Non-linear Dynamics Commons^™

The Sir Model When S(T) Is A Multi-Exponential Function., Teshome Mogessie Balkew

Electronic Theses and Dissertations

The SIR can be expressed either as a system of nonlinear ordinary differential equations or as a nonlinear Volterra integral equation. In general, neither of these can be solved in closed form. In this thesis, it is shown that if we assume S(t) is a finite multi-exponential, i.e. function of the form S(t) = a+ ∑ⁿ_k=1 r_ke^-σ_kt or a logistic function which is an infinite-multi-exponential, i.e. function of the form S(t) = c + a/b+e^wt, then …

All-Optical Control Of Nonlinear Self-Focusing In Plasmas Using Non-Resonantly Driven Plasma Wave, Serguei Y. Kalmykov, Bradley A. Shadwick, Michael C. Downer

Serge Youri Kalmykov

Excitation of plasma density perturbations by an initially bi-color laser pulse helps to control nonlinear refraction in the plasma and enables various types of laser self-guiding. In this report we consider a setup that not only makes possible the transport of laser energy over cm-long relatively dense plasmas (n_0 = 10^{18} cm^{−3}) but also transforms the pulse into the unique format inaccessible to the conventional amplification techniques (relativistically intense periodic trains of few-cycle spikes). This well focusable pulse train is a novel light source interesting for ultra-fast high-field science applications. The opposite case of suppression of nonlinear self-focusing and dynamical …

Electron Self-Injection Into An Evolving Plasma Bubble: The Way To A Dark Current Free Gev-Scale Laser Accelerator, Serguei Y. Kalmykov, Arnaud Beck, Sunghwan A. Yi, Vladimir N. Khudik, Bradley A. Shadwick, Erik Lefebvre, Michael C. Downer

Serge Youri Kalmykov

A time-varying electron density bubble created by the radiation pressure of a tightly focused petawatt laser pulse traps electrons of ambient rarefied plasma and accelerates them to a GeV energy over a few-cm distance. Expansion of the bubble caused by the shape variation of the self-guided pulse is the primary cause of electron self-injection in strongly rarefied plasmas (n_0 ~ 10^{17} cm^{−3}). Stabilization and contraction of the bubble extinguishes the injection. After the bubble stabilization, longitudinal non-uniformity of the accelerating gradient results in a rapid phase space rotation that produces a quasi-monoenergetic bunch well before the de-phasing limit. Combination of …

A Short-Distance Integral-Balance Solution To A Strong Subdiffusion Equation: A Weak Power-Law Profile, Jordan Hristov

Jordan Hristov

The work presents an integral solution of the time-fractional subdiffusion through a preliminary defined profile with unknown coefficients and the concept of penetration layer well known from the heat diffusion The profile satisfies the boundary conditions imposed at the boundary of the boundary layer in a weak form that allows its coefficients to be expressed through the boundary layer depth as unique parameter describing the profile. The technique is demonstrated by a solution of a time fractional subdiffusion equation in rectilinear 1-D conditions.

Some Features Of The Concentration Oscillations In The Phenylacetylene Oxidative Carbonylation Reaction (In Russian), Sergey N. Gorodsky

Sergey N. Gorodsky

Some modes of concentration oscillations in the homogeneous system KI-PdI2-CO-O2-CH3OH are described in this paper.

Neural Extensions To Robust Parameter Design, Bernard Jacob Loeffelholz

Theses and Dissertations

Robust parameter design (RPD) is implemented in systems in which a user wants to minimize the variance of a system response caused by uncontrollable factors while obtaining a consistent and reliable system response over time. We propose the use of artificial neural networks to compensate for highly non-linear problems that quadratic regression fails to accurately model. RPD is conducted under the assumption that the relationship between system response and controllable and uncontrollable variables does not change over time. We propose a methodology to find a new set of settings that will be robust to moderate system degradation while remaining robust …

Energetyka Niskoemisyjna, Wojciech M. Budzianowski

Wojciech Budzianowski

No abstract provided.

