Non-linear Dynamics Commons^™

Finding Unpredictable Behaviors Of Periodic Bouncing For Forced Nonlinear Spring Systems When Oscillating Time Is Large, Yanyue Ning

Honors Scholar Theses

The model of nonlinear spring systems can be applied to deal with different aspect of mechanical problems, such as oscillations in periodic flexing in bridges and ships. The concentration of this research is the bouncing behaviors of nonlinear spring system when the processing time is large, therefore nonlinear ordinary differential equations (ODE) are suitable since researchers can add different variables into the models and solve them by computational methods. Benefit from this, it is easy to check the oscillations or bouncing behaviors that each variable contributes to the model and find the relationship between some important factors: vibrating frequency, external …

G-Strands, Darryl Holm, Rossen Ivanov, James Percival

Articles

A G-strand is a map g(t,s): RxR --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3)_K-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3) …

Cyclic Universe With An Inflationary Phase From A Cosmological Model With Real Gas Quintessence, Rossen Ivanov, Emil Prodanov

Articles

Phase-plane stability analysis of a dynamical system describing the Universe as a two-fraction uid containing baryonic dust and real virial gas quintessence is presented. Existence of a stable periodic solution experiencing in ationary periods is shown. A van der Waals quintessence model is revisited and cyclic Universe solution again found.

An Exercise With The He’S Variation Iteration Method To A Fractional Bernoulli Equation Arising In A Transient Conduction With A Non-Linear Boundary Heat Flux, Jordan Hristov

Jordan Hristov

Surface temperature evolution of a body subjected to a nonlinear heat flux involving counteracting convection heating and radiation cooling has been solved by the variations iteration method (VIM) of He. The surface temperature equations comes as a combination of the time-fractional (half-time) subdiffusion model of the heat conduction and the boundary condition relating the temperature field gradient at the surface through the Riemann-Liouville fractional integral. The result of this equation is a Bernoulli-type ordinary fractional equation with a nonlinear term of 4th order. Two approaches in the identification of the general Lagrange multiplier and a consequent application of VIM have …

Synthesis Of Static And Dynamic Multiple-Input Translinear Element Networks, Bradley Minch

Bradley Minch

In this paper, we discuss the process of synthesizing static and dynamic multiple-input translinear element (MITE) networks systematically from high-level descriptions given in the time domain, in terms of static polynomial constraints and algebraic differential equations. We provide several examples, illustrating the process for both static and dynamic system constraints. Although our examples will all involve MITE networks, the early steps of the synthesis process are equally applicable to the synthesis of static and dynamic translinear-loop circuits.

Modeling And Mathematical Analysis Of Plant Models In Ecology, Eric A. Eager

Department of Mathematics: Dissertations, Theses, and Student Research

Population dynamics tries to explain in a simple mechanistic way the variations of the size and structure of biological populations. In this dissertation we use mathematical modeling and analysis to study the various aspects of the dynamics of plant populations and their seed banks.

In Chapter 2 we investigate the impact of structural model uncertainty by considering different nonlinear recruitment functions in an integral projection model for Cirsium canescens. We show that, while having identical equilibrium populations, these two models can elicit drastically different transient dynamics. We then derive a formula for the sensitivity of the equilibrium population to …

Integral-Balance Solution To The Stokes’ First Problem Of A Viscoelastic Generalized Second Grade Fluid, Jordan Hristov

Jordan Hristov

Integral balance solution employing entire domain approximation and the penetration dept concept to the Stokes’ first problem of a viscoelastic generalized second grade fluid has been developed. The solution has been performed by a parabolic profile with an unspecified exponent allowing optimization through minimization of the norm over the domain of the penetration depth. The closed form solution explicitly defines two dimensionless similarity variables and , responsible for the viscous and the elastic responses of the fluid to the step jump at the boundary. The solution was developed with three forms of the governing equation through its two dimensional forms …

Computationally Efficient Methods For Modelling Laser Wakefield Acceleration In The Blowout Regime, Benjamin M. Cowan, Serguei Y. Kalmykov, Arnaud Beck, Xavier Davoine, Kyle Bunkers, Agustin F. Lifschitz, Erik Lefebvre, David L. Bruhwiler, Bradley A. Shadwick, Donald P. Umstadter

