Dynamical Systems Theory, Delay Differential Equations

In this paper, we compare three types of dynamical systems used to describe tumor growth. These systems are defined as solutions to three delay differential equations: the logistic, the Gompertz and the Greenspan types. We present analysis of these systems and compare with experimental data for Ehrlich Ascites tumor in mice.

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On Smoother Attributions using Neural Stochastic Differential Equations

Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2021/73 ◽

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.

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Search for Periodic Orbits in Delay Differential Equations

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414500849 ◽

2014 ◽

Vol 24 (06) ◽

pp. 1450084 ◽

Cited By ~ 2

Author(s):

Romina Cobiaga ◽

Walter Reartes

Keyword(s):

Dynamical Systems ◽

Differential Equations ◽

Periodic Orbits ◽

Heuristic Search ◽

Delay Differential Equations ◽

Anharmonic Oscillator ◽

Analysis Method ◽

Van Der Pol ◽

Homotopy Analysis ◽

Delay Differential

In a previous paper, we developed a new way to apply the Homotopy Analysis Method (HAM) in the search for periodic orbits in dynamical systems modeled by ordinary differential equations. This method differs from the original in the heuristic search of the frequencies of the cycles. In this paper, we show that the method can be extended to the search for periodic orbits in delay differential equations. Herein, this methodology is applied twice, firstly in an equation of van der Pol type and secondly in an anharmonic oscillator, both systems with a delayed feedback.

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The Qualitative Analysis of Differential Equations and the Development of Dynamical Systems Theory

Brazilian Studies in Philosophy and History of Science - Boston Studies in the Philosophy of Science ◽

10.1007/978-90-481-9422-3_24 ◽

2011 ◽

pp. 313-324

Author(s):

Tatiana Roque

Keyword(s):

Dynamical Systems ◽

Differential Equations ◽

Qualitative Analysis ◽

Systems Theory ◽

Dynamical Systems Theory

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The role of genericity in the history of dynamical systems theory

10.1093/oxfordhb/9780198777267.013.10 ◽

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’.

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Delay Differential Equations and Dynamical Systems

10.1007/bfb0083474 ◽

1991 ◽

Cited By ~ 7

Keyword(s):

Dynamical Systems ◽

Differential Equations ◽

Delay Differential Equations ◽

Delay Differential

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Selection, Optimization, Compensation, and Equilibrium Dynamics

GeroPsych ◽

10.1024/1662-9647/a000081 ◽

2013 ◽

Vol 26 (1) ◽

pp. 61-73 ◽

Cited By ~ 10

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.

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