How Newton's Method can be used to find all roots of all polynomials

In this paper we will discuss Newton’s Method, its limitations and a theorem which deals with these limitations. We will be looking in particular at John Hubbard, Dierk Schleicher and Scott Sutherland’s theorem on finding all roots of complex polynomials by Newton’s method. It will look at constructing a finite set of points so that for every root of every polynomial of fixed degree, at least one of the points will converge to a root under Newton’s map

Biological control via "ecological" damping: An approach that attenuates non-target effects

In this work we develop and analyze a mathematical model of biological control to prevent or attenuate the explosive increase of an invasive species population in a three-species food chain. We allow for finite time blow-up in the model as a mathematical construct to mimic the explosive increase in population, enabling the species to reach "disastrous" levels, in a finite time. We next propose various controls to drive down the invasive population growth and, in certain cases, eliminate blow-up. The controls avoid chemical treatments and/or natural enemy introduction, thus eliminating various non-target effects associated with such classical methods. We refer to these new controls as "ecological damping", as their inclusion dampens the invasive species population growth. Further, we improve prior results on the regularity and Turing instability of the three-species model that were derived in earlier work. Lastly, we confirm the existence of spatio-temporal chaos

Second Order Neural Networks.

In this dissertation, a feedback neural network model has been proposed. This network uses a second order method of convergence based on the Newton-Raphson method. This neural network has both discrete as well as continuous versions. When used as an associative memory, the proposed model has been called the polynomial neural network (PNN). The memories of this network can be located anywhere in an n dimensional space rather than being confined to the corners of the latter. A method for storing memories has been proposed. This is a single step method unlike the currently known computationally intensive iterative methods. An energy function for the polynomial neural network has been suggested. Issues relating to the error-correcting ability of this network have been addressed. Additionally, it has been found that the attractor basins of the memories of this network reveal a curious fractal topology, thereby suggesting a highly complex and often unpredictable nature. The use of the second order neural network as a function optimizer has also been shown. While issues relating to the hardware realization of this network have only been addressed briefly, it has been indicated that such a network would have a large amount of hardware for its realization. This problem can be obviated by using a simplified model that has also been described. The performance of this simplified model is comparable to that of the basic model while requiring much less hardware for its realization

Fractal Nodal Band Structures

Non-Hermitian systems exhibit interesting band structures, where novel topological phenomena arise from the existence of exceptional points at which eigenvalues and eigenvectors coalesce. One important open question is how this would manifest at non-integer dimension. Here, we report on the appearance of fractal eigenvalue degeneracies and Fermi surfaces in Hermitian and non-Hermitian topological band structures. This might have profound implications on the physics of black holes and Fermi surface instability driven phenomena, such as superconductivity and charge density waves

A New Chaotic System with a Pear-shaped Equilibrium and its Circuit Simulation

This paper reports the finding a new chaotic system with a pear-shaped equilibrium curve and makes a valuable addition to existing chaotic systems with infinite equilibrium points in the literature. The new chaotic system has a total of five nonlinearities. Lyapunov exponents of the new chaotic system are studied for verifying chaos properties and phase portraits of the new system are unveiled. An electronic circuit simulation of the new chaotic system with pear-shaped equilibrium curve is shown using Multisim to check the model feasibility

Higher-order approximation of cubic–quintic duffing model

We apply an Artificial Parameter Lindstedt-Poincaré Method (APL-PM) to find improved approximate solutions for strongly nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. This approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution which makes it a unique solution. It is demonstrated that this method works very well for the whole range of parameters in the case of the cubic-quintic oscillator, and excellent agreement of the approximate frequencies with the exact one has been observed and discussed. Moreover, it is not limited to the small parameter such as in the classical perturbation method. Interestingly, This study revealed that the relative error percentage in the second-order approximate analytical period is less than 0.042% for the whole parameter values. In addition, we compared this analytical solution with the Newton– Harmonic Balancing Approach. Results indicate that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems. Utter simplicity of the solution procedure confirms that this method can be easily extended to other kinds of nonlinear evolution equations

Asymmetric collapse by dissolution or melting in a uniform flow

An advection--diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow is presented. By taking advantage of the conformal invariance of the model, a numerical method is introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of a variety of dissolving objects are shown, which shrink and then collapse to a single point in finite time. The simulations reveal a surprising exact relationship whereby the collapse point is the root of a non-analytic function given in terms of the flow velocity and the Laurent series coefficients describing the initial shape. This result is subsequently derived using residue calculus. The structure of the non-analytic function is examined for three different test cases, and a practical approach to determine the collapse point using a generalized Newton--Raphson root-finding algorithm is outlined. These examples also illustrate the possibility that the model breaks down in finite time prior to complete collapse, due to a topological singularity, as the dissolving boundary overlaps itself rather than breaking up into multiple domains (analogous to droplet pinch-off in fluid mechanics). In summary, the model raises fundamental mathematical questions about broken symmetries in finite-time singularities of both continuous and stochastic dynamical systems.Comment: 20 pages, 11 figure

Un algoritmo caótico híbrido mejorado de búsqueda por coordenadas cíclicas y técnicas de gradiente

En este artículo se presenta un algoritmo híbrido caótico que usa una búsqueda cíclica mejorada a lo largo de cada eje y el algoritmo BFGS para optimizar funciones no lineales. El método propuesto es una poderosa técnica de optimización; esto es demostrado al optimizar cuatro funciones benchmark con 30 dimensiones. La metodología propuesta es capaz de converger a una mejor solución, y más rápido que el algoritmo tradicional de optimización basado en caos, y otras técnicas competitivas