Donald P. Umstadter

Electron self-injection and acceleration until dephasing in the blowout regime is studied for a set of initial conditions typical of recent experiments with 100-terawatt-class lasers. Two different approaches to computationally efficient, fully explicit, 3D particle-in-cell modelling are examined. First, the Cartesian code VORPAL (Nieter, C. and Cary, J. R. 2004 VORPAL: a versatile plasma simulation code. J. Comput. Phys. 196, 538) using a perfect-dispersion electromagnetic solver precisely describes the laser pulse and bubble dynamics, taking advantage of coarser resolution in the propagation direction, with a proportionally larger time step. Using third-order splines for macroparticles helps suppress the sampling noise while …

Computationally Efficient Methods For Modelling Laser Wakefield Acceleration In The Blowout Regime, Benjamin M. Cowan, Serguei Y. Kalmykov, Arnaud Beck, Xavier Davoine, Kyle Bunkers, Agustin F. Lifschitz, Erik Lefebvre, David L. Bruhwiler, Bradley A. Shadwick, Donald P. Umstadter

Serge Youri Kalmykov

Electron self-injection and acceleration until dephasing in the blowout regime is studied for a set of initial conditions typical of recent experiments with 100-terawatt-class lasers. Two different approaches to computationally efficient, fully explicit, 3D particle-in-cell modelling are examined. First, the Cartesian code VORPAL (Nieter, C. and Cary, J. R. 2004 VORPAL: a versatile plasma simulation code. J. Comput. Phys. 196, 538) using a perfect-dispersion electromagnetic solver precisely describes the laser pulse and bubble dynamics, taking advantage of coarser resolution in the propagation direction, with a proportionally larger time step. Using third-order splines for macroparticles helps suppress the sampling noise while …

Thermal Impedance At The Interface Of Contacting Bodies: 1-D Example Solved By Semi-Derivatives, Jordan Hristov

Jordan Hristov

Simple 1-D semi-infinite heat conduction problems enable to demonstrate the potential of the fractional calculus in determination of transient thermal impedances of two bodies with different initial temperatures contacting at the interface ( ) at . The approach is purely analytic and uses only semi-derivatives (half-time) and semi-integrals in the Riemann-Liouville sense. The example solved clearly reveals that the fractional calculus is more effective in calculation the thermal resistances than the entire domain solutions

Variational Iteration Method For Q-Difference Equations Of Second Order, Guo-Cheng Wu

G.C. Wu

Recently, Liu extended He's variational iteration method to strongly nonlinear q-difference equations. In this study, the iteration formula and the Lagrange multiplier are given in a more accurate way. The q-oscillation equation of second order is approximately solved to show the new Lagrange multiplier's validness.

Generation Of Tunable, 100–800 Mev Quasi-Monoenergetic Electron Beams From A Laser-Wakefield Accelerator In The Blowout Regime, Sudeep Banerjee, Nathan D. Powers, Vidiya Ramanathan, Isaac Ghebregziabher, Kevin J. Brown, Chakra M. Maharjan, Shouyuan Chen, Arnaud Beck, Erik Lefebvre, Serguei Y. Kalmykov, Bradley A. Shadwick, Donald P. Umstadter

Donald P. Umstadter

In this paper, we present results on a scalable high-energy electron source based on laser wakefield acceleration. The electron accelerator using 30 - 80 TW, 30 fs laser pulses, operates in the blowout regime, and produces high-quality, quasi-monoenergetic electron beams in the range 100 - 800 MeV. These beams have angular divergence of 1 - 4 mrad, and 5 - 25 percent energy spread, with a resulting brightness 10^{11} electrons mm^{-2} MeV^{-1} mrad^{-2}. The beam parameters can be tuned by varying the laser and plasma conditions. The use of a high-quality laser pulse and appropriate target conditions enables optimization of …

Generation Of Tunable, 100–800 Mev Quasi-Monoenergetic Electron Beams From A Laser-Wakefield Accelerator In The Blowout Regime, Sudeep Banerjee, Nathan D. Powers, Vidiya Ramanathan, Isaac Ghebregziabher, Kevin J. Brown, Chakra M. Maharjan, Shouyuan Chen, Arnaud Beck, Erik Lefebvre, Serguei Y. Kalmykov, Bradley A. Shadwick, Donald P. Umstadter

