The Leray-Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems

1979 ◽  
pp. 326-409 ◽  
Author(s):  
Heinz-Otto Peitgen ◽  
Michael Prüfer
Keyword(s):  
Numerical Study ◽  
Continuation Method ◽  
Bifurcation Problems ◽  
Nonlinear Eigenvalue

NUMERICAL STUDY OF COEXISTING ATTRACTORS FOR THE HÉNON MAP

2013 ◽  
Vol 23 (07) ◽  
pp. 1330025 ◽  
Author(s):  
ZBIGNIEW GALIAS ◽  
WARWICK TUCKER
Keyword(s):  
Periodic Orbits ◽  
Parameter Space ◽  
Interval Analysis ◽  
Numerical Study ◽  
Continuation Method ◽  
Basins Of Attraction ◽  
Hénon Map ◽  
Coexisting Attractors ◽  
Henon Map ◽  
Parameter Values

The question of coexisting attractors for the Hénon map is studied numerically by performing an exhaustive search in the parameter space. As a result, several parameter values for which more than two attractors coexist are found. Using tools from interval analysis, we show rigorously that the attractors exist. In the case of periodic orbits, we verify that they are stable, and thus proper sinks. Regions of existence in parameter space of the found sinks are located using a continuation method; the basins of attraction are found numerically.

SYMMETRY REDUCTIONS AND A POSTERIORI FINITE ELEMENT ERROR ESTIMATORS FOR BIFURCATION PROBLEMS

2005 ◽  
Vol 15 (07) ◽  
pp. 2091-2107 ◽  
Author(s):  
C.-S. CHIEN ◽  
B.-W. JENG
Keyword(s):  
Finite Element ◽  
Eigenvalue Problems ◽  
Continuation Method ◽  
Discretization Scheme ◽  
Element Discretization ◽  
Nonlinear Eigenvalue Problems ◽  
Error Estimators ◽  
Symmetry Reductions ◽  
Relaxation Scheme ◽  
Nonlinear Eigenvalue

We discuss efficient continuation algorithms for solving nonlinear eigenvalue problems. First, we exploit the idea of symmetry reductions and discretize the problem on a symmetry cell by the finite element method. Then we incorporate the multigrid V-cycle scheme in the context of continuation method to trace solution branches of the discrete problems, where the preconditioned Lanczos method is used as the relaxation scheme. Next, we apply the symmetry reduction technique to the two-grid finite element discretization scheme [Chien & Jeng, 2005] to solve some nonlinear eigenvalue problems in physical science. The two-grid centered difference discretization scheme described therein was also implemented for comparison. Sample numerical results are reported.

Sign in / Sign up

Export Citation Format

Share Document