Non-Classical Symmetry Solutions To The Fitzhugh Nagumo Equation., Arash Mehraban

Electronic Theses and Dissertations

In Reaction-Diffusion systems, some parameters can influence the behavior of other parameters in that system. Thus reaction diffusion equations are often used to model the behavior of biological phenomena. The Fitzhugh Nagumo partial differential equation is a reaction diffusion equation that arises both in population genetics and in modeling the transmission of action potentials in the nervous system. In this paper we are interested in finding solutions to this equation. Using Lie groups in particular, we would like to find symmetries of the Fitzhugh Nagumo equation that reduce this non-linear PDE to an Ordinary Differential Equation. In order to accomplish …

Nonlinear Acoustics Of Piston-Driven Gas-Column Oscillations, Andrew William Wilson

Masters Theses

The piston-driven oscillator is traditionally modeled by directly applying boundary conditions to the acoustic wave equations; with better models re-deriving the wave equations but retaining nonlinear and viscous effects. These better models are required as the acoustic solution exhibits singularity near the natural frequencies of the cavity, with an unbounded (and therefore unphysical) solution. Recently, a technique has been developed to model general pressure oscillations in propulsion systems and combustion devices. Here, it is shown that this technique applies equally well to the piston-driven gas-column oscillator; and that the piston experiment provides strong evidence for the validity of the general …

Optimal Control Of A Switched System In Microbial Fed-Batch Fermentation Process, Chongyang Liu, Zhaohua Gong, Enmin Feng

Chongyang Liu

The main control goal in fed-batch fermentation process is to get a high concentration of production. In this paper, by taking the feed rate of glycerol as the control function, a nonlinear switched system is proposed to formulate the fed-batch fermentation process of glycerol to 1,3-propanediol (1,3-PD). To maximize the concentration of 1,3-PD at the terminal time, an optimal switching control model subject to constraints of continuous state inequality and control function is presented. A computational approach is developed to seek the optimal solution in two aspects. On the one hand, the control parametrization enhancing transform together with the control …

Heat-Balance Integral To Fractional (Half-Time) Heat Diffusion Sub-Model, Jordan Hristov

Jordan Hristov

The fractional (half-time) sub-model of the heat diffusion equation, known as Dirac-like evolution diffusion equation has been solved by the heat-balance integral method and a parabolic pro file with unspecified exponent. The fractional heat-balance integral method has been tested with two classic examples: fixed temperature and fixed flux at the boundary. The heat-balance technique allows easily the convolution integral of the fractional half-time derivative to be solved as a convolution of the time-independent approximating function. The fractional sub-model provides an artificial boundary condition at the boundary that closes the set of the equations required to express all parameters of the …

The 1905 Einstein Equation In A General Mathematical Analysis Model Of Quasars, Byron E. Bell

Byron E. Bell

Numerical Modelling Of A 10-Cm-Long Multi-Gev Laser Wakefield Accelerator Driven By A Self-Guided Petawatt Pulse, Serguei Y. Kalmykov, Sunghwan A. Yi, Arnaud Beck, Agustin F. Lifschitz, Xavier Davoine, Erik Lefebvre, Alexander Pukhov, Vladimir N. Khudik, Gennady Shvets, Steven A. Reed, Peng Dong, Xiaoming Wang, Dongsu Du, Stefan Bedacht, Rafal B. Zgadzaj, Watson Henderson, Aaron Bernstein, Gilliss Dyer, Mikael Martinez, Erhard Gaul, Todd Ditmire, Michael C. Downer

Serge Youri Kalmykov

Holographic Visualization Of Laser Wakefields, Peng Dong, Steven A. Reed, Sunghwan A. Yi, Serguei Y. Kalmykov, Zhengyan Y. Li, Gennady Shvets, Nicholas H. Matlis, Christopher Mcguffey, Stepan S. Bulanov, Vladimir Chvykov, Galina Kalintchenko, Karl Krushelnick, Anatoly Maksimchuk, Takeshi Matsuoka, Alexander G. R. Thomas, Victor Yanovsky, Michael C. Downer

