
 Abstract / Description

The dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question...
Show moreThe dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question. This theory is applied to (bio)nano particle con gurations. We also investigate Hopf bifurcation surfaces of various dynamical systems.
Show less  Date Issued
 2011
 Identifier
 CFE0003933, ucf:48689
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0003933
 Title
 VARIATIONAL EMBEDDED SOLITONS, AND TRAVELING WAVETRAINS GENERATED BY GENERALIZED HOPF BIFURCATIONS, IN SOME NLPDE SYSTEMS.
 Creator

Smith, Todd, Choudhury, Roy, University of Central Florida
 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...
Show moreIn 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 predatorprey model with general predator death rate and prey birth rate terms, ii. a laserdiode system, and iii. travelingwave solutions of twospecies predatorprey/reactiondiusion equations with arbitrary nonlinear/reaction terms. For speci c choices of the nonlinear terms, the quasiperiodic orbit in the postbifurcation 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 postbifurcation 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 predatorprey models yield a variety of behaviors in the postbifurcation regime, the laserdiode evinces extremely stable quasiperiodic solutions over a wide range of parameters, which is very desirable for robust operation of the system in oscillator mode.
Show less  Date Issued
 2011
 Identifier
 CFE0003634, ucf:48887
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0003634
 Title
 DISSIPATIVE SOLITONS IN THE CUBIC–QUINTIC COMPLEX GINZBURG–LANDAU EQUATION:BIFURCATIONS AND SPATIOTEMPORAL STRUCTURE.
 Creator

Mancas, Ciprian, Choudhury, Roy S., University of Central Florida
 Abstract / Description

Comprehensive numerical simulations (reviewed in Dissipative Solitons, Akhmediev and Ankiewicz (Eds.), Springer, Berlin, 2005) of pulse solutions of the cubicquintic GinzburgLandau equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping,...
Show moreComprehensive numerical simulations (reviewed in Dissipative Solitons, Akhmediev and Ankiewicz (Eds.), Springer, Berlin, 2005) of pulse solutions of the cubicquintic GinzburgLandau equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping, snake, erupting, and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and nonintegrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulsetype structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied. In this dissertation, we develop a theoretical framework for these novel classes of solutions. In the first part, we use a traveling wave reduction or a socalled spatial approximation to comprehensively investigate the bifurcations of plane wave and periodic solutions of the CGLE. The primary tools used here are Singularity Theory and Hopf bifurcation theory respectively. Generalized and degenerate Hopf bifurcations have also been considered to track the emergence of global structure such as homoclinic orbits. However, these results appear difficult to correlate to the numerical bifurcation sequences of the dissipative solitons. In the second part of this dissertation, we shift gears to focus on the issues of central interest in the area, i.e., the conditions for the occurrence of the five categories of dissipative solitons, as well the dependence of both their shape and their stability on the various parameters of the CGLE, viz. the nonlinearity, dispersion, linear and nonlinear gain, loss and spectral filtering parameters. Our predictions on the variation of the soliton amplitudes, widths and periods with the CGLE parameters agree with simulation results. For this part, we develop and discuss a variational formalism within which to explore the various classes of dissipative solitons. Given the complex dynamics of the various dissipative solutions, this formulation is, of necessity, significantly generalized over all earlier approaches in several crucial ways. Firstly, the two alternative starting formulations for the Lagrangian are recent and not well explored. Also, after extensive discussions with David Kaup, the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple (and the trial function integrable!) while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses. In addition, the resulting EulerLagrange equations are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the wellknown stationary solitons or plain pulses, we use dynamical systems theory to focus on more complex attractors viz. periodic, quasiperiodic, and chaotic ones. Periodic evolution of the trial function parameters on stable periodic attractors constructed via the method of multiple scales yield solitons whose amplitudes are nonstationary or time dependent. In particular, pulsating, snake (and, less easily, creeping) dissipative solitons may be treated in this manner. Detailed results are presented here for the pulsating solitary waves  their regimes of occurrence, bifurcations, and the parameter dependences of the amplitudes, widths, and periods agree with simulation results. Finally, we elucidate the Hopf bifurcation mechanism responsible for the various pulsating solitary waves, as well as its absence in Hamiltonian and integrable systems where such structures are absent. Results will be presented for the pulsating and snake soliton cases. Chaotic evolution of the trial function parameters in chaotic regimes identified using dynamical systems analysis would yield chaotic solitary waves. The method also holds promise for detailed modeling of chaotic solitons as well. This overall approach fails only to address the fifth class of dissipative solitons, viz. the exploding or erupting solitons.
Show less  Date Issued
 2007
 Identifier
 CFE0001571, ucf:47116
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0001571
 Title
 EPIDEMIOLOGICAL MODELS FOR MUTATING PATHOGENS WITH TEMPORARY IMMUNITY.
 Creator

