4. Do you know the way to solve equations? Spinning tops and chaotic rabbits

2018 ◽  
pp. 48-69
Author(s):  
Alain Goriely
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Numerical Solution ◽  
Applied Mathematics ◽  
Scientific Models ◽  
Henri Poincaré ◽  
Principal Tool ◽  
The Difference ◽  
The Way

In applied mathematics it is of the greatest importance to solve equations. These solutions provide information on key quantities and allow us to give specific answers to scientific problems. ‘Do you know the way to solve equations? Spinning tops and chaotic rabbits’ describes the ways to solve equations and differential equations, outlining the key work of mathematicians Sofia Kovalevskaya, Pierre-Simon Laplace, Paul Painlevé, and Henri Poincaré, whose discovery led to the birth of the theory of chaos and dynamical systems. The difference between an exact and a numerical solution is also explained. Numerical analysis has become the principal tool for querying and solving scientific models.

THE NUMERICAL SOLUTION OF NONLINEAR STOCHASTIC DYNAMICAL SYSTEMS: A BRIEF INTRODUCTION

1991 ◽  
Vol 01 (02) ◽  
pp. 277-286 ◽  
Author(s):  
P. E. KLOEDEN ◽  
E. PLATEN ◽  
H. SCHURZ
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Numerical Solution ◽  
Stochastic Differential Equations ◽  
Taylor Expansion ◽  
Rapid Development ◽  
Stochastic Calculus ◽  
Stochastic Dynamical Systems ◽  
The Subject

The numerical analysis of stochastic differential equations, currently undergoing rapid development, differs significantly from its deterministic counterpart due to the peculiarities of stochastic calculus. This article presents a brief, pedagogical introduction to the subject from the perspective of stochastic dynamical systems. The key tool is the stochastic Taylor expansion. Strong, pathwise approximations are distinguished from weak, functional approximations, and their role in stability with Lyapunov exponents and stiffness is discussed.

scholarly journals ABOUT SOME SUPPLEMENTARY POSSIBILITY FOR NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS

2016 ◽  
Vol 11 (1) ◽  
pp. 75-78
Author(s):  
A. B. Chaadaev
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Numerical Solution ◽  
Third Order ◽  
Laplace Operators ◽  
Partial Differential ◽  
The Difference

A substitution of an non-homogeneous term and of a differential operator by the difference of Laplace operators in the direct co-ordinate system and in the turned one in the partial differential equations of first, second and third order is proposed. The numerical solution obtained by solving the substituting equation corresponds to the exact solution of the initial equations.

The role of genericity in the history of dynamical systems theory

2017 ◽  
Author(s):  
Tatiana Roque
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Systems Theory ◽  
Singularity Theory ◽  
Dynamical Systems Theory ◽  
Henri Poincaré ◽  
New Methods ◽  
History Of ◽  
Equation Différentielle

This article examines the role of genericity in the development of dynamical systems theory. In his memoir ‘Sur les courbes définies par une équation différentielle’, published in four parts between 1881 and 1886, Henri Poincaré studied the behavior of curves that are solutions for certain types of differential equations. He successfully classified them by focusing on singular points, described the trajectories’ behavior in important particular cases and provided new methods that proved to be extremely useful. This article begins with a discussion of singularity theory and its influence on the first definitions of genericity, along with the application of the notions of structural stability and genericity to understand dynamical systems. It also analyzes the Smale conjecture and how it was proven wrong and concludes with an overview of changes in the definitions of genericity meant to describe the ‘dark realm of dynamics’.

scholarly journals Simulating deformable objects for computer animation: A numerical perspective

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Uri M. Ascher ◽  
Egor Larionov ◽  
Seung Heon Sheen ◽  
Dinesh K. Pai
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Computer Animation ◽  
Applied Mathematics ◽  
A Priori ◽  
Numerical Algorithms ◽  
Complex Geometries ◽  
Deformable Objects ◽  
Highly Oscillatory ◽  
The Mathematical Model

<p style='text-indent:20px;'>We examine a variety of numerical methods that arise when considering dynamical systems in the context of physics-based simulations of deformable objects. Such problems arise in various applications, including animation, robotics, control and fabrication. The goals and merits of suitable numerical algorithms for these applications are different from those of typical numerical analysis research in dynamical systems. Here the mathematical model is not fixed <i>a priori</i> but must be adjusted as necessary to capture the desired behaviour, with an emphasis on effectively producing lively animations of objects with complex geometries. Results are often judged by how realistic they appear to observers (by the "eye-norm") as well as by the efficacy of the numerical procedures employed. And yet, we show that with an adjusted view numerical analysis and applied mathematics can contribute significantly to the development of appropriate methods and their analysis in a variety of areas including finite element methods, stiff and highly oscillatory ODEs, model reduction, and constrained optimization.</p>

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