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Computing the differential of an unfolded contact diffeomorphism

Klaus Böhmer, Drahoslava Janovská, Vladimír Janovský (2003)

Applications of Mathematics

Consider a bifurcation problem, namely, its bifurcation equation. There is a diffeomorphism Φ linking the actual solution set with an unfolded normal form of the bifurcation equation. The differential D Φ ( 0 ) of this diffeomorphism is a valuable information for a numerical analysis of the imperfect bifurcation. The aim of this paper is to construct algorithms for a computation of D Φ ( 0 ) . Singularity classes containing bifurcation points with c o d i m 3 , c o r a n k = 1 are considered.

Continuation of invariant subspaces via the Recursive Projection Method

Vladimír Janovský, O. Liberda (2003)

Applications of Mathematics

The Recursive Projection Method is a technique for continuation of both the steady states and the dominant invariant subspaces. In this paper a modified version of the RPM called projected RPM is proposed. The modification underlines the stabilization effect. In order to improve the poor update of the unstable invariant subspace we have applied subspace iterations preconditioned by Cayley transform. A statement concerning the local convergence of the resulting method is proved. Results of numerical...

Epidemiology of Dengue Fever: A Model with Temporary Cross-Immunity and Possible Secondary Infection Shows Bifurcations and Chaotic Behaviour in Wide Parameter Regions

Maíra Aguiar, Bob Kooi, Nico Stollenwerk (2008)

Mathematical Modelling of Natural Phenomena

Basic models suitable to explain the epidemiology of dengue fever have previously shown the possibility of deterministically chaotic attractors, which might explain the observed fluctuations found in empiric outbreak data. However, the region of bifurcations and chaos require strong enhanced infectivity on secondary infection, motivated by experimental findings of antibody-dependent-enhancement. Including temporary cross-immunity in such models, which is common knowledge among field researchers...

Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions

Edward J. Allen, John A. Burns, David S. Gilliam (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Using Burgers’ equation with mixed Neumann–Dirichlet boundary conditions, we highlight a problem that can arise in the numerical approximation of nonlinear dynamical systems on computers with a finite precision floating point number system. We describe the dynamical system generated by Burgers’ equation with mixed boundary conditions, summarize some of its properties and analyze the equilibrium states for finite dimensional dynamical systems that are generated by numerical approximations of this...

One-Parameter Bifurcation Analysis of Dynamical Systems using Maple

Borisov, Milen, Dimitrova, Neli (2010)

Serdica Journal of Computing

This paper presents two algorithms for one-parameter local bifurcations of equilibrium points of dynamical systems. The algorithms are implemented in the computer algebra system Maple 13 © and designed as a package. Some examples are reported to demonstrate the package’s facilities.* This paper is partially supported by the Bulgarian Science Fund under grant Nr. DO 02–359/2008.

Rigorous numerics for symmetric homoclinic orbits in reversible dynamical systems

Yasuaki Hiraoka (2007)


We propose a new rigorous numerical technique to prove the existence of symmetric homoclinic orbits in reversible dynamical systems. The essential idea is to calculate Melnikov functions by the exponential dichotomy and the rigorous numerics. The algorithm of our method is explained in detail by dividing into four steps. An application to a two dimensional reversible system is also treated and the existence of a symmetric homoclinic orbit is rigorously verified as an example.

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