Singh, Neeta, Rollins, David, University of Central Florida
 Abstract / Description

Significant progress has been made in understanding different scenarios for disease transmissions and behavior of epidemics in recent years. A considerable amount of work has been done in modeling the dynamics of diseases by systems of ordinary differential equations. But there are very few mathematical models that deal with the genetic mutations of a pathogen. Infact, not much has been done to model the dynamics of mutations of pathogen explaining its effort to escape the host's immune...
Show moreSignificant progress has been made in understanding different scenarios for disease transmissions and behavior of epidemics in recent years. A considerable amount of work has been done in modeling the dynamics of diseases by systems of ordinary differential equations. But there are very few mathematical models that deal with the genetic mutations of a pathogen. Infact, not much has been done to model the dynamics of mutations of pathogen explaining its effort to escape the host's immune defense system after it has infected the host. In this dissertation we develop an SIR model with variable infection age for the transmission of a pathogen that can mutate in the host to produce a second infectious mutant strain. We assume that there is a period of temporary immunity in the model. A temporary immunity period along with variable infection age leads to an integrodifferentialdifference model. Previous efforts on incorporating delays in epidemic models have mainly concentrated on inclusion of latency periods (this assumes that the force of infection at a present time is determined by the number of infectives in the past). We begin with reviewing some basic models. These basic models are the building blocks for the later, more detailed models. Next we consider the model for mutation of pathogen and discuss its implications. Finally, we improve this model for mutation of pathogen by incorporating delay induced by temporary immunity. We examine the influence of delay as we establish the existence, and derive the explicit forms of diseasefree, boundary and endemic equilibriums. We will also investigate the local stability of each of these equilibriums. The possibility of Hopf bifurcation using delay as the bifurcation parameter is studied using both analytical and numerical solutions.
Show less  Date Issued
 2006
 Identifier
 CFE0001043, ucf:46801
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0001043
 Title
 NUMERICAL MODELING OF THE SHOCK TUBE FLOW FIELDS BEFORE ANDDURING IGNITION DELAY TIME EXPERIMENTS AT PRACTICAL CONDITIONS.
 Creator

lamnaouer, mouna, Kassab, Alain, University of Central Florida
 Abstract / Description

An axisymmetric shocktube model has been developed to simulate the shockwave propagation and reflection in both nonreactive and reactive flows. Simulations were performed for the full shocktube geometry of the highpressure shock tube facility at Texas A&M University. Computations were carried out in the CFD solver FLUENT based on the finite volume approach and the AUSM+ flux differencing scheme. Adaptive mesh refinement (AMR) algorithm was applied to the timedependent flow fields to...
Show moreAn axisymmetric shocktube model has been developed to simulate the shockwave propagation and reflection in both nonreactive and reactive flows. Simulations were performed for the full shocktube geometry of the highpressure shock tube facility at Texas A&M University. Computations were carried out in the CFD solver FLUENT based on the finite volume approach and the AUSM+ flux differencing scheme. Adaptive mesh refinement (AMR) algorithm was applied to the timedependent flow fields to accurately capture and resolve the shock and contact discontinuities as well as the very fine scales associated with the viscous and reactive effects. A conjugate heat transfer model has been incorporated which enhanced the credibility of the simulations. The multidimensional, timedependent numerical simulations resolved all of the relevant scales, ranging from the size of the system to the reaction zone scale. The robustness of the numerical model and the accuracy of the simulations were assessed through validation with the analytical ideal shocktube theory and experimental data. The numerical method is first applied to the problem of axisymmetric inviscid flow then viscous effects are incorporated through viscous modeling. The nonidealities in the shock tube have been investigated and quantified, notably the nonideal transient behavior in the shock tube nozzle section, heat transfer effects from the hot gas to the shock tube side walls, the reflected shock/boundary layer interactions or what is known as bifurcation, and the contact surface/bifurcation interaction resulting into driver gas contamination. The nonreactive model is shown to be capable of accurately simulating the shock and expansion wave propagations and reflections as well as the flow nonuniformities behind the reflected shock wave. Both the inviscid and the viscous nonreactive models provided a baseline for the combustion model iii which involves elementary chemical reactions and requires the coupling of the chemistry with the flow fields adding to the complexity of the problem and thereby requiring tremendous computational resources. Combustion modeling focuses on the ignition process behind the reflected shock wave in undiluted and diluted Hydrogen test gas mixtures. Accurate representation of the Shock Ã‚Â–tube reactive flow fields is more likely to be achieved by the means of the LES model in conjunction with the EDC model. The shocktube CFD model developed herein provides valuable information to the interpretation of the shocktube experimental data and to the understanding of the impact the facilitydependent nonidealities can have on the ignition delay time measurements.
Show less  Date Issued
 2010
 Identifier
 CFE0003011, ucf:48366
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0003011
 Title
 Ignition Studies of OxySyngas/CO2 Mixtures Using Shock Tube for Cleaner Combustion Engines.
 Creator

