scholarly journals On the sum of positive divisors functions

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Radek Erban ◽  
Robert A. Van Gorder
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
System Analysis ◽  
Divisor Functions ◽  
Recursive Formulas ◽  
Solution Techniques

AbstractProperties of divisor functions $$\sigma _k(n)$$ σ k ( n ) , defined as sums of k-th powers of all divisors of n, are studied through the analysis of Ramanujan’s differential equations. This system of three differential equations is singular at $$x=0$$ x = 0 . Solution techniques suitable to tackle this singularity are developed and the problem is transformed into an analysis of a dynamical system. Number theoretical consequences of the presented dynamical system analysis are then discussed, including recursive formulas for divisor functions.

scholarly journals Dynamical system analysis of FLRW models with Modified Chaplygin gas

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Ali Osman Yılmaz ◽  
Ertan Güdekli
Keyword(s):  
Dynamical System ◽  
Equation Of State ◽  
Energy Density ◽  
System Analysis ◽  
Chaplygin Gas ◽  
Modified Chaplygin Gas ◽  
State Spaces ◽  
The Universe ◽  
Matter Energy ◽  
Far Future

AbstractWe investigate Friedmann–Lamaitre–Robertson–Walker (FLRW) models with modified Chaplygin gas and cosmological constant, using dynamical system methods. We assume $$p=(\gamma -1)\mu -\dfrac{A}{\mu ^\alpha }$$ p = ( γ - 1 ) μ - A μ α as equation of state where $$\mu$$ μ is the matter-energy density, p is the pressure, $$\alpha$$ α is a parameter which can take on values $$0<\alpha \le 1$$ 0 < α ≤ 1 as well as A and $$\gamma$$ γ are positive constants. We draw the state spaces and analyze the nature of the singularity at the beginning, as well as the fate of the universe in the far future. In particular, we address the question whether there is a solution which is stable for all the cases.

scholarly journals Trunk control during standing reach: A dynamical system analysis of movement strategies in patients with mechanical low back pain

2009 ◽  
Vol 29 (3) ◽  
pp. 370-376 ◽  
Author(s):  
Sheri P. Silfies ◽  
Anand Bhattacharya ◽  
Scott Biely ◽  
Sue S. Smith ◽  
Simon Giszter
Keyword(s):  
Dynamical System ◽  
Low Back Pain ◽  
Back Pain ◽  
System Analysis ◽  
Low Back ◽  
Trunk Control ◽  
Movement Strategies

scholarly journals Dynamical system analysis of three-form field dark energy model with baryonic matter

2020 ◽  
Vol 80 (9) ◽  
Author(s):  
Soumya Chakraborty ◽  
Sudip Mishra ◽  
Subenoy Chakraborty
Keyword(s):  
Dynamical System ◽  
Dark Energy ◽  
Cold Dark Matter ◽  
System Analysis ◽  
Evolution Equations ◽  
Matter Field ◽  
Baryonic Matter ◽  
Form Field ◽  
The Stability ◽  
Manifold Theory

AbstractA cosmological model having matter field as (non) interacting dark energy (DE) and baryonic matter and minimally coupled to gravity is considered in the present work with flat FLRW space time. The DE is chosen in the form of a three-form field while radiation and dust (i.e; cold dark matter) are the baryonic part. The cosmic evolution is studied through dynamical system analysis of the autonomous system so formed from the evolution equations by suitable choice of the dimensionless variables. The stability of the non-hyperbolic critical points are examined by Center manifold theory and possible bifurcation scenarios have been examined.

Computer Aided Dynamical System Analysis and Control for Undergraduate Courses

1977 ◽  
Vol 10 (2) ◽  
pp. 44-50 ◽  
Author(s):  
C. McCorkell ◽  
N. Wilson
Keyword(s):  
Dynamical System ◽  
Northern Ireland ◽  
System Analysis ◽  
Computer Usage ◽  
Synthesis Techniques ◽  
Initial Stage ◽  
Control Options ◽  
Computer Aided ◽  
Computational Requirement ◽  
And Control

Dynamical system analysis is included in undergraduate courses in the Northern Ireland Polytechnic, as part of a presentation of general engineering methodology and more particularly, accompanied by synthesis techniques, in control options at final year honours level. Such is the extent of the computational requirement, necessary for a non-trivial treatment, that steps have been taken to introduce computer usage where possible. Included is information on the initial stage of a project undertaken to provide for the computational needs of undergraduates involved in dynamical problems in the laboratory.

scholarly journals Dynamical system analysis of a five-dimensional cosmological model

2018 ◽  
Vol 363 (10) ◽  
Author(s):  
A. Savaş Arapoğlu ◽  
Ezgi Yalçınkaya ◽  
A. Emrah Yükselci
Keyword(s):  
Dynamical System ◽  
Cosmological Model ◽  
System Analysis

scholarly journals On asymptotically stable sets

1970 ◽  
Vol 17 (2) ◽  
pp. 181-186 ◽  
Author(s):  
D. Desbrow
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Metric Space ◽  
Physical System ◽  
Asymptotically Stable ◽  
Stable Sets ◽  
Compact Closure ◽  
First Order ◽  
Closed Sets ◽  
The Future

In this paper we study closed sets having a neighbourhood with compact closure which are positively asymptotically stable under a flow on a metric space X. For an understanding of this and the rest of the introduction it is sufficient for the reader to have in mind as an example of a flow a system of first order, autonomous ordinary differential equations describing mathematically a time-independent physical system; in short a dynamical system. In a flow a set M is positively stable if the trajectories through all points sufficiently close to M remain in the future in a given neighbourhood of M. The set M is positively asymptotically stable if it is positively stable and, in addition, trajectories through all points of some neighbourhood of M approach M in the future.

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