# Mathematical Modelling of Natural Phenomena (Math Model Nat Phenom)

## Journal description

The Mathematical Modelling of Natural Phenomena (MMNP) is an international research journal, which publishes top-level original and review papers, short communications and proceedings on mathematical modelling in biology, medicine, chemistry, physics, and other areas.

## Current impact factor: 0.81

## Impact Factor Rankings

2016 Impact Factor | Available summer 2017 |
---|---|

2014 / 2015 Impact Factor | 0.813 |

2013 Impact Factor | 0.725 |

2012 Impact Factor | 0.558 |

2011 Impact Factor | 0.633 |

2010 Impact Factor | 0.714 |

## Impact factor over time

Impact factor

Year

## Additional details

5-year impact | 0.81 |
---|---|

Cited half-life | 3.70 |

Immediacy index | 0.74 |

Eigenfactor | 0.00 |

Article influence | 0.42 |

Website | Mathematical Modelling of Natural Phenomena website |

ISSN | 1760-6101 |

## Publisher details

- Pre-print
- Author can archive a pre-print version

- Post-print
- Author can archive a post-print version

- Conditions
- On author's personal website or institutional website or OAI compliant website
- Some journals require an embargo for deposit in funder's designated repositories (see journal)
- Publisher's version/PDF may be used (see journal)
- Must link to publisher version
- Publisher copyright and source must be acknowledged
- Non-commercial

- Classificationgreen

## Publications in this journal

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**ABSTRACT:**This paper reviews earlier results of the author regarding the hydrodynamic limit problem for the Boltzmann equation. In particular the key points are that the work of Gorban&Karlin suggests that Korteweg hydrodynamics is implied by the Boltzmann equation and this correspondence is validated by comparison with experimental, analytical, and numerical results. But if the correspondence is indeed valid then passage to limiting compressible Euler equations will not be generally possible after any time for which the the Euler system fails to have smooth solutions. Dedicated to my friend Alexander Gorban on the occasion of his sixtieth birthday - [Show abstract] [Hide abstract]

**ABSTRACT:**Differential equation models of chemical or biochemical systems usually display multiple, widely varying time scales, i.e. they are stiff. After the decay of transients, trajectories of these systems approach low-dimensional invariant manifolds on which the eventual attractor (an equilibrium point in a closed system) is approached, and in which this attractor is embedded. Computing one of these slow invariant manifolds (SIMs) results in a reduced model of dimension equal to the dimension of the SIM. Another approach to model reduction involves lumping, the formulation of a reduced set of variables that combine the original model variables and in terms of which the reduced model is framed. In this study, we combine lumping with a constructive method for SIMs based on the iterative solution of the invariance equation. We illustrate these methods using a simple model of a linear metabolic pathway, and a model for hydrogen oxidation. The former is treated with a linear lumping function, while a nonlinear lumping function based on a Lyapunov function is used in the latter. -
##### Article: Flow Structure Identification for Nonlinear Dynamical Systems via Finite-Time Lyapunov Analysis

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**ABSTRACT:**Identifying and characterizing geometric structure in the flow of a nonlinear dynamical system can facilitate understanding, model simplification, and solution approximation. The approach addressed in this paper uses information from finite-time Lyapunov exponents and vectors associated with the tangent linear dynamics. We refer to this approach as finite-time Lyapunov analysis (FTLA). FTLA identifies the potential for flow structure based on the stability and timescales implied by the spectrum of finite-time Lyapunov exponents. The corresponding Lyapunov vectors provide a basis for representing a splitting of the tangent space at phase points consistent with the splitting of the spectrum. A key property that makes FTLA viable is the exponential convergence of the splitting as the finite time increases. Tangency conditions for the vector field are used to determine points on manifolds of interest. The benefits of the FTLA approach are the dynamical model need not be in a special normal form, the manifolds of interest need not be attracting nor of known dimension, and the manifolds need not be associated with a fixed point or periodic orbit. After a brief review of FTLA, it is applied to spacecraft stationkeeping around a libration point in the circular restricted three-body problem. This application requires locating the stable and unstable subspaces at points on periodic and aperiodic orbits. For the periodic case, the FTLA subspaces are shown to agree with the Floquet subspaces; for the quasi-periodic case, the accuracy of the FTLA subspaces is demonstrated by simulation. - [Show abstract] [Hide abstract]

**ABSTRACT:**The Computational Singular Perturbation (CSP) method, developed by Lam and Goussis [Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 931-941], is a commonly-used method for finding approximations of slow manifolds in systems of ordinary differential equations (ODEs) with multiple time scales. The validity of the CSP method was established for fast-slow systems with a small parameter ε by the authors in [Journal of Nonlinear Science, 14 (2004), 59-91]. In this article, we consider a more general class of ODEs which lack an explicit small parameter ε, but where fast and slow variables are nevertheless separated by a spectral gap. First, we show that certain key quantities used in the CSP method are tensorial and thus invariant under coordinate changes in the state space. Second, we characterize the slow manifold in terms of these key quantities and explain how these characterizations are related to the invariance equation. The implementation of the CSP method can be either as a one-step or as a two-step procedure. The one-step CSP method aims to approximate the slow manifold; the two-step CSP method goes one step further and aims to decouple the fast and slow variables at each point in the state space. We show that, in either case, the operations of changing coordinates and performing one iteration of the CSP method commute. We use the commutativity property to give a new, concise proof of the validity of the CSP method for fast-slow systems and illustrate with an example due to Davis and Skodje. -
##### Article: Entropy-inspired Lyapunov Functions and Linear First Integrals for Positive Polynomial Systems

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**ABSTRACT:**Two partially overlapping classes of positive polynomial systems, chemical reaction networks with mass action law (MAL-CRNs) and quasi-polynomial systems (QP systems) are considered. Both of them have an entropy-like Lyapunov function associated to them which are similar but not the same. Inspired by the work of Prof. Gorban [12] on the entropy-functionals for Markov chains, and using results on MAL-CRN and QP-systems theory we characterize MAL-CRNs and QP systems that enable both types of entropy-like Lyapunov functions. The starting point of the analysis is the class of linear weakly reversible MAL-CRNs that are mathematically equivalent to Markov chains with an equilibrium point where various entropy level set equivalent Lyapunov functions are available. We show that non-degenerate linear kinetic systems with a linear first integral (that corresponds to conservation) can be transformed to linear weakly reversible MAL-CRNs using linear diagonal transformation, and the coefficient matrix of this system is diagonally stable. This implies the existence of the weighted version of the various entropy level set equivalent Lyapunov functions for non-degenerate linear kinetic systems with a linear first integral. Using translated X-factorable phase space transformations and nonlinear variable transformations a dynamically similar linear ODE model is associated to the QP system models with a positive equilibrium point. The non-degenerate kinetic property together with the existence of positive equilibrium point form a sufficient condition of the existence of the weighted version of the various entropy level set equivalent Lyapunov functions in this case. Further extension has been obtained by using the time re-parametrization transformation defined for QP models. -
##### Article: Calculation of electromagnetic wave scattering by a small impedance particle of an arbitrary shape

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**ABSTRACT:**Scattering of electromagnetic (EM) waves by small (ka 1) impedance particle D of an arbitrary shape, embedded in a homogeneous medium, is studied. Analytic, closed form, formula for the scattered field is derived. The scattered field is of the order O(a2-κ), where κ [0,1) is a number. This field is much larger than in the case of Rayleigh-type scattering. The numerical results demonstrate a wide range of applicability of the analytic formula for the scattered field. Comparison with Mie-type solution is carried out for various boundary impedances and radii of the particle.

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.