Serge Youri Kalmykov

We report ‘snapshots’ of laser-generated plasma accelerator structures acquired by frequency domain holography (FDH) and frequency domain shadowgraphy (FDS), techniques for visualizing quasi-static objects propagating near the speed of light. FDH captures images of sinusoidal wakes in mm-length plasmas of density 1 < n_{e} < 5 x 10^{18} cm^{−3} from phase modulations they imprint on co-propagating probe pulses. Changes in the wake structure (such as the curvature of the wavefront), caused by the laser and plasma parameter variations from shot to shot, were observed. FDS visualizes lasergenerated electron density bubbles in mm-length plasmas of density n_{e} > 10^{19} cm^{−3} using amplitude modulations they imprint on co-propagating probe pulses. Variations in the spatio-temporal structure of bubbles are inferred from corresponding variations in the shape of ‘bullets’ of probe light trapped inside them and correlated with mono-energetic electron generation. Both FDH and FDS average over structural variations that occur during propagation through the plasma medium. We explore …

Codes From Riemann-Roch Spaces For Y2 = Xp - X Over Gf(P), Darren B. Glass, David Joyner, Amy Ksir

Math Faculty Publications

Let Χ denote the hyperelliptic curve y2 = xp - x over a field F of characteristic p. The automorphism group of Χ is G = PSL(2, p). Let D be a G-invariant divisor on Χ(F). We compute explicit F-bases for the Riemann-Roch space of D in many cases as well as G-module decompositions. AG codes with good parameters and large automorphism group are constructed as a result. Numerical examples using GAP and SAGE are also given.

Formation Of Optical Bullets In Laser-Driven Plasma Bubble Accelerators, Peng Dong, Steven A. Reed, Sunghwan A. Yi, Serguei Y. Kalmykov, Gennady Shvets, Michael C. Downer, Nicholas H. Matlis, Wim P. Leemans, Christopher Mcguffey, Stepan S. Bulanov, Vladimir Chvykov, Galina Kalintchenko, Karl Krushelnick, Anatoly Maksimchuk, Takeshi Matsuoka, Alexander G. R. Thomas, Victor Yanovsky

Serge Youri Kalmykov

Electron density bubbles—wake structures generated in plasma of density n_{e} ~ 10^{19} cm^{-3} by the light pressure of intense ultrashort laser pulses—are shown to reshape weak copropagating probe pulses into optical ‘‘bullets.’’ The bullets are reconstructed using frequency-domain interferometric techniques in order to visualize bubble formation. Bullets are confined in three dimensions to plasma-wavelength size, and exhibit higher intensity, broader spectrum and flatter temporal phase than surrounding probe light, evidence of their compression by the bubble. Bullets observed at 0.8 < n_{e} < 1.2 x 10^{19} cm^{-3} provide the first observation of bubble formation below the electron capture threshold. At higher n_{e}, bullets appear with high shot-to-shot stability together with relativistic electrons that vary widely in spectrum, and help relate bubble formation to fast electron generation.

Morphological Evolution Of Single-Crystal Ultrathin Solid Films, Mikhail Khenner

Mathematics Faculty Publications

An introduction to mathematical modeling of ultrathin solid films and the role of such modeling in nanotechnologies: Educational presentation for senior physics majors

Morphological Evolution Of Single-Crystal Ultrathin Solid Films, Mikhail Khenner

Mathematics Faculty Publications

An introduction to mathematical modeling of ultrathin solid films and the role of such modeling in nanotechnologies: Educational/Research presentation for senior physics majors