Barak, Samuel, Vasu Sumathi, Subith, Kapat, Jayanta, Ahmed, Kareem, University of Central Florida
 Abstract / Description

In this study, syngas combustion was investigated behind reflected shock waves in order to gain insight into the behavior of ignition delay times and effects of the CO2 dilution. Pressure and light emissions timehistories measurements were taken at a 2 cm axial location away from the end wall. Highspeed visualization of the experiments from the end wall was also conducted. Oxysyngas mixtures that were tested in the shock tube were diluted with CO2 fractions ranging from 60%  85% by volume...
Show moreIn this study, syngas combustion was investigated behind reflected shock waves in order to gain insight into the behavior of ignition delay times and effects of the CO2 dilution. Pressure and light emissions timehistories measurements were taken at a 2 cm axial location away from the end wall. Highspeed visualization of the experiments from the end wall was also conducted. Oxysyngas mixtures that were tested in the shock tube were diluted with CO2 fractions ranging from 60%  85% by volume. A 10% fuel concentration was consistently used throughout the experiments. This study looked at the effects of changing the equivalence ratios (?), between 0.33, 0.5, and 1.0 as well as changing the fuel ratio (?), hydrogen to carbon monoxide, from 0.25, 1.0 and 4.0. The study was performed at 1.611.77 atm and a temperature range of 10061162K. The highspeed imaging was performed through a quartz end wall with a Phantom V710 camera operated at 67,065 frames per second. From the experiments, when increasing the equivalence ratio, it resulted in a longer ignition delay time. In addition, when increasing the fuel ratio, a lower ignition delay time was observed. These trends are generally expected with this combustion reaction system. The highspeed imaging showed nonhomogeneous combustion in the system, however, most of the light emissions were outside the visible light range where the camera is designed for. The results were compared to predictions of two combustion chemical kinetic mechanisms: GRI v3.0 and AramcoMech v2.0 mechanisms. In general, both mechanisms did not accurately predict the experimental data. The results showed that current models are inaccurate in predicting CO2 diluted environments for syngas combustion.
Show less  Date Issued
 2018
 Identifier
 CFE0006974, ucf:52909
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0006974
 Title
 Shock Tube Investigations of Novel Combustion Environments Towards a CarbonNeutral Future.
 Creator

Barak, Samuel, Vasu Sumathi, Subith, Kapat, Jayanta, Ahmed, Kareem, University of Central Florida
 Abstract / Description

Supercritical carbon dioxide (sCO2) cycles are being investigated for the future of power generation. These cycles will contribute to a carbonneutral future to combat the effects of climate change. These directfired closed cycles will produce power without adding significant pollutants to the atmosphere. For these cycles to be efficient, they will need to operate at significantly higher pressures (e.g., 300 atm for Allam Cycle) than existing systems (typically less than 40 atm). There is...
Show moreSupercritical carbon dioxide (sCO2) cycles are being investigated for the future of power generation. These cycles will contribute to a carbonneutral future to combat the effects of climate change. These directfired closed cycles will produce power without adding significant pollutants to the atmosphere. For these cycles to be efficient, they will need to operate at significantly higher pressures (e.g., 300 atm for Allam Cycle) than existing systems (typically less than 40 atm). There is limited knowledge on combustion at these pressures or at the high dilution of carbon dioxide. Nominal fuel choices for gas turbines include natural gas and syngas (mixture of CO and H2). Shock tubes study these problems in order to understand the fundamentals and solve various challenges. Shock tube experiments have been studied by the author in the sCO2 regime for various fuels including natural gas, methane and syngas. Using the shock tube to take measurements, pressure and light emissions timehistories measurements were taken at a 2cm axial location away from the end wall. Experiments for syngas at lower pressure utilized highspeed imaging through the end wall to investigate the effects of bifurcation. It was found that carbon dioxide created unique interactions with the shock tube compared to tradition bath gasses such as argon. The experimental results were compared to predictions from leading chemical kinetic mechanisms. In general, mechanisms can predict the experimental data for methane and other hydrocarbon fuels; however, the models overpredict for syngas mixtures. Reaction pathway analysis was evaluated to determine where the models need improvements. A new shock tube has been designed and built to operate up to 1000 atm pressures for future highpressure experiments. Details of this new facility are included in this work. The experiments in this work are necessary for mechanism development to design an efficient combustor operate these cycles.
Show less  Date Issued
 2019
 Identifier
 CFE0007781, ucf:52359
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007781