Serge Youri Kalmykov

In this paper, we present results on a scalable high-energy electron source based on laser wakefield acceleration. The electron accelerator using 30 - 80 TW, 30 fs laser pulses, operates in the blowout regime, and produces high-quality, quasi-monoenergetic electron beams in the range 100 - 800 MeV. These beams have angular divergence of 1 - 4 mrad, and 5 - 25 percent energy spread, with a resulting brightness 10^{11} electrons mm^{-2} MeV^{-1} mrad^{-2}. The beam parameters can be tuned by varying the laser and plasma conditions. The use of a high-quality laser pulse and appropriate target conditions enables optimization of …

The Discrete Yang-Fourier Transforms In Fractal Space, Yang Xiao-Jun

Xiao-Jun Yang

The Yang-Fourier transform (YFT) in fractal space is a generation of Fourier transform based on the local fractional calculus. The discrete Yang-Fourier transform (DYFT) is a specific kind of the approximation of discrete transform, used in Yang-Fourier transform in fractal space. This paper points out new standard forms of discrete Yang-Fourier transforms (DYFT) of fractal signals, and both properties and theorems are investigated in detail.

Expression Of Generalized Newton Iteration Method Via Generalized Local Fractional Taylor Series, Yang Xiao-Jun

Xiao-Jun Yang

Local fractional derivative and integrals are revealed as one of useful tools to deal with everywhere continuous but nowhere differentiable functions in fractal areas ranging from fundamental science to engineering. In this paper, a generalized Newton iteration method derived from the generalized local fractional Taylor series with the local fractional derivatives is reviewed. Operators on real line numbers on a fractal space are induced from Cantor set to fractional set. Existence for a generalized fixed point on generalized metric spaces may take place.

Laser Plasma Acceleration With A Negatively Chirped Pulse: All-Optical Control Over Dark Current In The Blowout Regime, Serguei Y. Kalmykov, Arnaud Beck, Xavier Davoine, Erik Lefebvre, Bradley A. Shadwick

Serge Youri Kalmykov

Recent experiments with 100 terawatt-class, sub-50 femtosecond laser pulses show that electrons self-injected into a laser-driven electron density bubble can be accelerated above 0.5 gigaelectronvolt energy in a sub-centimetre length rarefied plasma. To reach this energy range, electrons must ultimately outrun the bubble and exit the accelerating phase; this, however, does not ensure high beam quality. Wake excitation increases the laser pulse bandwidth by red-shifting its head, keeping the tail unshifted. Anomalous group velocity dispersion of radiation in plasma slows down the red-shifted head, compressing the pulse into a few-cycle-long piston of relativistic intensity. Pulse transformation into a piston causes …

The Zero-Mass Renormalization Group Differential Equations And Limit Cycles In Non-Smooth Initial Value Problems, Yang Xiaojun

Xiao-Jun Yang

In the present paper, using the equation transform in fractal space, we point out the zero-mass renormalization group equations. Under limit cycles in the non-smooth initial value, we devote to the analytical technique of the local fractional Fourier series for treating zero-mass renormalization group equations, and investigate local fractional Fourier series solutions.

A Novel Approach To Processing Fractal Dynamical Systems Using The Yang-Fourier Transforms, Yang Xiaojun

Xiao-Jun Yang

In the present paper, local fractional continuous non-differentiable functions in fractal space are investigated, and the control method for processing dynamic systems in fractal space are proposed using the Yang-Fourier transform based on the local fractional calculus. Two illustrative paradigms for control problems in fractal space are given to elaborate the accuracy and reliable results.

On Soliton Interactions For A Hierarchy Of Generalized Heisenberg Ferromagnetic Models On Su(3)/S(U(1) $\Times$ U(2)) Symmetric Space, Vladimir Gerdjikov, Georgi Grahovski, Alexander Mikhailov, Tihomir Valchev

Articles

We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator L. The Lax representation is Z2 \times Z2 reduced and is naturally associated with the symmetric space SU(3)/S(U(1) \times U(2)). The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the N-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one-soliton solutions of NLEEs with even and odd dispersion laws have different properties. In …

Singular Solutions Of Coss-Coupled Epdiff Equations: Waltzing Peakons And Compacton Pairs, Colin Cotter, Darryl Holm, Rossen Ivanov, James Percival

Conference papers

We introduce EPDiff equations as Euler-Poincare´ equations related to Lagrangian provided by a metric, invariant under the Lie Group Diff(Rⁿ). Then we proceed with a particular form of EPDiff equations, a cross coupled two-component system of Camassa-Holm type. The system has a new type of peakon solutions, 'waltzing' peakons and compacton pairs.