Morphological Evolution Of Single-Crystal Ultrathin Solid Films, Mikhail Khenner

Mikhail Khenner

An introduction to mathematical modeling of ultrathin solid films and the role of such modeling in nanotechnologies: Educational/Research presentation for senior physics majors

The Camassa-Holm Hierarchy And Soliton Perturbations, Georgi Grahovski, Rossen Ivanov

Conference papers

The theory of soliton perturbations is considered. The focus is on the usefulness of the so-called Generalised Fourier Transform (GFT). This is a method that involves expansions over the complete basis of “squared solutions” of the spectral problem, associated to the soliton equation. The Inverse Scattering Transform for the corresponding hierarchy of soliton equations can be viewed as a GFT where the expansions of the solutions have generalised Fourier coefficients given by the scattering data. The GFT provides a natural setting for the analysis of small perturbations to an integrable equation: starting from a purely soliton solution one can ’modify’ …

Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

Mathematics Faculty Publications

We study long-wave Marangoni convection in a layer heated from below. Using the scaling k=O Bi, where k is the wave number and Bi is the Biot number, we derive a set of amplitude equations. Analysis of this set shows presence of monotonic and oscillatory modes of instability. Oscillatory mode has not been previously found for such direction of heating. Studies of weakly nonlinear dynamics demonstrate that stable steady and oscillatory patterns can be found near the stability threshold.

Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

Mathematics Faculty Publications

We study long-wave Marangoni convection in a layer heated from below. Using the scaling k=O Bi, where k is the wave number and Bi is the Biot number, we derive a set of amplitude equations. Analysis of this set shows presence of monotonic and oscillatory modes of instability. Oscillatory mode has not been previously found for such direction of heating. Studies of weakly nonlinear dynamics demonstrate that stable steady and oscillatory patterns can be found near the stability threshold.

Local Fractional Fourier’S Transform Based On The Local Fractional Calculus, Yang Xiao-Jun

Xiao-Jun Yang

A new modeling for the local fractional Fourier’s transform containing the local fractional calculus is investigated in fractional space. The properties of the local fractional Fourier’s transform are obtained and two examples for the local fractional systems are investigated in detail.

Exact Three-Wave Solutions For High Nonlinear Form Of Benjamin-Bona-Mahony-Burgers Equations, Mohammad Taghi Darvishi, Mohammad Najafi M.Najafi, Maliheh Najafi

mohammad najafi

By means of the idea of three-wave method, we obtain some analytic solutions for high nonlinear form of Benjamin-Bona-Mahony-Burgers (shortly BBMB) equations in its bilinear form.

Modeling And Optimal Control Of A Nonlinear Dynamical System In Microbial Fed-Batch Fermentation Process, Chongyang Liu, Zhaohua Gong, Enmin Feng

Chongyang Liu

The mathematical model and optimal control of microbial fed-batch fermentation is considered in this paper. Since it is decisive for increasing the productivity of 1,3-propanediol (1,3-PD) to optimize the feeding rate of glycerol and the switching instants between the batch and feeding processes in the fermentation process, we propose a new nonlinear dynamical system to formulate the process. In the system, the switching instants are variable and the feeding rate of glycerol is regarded as the control function. Some important properties of the proposed system and its solution are then discussed. To maximize the concentration of 1,3-PD at the terminal …

Improvement Of The Stoichiometric Network Analysis For Determination Of Instability Conditions Of Complex Nonlinear Reaction Systems, Zeljko D. Cupic

Zeljko D Cupic

No abstract provided.

Grafika Inżynierska Ćw., Wojciech M. Budzianowski

Wojciech Budzianowski

No abstract provided.

Projektowanie Procesów Biotechnologicznych Proj., Wojciech M. Budzianowski

Wojciech Budzianowski

No abstract provided.

Projektowanie I Optymalizacja Procesów Proj., Wojciech M. Budzianowski

Wojciech Budzianowski

No abstract provided.

Metody Numeryczne Lab., Wojciech M. Budzianowski

Wojciech Budzianowski

No abstract provided.