Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

2013 ◽  
Keyword(s):  
Dynamical Systems ◽  
Fractal Geometry ◽  
Applied Mathematics

scholarly journals Graph-facilitated resonant mode counting in stochastic interaction networks

2017 ◽  
Vol 14 (137) ◽  
pp. 20170447 ◽  
Author(s):  
Michael F. Adamer ◽  
Thomas E. Woolley ◽  
Heather A. Harrington
Keyword(s):  
Dynamical Systems ◽  
Applied Mathematics ◽  
Direct Calculation ◽  
Resonant Mode ◽  
Interaction Networks ◽  
Deterministic System ◽  
Stochastic Dynamical Systems ◽  
Graph Theoretic ◽  
Resonant Modes ◽  
Parameter Ranges

Oscillations in dynamical systems are widely reported in multiple branches of applied mathematics. Critically, even a non-oscillatory deterministic system can produce cyclic trajectories when it is in a low copy number, stochastic regime. Common methods of finding parameter ranges for stochastically driven resonances, such as direct calculation, are cumbersome for any but the smallest networks. In this paper, we provide a systematic framework to efficiently determine the number of resonant modes and parameter ranges for stochastic oscillations relying on real root counting algorithms and graph theoretic methods. We argue that stochastic resonance is a network property by showing that resonant modes only depend on the squared Jacobian matrix J 2 , unlike deterministic oscillations which are determined by J . By using graph theoretic tools, analysis of stochastic behaviour for larger interaction networks is simplified and stochastic dynamical systems with multiple resonant modes can be identified easily.

On State-Space Modeling and Signal Localization in Dynamical Systems

2021 ◽  
Vol 2 (1) ◽  
Author(s):  
Asok Ray
Keyword(s):  
Dynamical Systems ◽  
State Space ◽  
Applied Mathematics ◽  
System Model ◽  
Theoretical Physics ◽  
Vector Spaces ◽  
Real Field ◽  
Finite Dimensional ◽  
Space Modeling ◽  
And Control

Abstract This letter focuses on two topics in engineering analysis, which are (1) degree-of-freedom (DOF) in modeling of dynamical systems and (2) simultaneous time and frequency localization of signals. These issues are explained from the perspectives of decision and control by making use of concepts from applied mathematics and theoretical physics. Specifically, a new definition is proposed to clarify the notion of “DOF,” which is consistent with the dimension of the state space of the dynamical system model. Relevant examples are presented on (finite-dimensional) vector spaces over the real field R and/or the complex field C.

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.

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