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’.

The Oxford Handbook of Generality in Mathematics and the Sciences

2017 ◽  
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Mathematical Physics ◽  
Dynamical Systems Theory ◽  
Evolutionary Strategies ◽  
René Descartes ◽  
History Of ◽  
Recurrence Theorem ◽  
Different Levels

This handbook examines how actors have valued generality in mathematics and the sciences and how they worked with specific types of “general” entities, procedures, and arguments. It argues that actors have shaped these various types of generality, mainly by introducing specific terminologies to distinguish between different levels or forms of generality, as well as designing means to work with them, or to work in relation to them. The book is organized into three parts. Part I deals with the meaning and value of generality, and more specifically the value of generality in Michel Chasles’s historiography of geometry and generality in Gottfried Leibniz’s mathematics. Part II focuses on statements and concepts that make up the general, covering topics such as Henri Poincaré’s work on the recurrence theorem and the role of genericity in the history of dynamical systems theory. Part III explores the practices of generality, including the dispute over tangents between René Descartes and Pierre de Fermat, generality in James Clerk Maxwell’s theory of electromagnetism, and practices of generalization in mathematical physics, biology, and evolutionary strategies.

scholarly journals Some elements for a history of the dynamical systems theory

2021 ◽  
Vol 31 (5) ◽  
pp. 053110
Author(s):  
Christophe Letellier ◽  
Ralph Abraham ◽  
Dima L. Shepelyansky ◽  
Otto E. Rössler ◽  
Philip Holmes ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Dynamical Systems Theory ◽  
History Of

scholarly journals On Smoother Attributions using Neural Stochastic Differential Equations

2021 ◽  
Author(s):  
Sumit Jha ◽  
Rickard Ewetz ◽  
Alvaro Velasquez ◽  
Susmit Jha
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Systems Theory ◽  
Dynamical Systems Theory ◽  
Small Perturbations

Several methods have recently been developed for computing attributions of a neural network's prediction over the input features. However, these existing approaches for computing attributions are noisy and not robust to small perturbations of the input. This paper uses the recently identified connection between dynamical systems and residual neural networks to show that the attributions computed over neural stochastic differential equations (SDEs) are less noisy, visually sharper, and quantitatively more robust. Using dynamical systems theory, we theoretically analyze the robustness of these attributions. We also experimentally demonstrate the efficacy of our approach in providing smoother, visually sharper and quantitatively robust attributions by computing attributions for ImageNet images using ResNet-50, WideResNet-101 models and ResNeXt-101 models.

Selection, Optimization, Compensation, and Equilibrium Dynamics

GeroPsych ◽  
2013 ◽  
Vol 26 (1) ◽  
pp. 61-73 ◽  
Author(s):  
Steven M. Boker
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Human Development ◽  
Systems Theory ◽  
Later Life ◽  
Dynamical Systems Theory ◽  
Intraindividual Variability ◽  
Interindividual Differences ◽  
Equilibrium Dynamics ◽  
Current Article

One of the major theoretic frameworks through which human development is studied is a process-oriented model involving selection, optimization, and compensation. These three processes each provide accounts for methods by which gains are maximized and losses minimized throughout the lifespan, in particular during later life. These processes can be cast within the framework of dynamical systems theory and then modeled using differential equations. The current article will review basic tenets of selection, optimization, and compensation while introducing language and concepts from dynamical systems. Four categories of interindividual differences and intraindividual variability in dynamics are then described and discussed in the context of selection, optimization, and compensation.

Cartographic production in Russia: main stages and trends

2020 ◽  
Vol 963 (9) ◽  
pp. 30-43
Author(s):  
M.Yu. Orlov
Keyword(s):  
Chronological Order ◽  
Xviii Century ◽  
New Methods ◽  
Current State ◽  
Factors Influencing ◽  
The Future ◽  
History Of ◽  
State Enterprises ◽  
Further Development

Studying the current state of cartography and ways of further developing the industry, the role of the map in the future of the society, new methods of promoting cartographic products is impossible without a deep scientific analyzing all the paths, events and factors influencing its formation and development throughout all the historic steps of cartographic production in Russia. In the article, the history of cartographic production in Russia is considered together with the development of private, state and military cartography, since, despite some differences, they have a common technical, technological and production basis. The author describes the stages of originating, formation and growth of industrial cartographic production from the beginning of the XVIII century until now. The connection between the change of political formations and technological structures with the mentioned stages of maps and atlases production is considered. Each stage is studied in detail, a step-by-step analysis was carried out, and the characteristics of each stage are described. All the events and facts are given in chronological order, highlighting especially significant moments influencing the evolution of cartographic production. The data on the volumes of printing and sales of atlases and maps by commercial and state enterprises are presented. The main trends and lines of further development of cartographic production in Russia are studied.

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