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Speedpro SpeedPro Series Online. Read PDF Leadership 2. Reflections of the Incarnation in Art and Music Online. The Lost Paintings Series Online. Juicing Recipes for Weight Loss Online. The Cinema of Akira Kurosawa Online. Read Skills for Success: In this course, we learn the origin of the Fibonacci numbers and the golden ratio, and derive a formula to compute any Fibonacci number from powers of the golden ratio. This course covers mathematical topics in college algebra, with an emphasis on functions. The course is designed to help prepare students to enroll for a first semester course in single variable calculus.

This course teaches a calculus that enables precise quantitative predictions of large combinatorial structures. In addition, this course covers generating functions and real asymptotics and then introduces the symbolic method in the context of applications in the analysis of algorithms and basic structures such as permutations, trees, strings, words, and mappings. Calculus is about the very large, the very small, and how things change. The surprise is that something seemingly so abstract ends up explaining the real world.

Calculus plays a starring role in the biological, physical, and social sciences. By focusing outside of the classroom, we will see examples of calculus appearing in daily life. The goal of the course is to explain the fundamental ideas of linear algebra and how to use them to find easy solutions of hard problems.

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This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable. One of this theory's main characteristics lies in its interdisciplinary nature Strogatz, because it is not a pure mathematics theory but a mathematical theory in constant interaction with many disciplines from physical sciences physics, fluid mechanics , engineering sciences au-tomatic regulation and life sciences population dynamics, epidemiology Aubin and Dalmedico, Indeed, a dynamic system is a mathematic model which is difficult or impossible to solve analytically in most cases, due to the pertinent equations' nonlinearity; traditional analytical methods cannot be easily applied or they do not exist.

Computer-based simulation is a way to overcome such difficulties Cellier and Kofman, as it allows numerical approximation of solutions. However, solving a mathematical model numerically requires knowledge of programming languages and numerical algorithms. The DSamala toolbox was thus developed to simulate and numerically analyse discrete, continuous, stochastic dynamic systems.

Handling this software does not require a user to be an expert in programming languages or numerical algorithms. DSamala toolbox has a very simple, easily-used graphical user interface GUI , a set of functions through which a user can interact with a code, and it enables obtaining efficient, effective results when studying a dynamic system.

The DSamala toolbox enables iterating functions, analysing orbits, determining and classifying fixed and periodic points, plotting time series, bifurcation diagrams, providing solutions for a system, phase portraits and vector fields, identifying and classifying equilibrium points, calculating Lyapunov exponents, determining Hamiltonian functions and analysing stability i.

Lyapunov's direct method.

This tool includes some classic dynamic systems involving chaotic behaviour, such as Mandelbrot set Mandelbrot, , logistic equation Holmgren, and the Lorenz system Lorenz, , which would be very costly in terms of time and effort when only performing an analytical study without the aid of a computer, due to behaviour complexity. A dynamic system is one whose state changes as time elapses Arrowsmith and Place, It is represented by a mathematical model Campos and Isaza, which can be determined by difference equations, ordinary differential equations and stochastic differential equations.

Such mathematical models are usually nonlinear and it is well-known that a nonlinear dynamical model can involve very complicated behaviour Perko, The Runge Kutta 4th order numerical method Mathews and Fink, is used to obtain a phase portrait and system solutions in the numerical simulation of such systems; Ramasubramanian and Sriram's method Ramasubramanian and Sriram, is used for calculating Lyapunov exponents. A stochastic differential equation's general expression is given by equation 3 , Honerkamp, :. Very few computational tools are currently available for simulating and analysing dynamic systems; some tools are complex in their use, requiring a commercial licence and are mostly developed just for a particular application.

Tools of this type would be the graphic bifurcation identifier GBI Tovar, , using MATLAB, which locates and identifies local bifurcations in discrete and continuous dynamic systems, autonomous and multiparametric dynamic systems and multidi-mensional dynamic systems. Its graphical interface is complicated for a user lacking MATLAB experience or without very detailed help on how to use the software. Java PHASER Glick and Kocak, simulates discrete and continuous dynamic systems using real variables; its disadvantage lies in it being commercial software Phaser Scientific Software, LLC whose code is not available to a user and obviously is not free software last released It has basic functions for simulating a system but it cannot fully analyse a dynamic system; it is found in versions for Java and MATLAB.

It enables computational analysis of steady-state solu-tion continuity and stability. Compared to the aforementioned methods, DSamala toolbox is a complete tool because it has a variety of features, like its graphical user interface, which is very simple and easy to use and has a set of functions toolbox where the user can control the numerical code developed, thereby allowing better analysis and simulation of a dynamic system. DSamala toolbox can be used from the functions contained in the toolbox if a user has knowledge of MATLAB; if not, then through the graphical interface.

This tool is complemented by a graphical interface manual and toolbox function reference manual. The DSamala toolbox thus represents a new alternative for software for teaching and research in the area of dynamic systems. The overall model fits dynamic systems which are broken down into three models: discrete, continuous and stochastic systems. Each has a list of features which are characteristics of the software Figure 1 :. Graph iteration displays a discrete system's behaviour and is ana-lysed by returning fixed points and classifying them as being repeller, attractor or not hyperbolic.

Time series graph a system's time series and returns values for such series. This graph shows whether a system has periodic points and their stability. Function iteration leads to obtaining fixed points for a selected period and the graph of their behaviour.

A bifurcation diagram shows a graph observing changes in a system's dynamics i. Newton's method is a basic tool for finding fixed points, graphically showing the value found as the method's convergence. Analysis with parameters analyses a parameterised system's dynamics. It establishes fixed points for the system in terms of parameters and may give these parameters values for generating graphical behaviour. Lyapunov exponents analyse system stability, thereby determining whether a system is chaotic; bifurcation points are obtained for a parameterised family.

Applications interact with some default examples logistic equation, Julia set and Mandelbrot set where a user enters the data, simulation is performed and system behaviour observed. Behaviour involves simulating a system and phase portrait, solution curves, vector field, nullclinales and points of equilibrium, along with their respective classification. Bifurcation diagram shows changes in continuous system dynamics, parameterised as parameter value varies.

## MIGUEL DE GUZMÁN

Hamiltonian systems check whether a system is conservative; if so, the Hamiltonian function is obtained and its graph, together with contour lines. Lyapunov exponents give exact values, unlike discrete systems which only generate a graph of the evolution of these exponents as time elapses. Lyapunov function allows analysing system stability; it is essential when this cannot be done from behaviour functionality, for example, in the case of zero eigenvalues. Analysis with parameters allows analysing parameterised systems, indicating points of equilibrium and the type of behaviour.

Application interacts with some default examples Lorenz system or harmonic oscillator where a user can enter data, simulate and observe system behaviour. Graphic functions graph a stochastic function and can analyse systems in one and two dimensions. Autonomous one-dimensional systems graph this type of system's behaviour with constant deterministic functions.

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One-dimensional dynamic systems graph one-dimensional system's solution having any deterministic function. Two-dimensional systems plot 2D systems' phase portrait and solution curves.

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Applications interact with some default examples Lotka-Volterra stochastic system and SI stochastic epidemic model where a user can enter data, simulate and observe behav-iour. MATLAB version Ra must be installed or any higher version for its operation with the image processing toolbox and symbolic math toolbox. Figure 2 shows the general running of the DSamala toolbox graphical user interface; the type of dynamic system can be chosen discrete, continuous or stochastic.

The required functionality, data is then entered in the current application and output data displayed a graph or data analysis of the selected system. This can be done as often as desired until is all active windows are closed.