scholarly journals APPLICATION OF CENTRE MANIFOLD THEORY IN GENERALIZED MAXWELL-BLOCH LASER EQUATIONS

1996 ◽  
Vol 45 (1) ◽  
pp. 46
Author(s):  
WANG KAI-GE ◽  
WANG YU-LONG ◽  
SUN YIN-GUAN
Keyword(s):  
Centre Manifold ◽  
Manifold Theory

A centre theorem for two-dimensional complex holomorphic systems and its generalization

1995 ◽  
Vol 450 (1939) ◽  
pp. 225-232 ◽  
Keyword(s):  
Singular Point ◽  
Equilibrium Point ◽  
Point Theory ◽  
Two Dimensional ◽  
Imaginary Eigenvalue ◽  
Centre Manifold ◽  
Manifold Theory

We consider a two-dimensional complex holomorphic system. In particular, we use the centre manifold theory together with the singular point theory of C. H. Briot & J. C. Bouquet ( J . Éc . imp . Polyt . 21, 133 (1856)) to establish a centre theorem concerning the behaviour of the phase paths of the system in the neighbourhood of an equilibrium point having a single purely imaginary eigenvalue. An extended centre theorem is established for the corresponding N -dimensional complex holomorphic system ( N ≥ 3).

scholarly journals Controlling the Dynamical Spread of Coronavirus Disease (COVID-19) in a Population

2020 ◽  
Author(s):  
Akanni John Olajide
Keyword(s):  
Recovery Rate ◽  
Lyapunov Functions ◽  
Progression Rate ◽  
Contact Rate ◽  
Equilibrium Solutions ◽  
Centre Manifold ◽  
Effective Contact ◽  
Manifold Theory ◽  
Theoretical Results ◽  
Positivity And Boundedness

In the paper, a model governed by a system of ordinary differential equations was considered; the whole population was divided into Susceptible individuals (S), Exposed individuals (E), Infected individuals (I), Quarantined individuals (Q) and Recovered individuals (R). The well-posedness of the model was investigated by the theory of positivity and boundedness. Analytically, the equilibrium solutions were examined. A key threshold which measures the potential spread of the Coronavirus in the population is derived using the next generation method. Bifurcation analysis and global stability of the model were carried out using centre manifold theory and Lyapunov functions respectively. The effects of some parameters such as Progression rate of exposed class to infectious class, Effective contact rate, Modification parameter, Quarantine rate of infectious class, Recovery rate of infectious class and Recovery rate of quarantined class on R0 were explored through sensitivity analysis. Numerical simulations were carried out to support the theoretical results, to reduce the burden of COVID 19 disease in the population and significant in the spread of it in the population.

scholarly journals Boundary conditions for approximate differential equations

1992 ◽  
Vol 34 (1) ◽  
pp. 54-80 ◽  
Author(s):  
A. J. Roberts
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Initial Conditions ◽  
Higher Order ◽  
Centre Manifold ◽  
Systematic Fashion ◽  
Model Equations ◽  
Manifold Theory ◽  
Finite Domains ◽  
Correct Boundary Conditions

AbstractA large number of mathematical models are expressed as differential equations. Such models are often derived through a slowly-varying approximation under the assumption that the domain of interest is arbitrarily large; however, typical solutions and the physical problem of interest possess finite domains. The issue is: what are the correct boundary conditions to be used at the edge of the domain for such model equations? Centre manifold theory [24] and its generalisations may be used to derive these sorts of approximations, and higher-order refinements, in an appealing and systematic fashion. Furthermore, the centre manifold approach permits the derivation of appropriate initial conditions and forcing for the models [25, 7]. Here I show how to derive asymptotically-correct boundary conditions for models which are based on the slowly-varying approximation. The dominant terms in the boundary conditions typically agree with those obtained through physical arguments. However, refined models of higher order require subtle corrections to the previously-deduced boundary conditions, and also require the provision of additional boundary conditions to form a complete model.

scholarly journals The application of centre-manifold theory to the evolution of system which vary slowly in space

1988 ◽  
Vol 29 (4) ◽  
pp. 480-500 ◽  
Author(s):  
A. J. Roberts
Keyword(s):  
Multiple Scales ◽  
Limited Range ◽  
Future Research ◽  
Centre Manifold ◽  
Space And Time ◽  
Physical Problems ◽  
Formal Procedure ◽  
Slow Evolution ◽  
Manifold Theory ◽  
Correction Terms

AbstractIn many physical problems, the system tends quickly to a particular structure, which then evolves relatively slowly in space and time. Various methods exist to derive equations describing the slow evolution of the particular structure; for example, the method of multiple scales. However, the resulting equations are typically valid only for a limited range of the parameters. In order to extend the range of validity and to improve the accuracy, correction terms must be found for the equations. Here we describe a procedure, inspired by centre-manifold theory, which provides a systematic approach to calculating a sequence of successively more accurate approximations to the evolution of the principal structure in space and time.The formal procedure described here raises a number of questions for future research. For example: what sort of error bounds can be obtained, do the approximations converge or are they strictly asymptotic, and what sort of boundary conditions are appropriate in a given problem?

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