Nonlinear Waves, Dynamical Systems and Other Applied Mathematics Programs

1991 ◽  
Author(s):  
Jr. Miller ◽  
Friedman Willard ◽  
Avner
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Waves ◽  
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.

SOME COMMON EXPRESSIONS AND NEW BIFURCATION PHENOMENA FOR NONLINEAR WAVES IN A GENERALIZED mKdV EQUATION

2013 ◽  
Vol 23 (03) ◽  
pp. 1330007 ◽  
Author(s):  
RUI LIU ◽  
WEIFANG YAN
Keyword(s):  
Dynamical Systems ◽  
Solitary Waves ◽  
Nonlinear Waves ◽  
Blow Up ◽  
Mkdv Equation ◽  
Bifurcation Method ◽  
The Third ◽  
Bifurcation Phenomena ◽  
Kink Waves ◽  
The Common

Using the bifurcation method of dynamical systems, we study nonlinear waves in the generalized mKdV equation ut + a(1 + bu2)u2ux + uxxx = 0. (i) We obtain four types of new expressions. The first type is composed of four common expressions of the symmetric solitary waves, the kink waves and the blow-up waves. The second type includes four common expressions of the anti-symmetric solitary waves, the kink waves and the blow-up waves. The third type is made of two trigonometric expressions of periodic-blow-up waves. The fourth type is composed of two fractional expressions of 1-blow-up waves. (ii) We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) We reveal two kinds of new bifurcation phenomena. The first phenomenon is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, blow-up waves, tall-kink waves and anti-symmetric solitary waves. The second phenomenon is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves. We also show that the common expressions include many results given by pioneers.

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.

Bifurcations of Nonlinear Waves in the Generalized KdV–mKdV-Like Equation

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Yiren Chen ◽  
Wensheng Chen
Keyword(s):  
Dynamical Systems ◽  
Solitary Waves ◽  
Nonlinear Waves ◽  
Blow Up ◽  
Analytic Method ◽  
Periodic Waves ◽  
Bifurcation Phenomena ◽  
Kink Waves ◽  
Explicit Expressions

Using bifurcation analytic method of dynamical systems, we investigate the nonlinear waves and their bifurcations of the generalized KdV–mKdV-like equation. We obtain the following results : (i) Three types of new explicit expressions of nonlinear waves are obtained. They are trigonometric expressions, exp-function expressions, and hyperbolic expressions. (ii) Under different parameteric conditions, these expressions represent different waves, such as solitary waves, kink waves, 1-blow-up waves, 2-blow-up waves, smooth periodic waves and periodic blow-up waves. (iii) Two kinds of new interesting bifurcation phenomena are revealed. The first phenomenon is that the single-sided periodic blow-up waves can bifurcate from double-sided periodic blow-up waves. The second phenomenon is that the double-sided 1-blow-up waves can bifurcate from 2-blow-up waves. Furthermore, we show that the new expressions encompass many existing results.

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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