Numerical computation of bifurcations in large equilibrium systems in matlab

Abstract

a b s t r a c tThe Continuation of Invariant Subspaces (CIS) algorithm produces a smoothly-varying basis for an invariant subspace R(s) of a parameter-dependent matrix A(s). We have incorporated the CIS algorithm into Cl_matcont, a Matlab package for the study of dynamical systems and their bifurcations. Using subspace reduction, we extend the functionality of Cl_matcont to large-scale computations of bifurcations of equilibria. In this paper, we describe the algorithms and functionality of the resulting Matlab bifurcation package Cl_matcontL. The novel features include: new CIS-based, continuous, well-scaled test functions for codimension 1 and 2 bifurcations; detailed description of locators for large problems; and examples of bifurcation analysis in large sparse problems.

Degenerate Bogdanov–Takens bifurcations in a one-dimensional transport model of a fusion plasma

“…In a continuation algorithm (see e.g. [7]), the border v bor is typically set to solution v of (7) computed at the previous step and the border w bor is set to w, where w is the solution of the…”

“…Compared to the detection of Hopf-bifurcations in [7], the second test function ψ (2) H was changed to…”

“…Section 5 contains a discussion of the results. Appendix discusses new features of the code of cl_matcontL compared to the description of its first version in [7].…”

“…The solutions of these PDEs are typically studied by considering their spatial discretization, which typically results in large systems of ODEs with large and sparse Jacobian matrices. Information about bifurcations of equilibria is typically contained in a small invariant subspace, corresponding to rightmost eigenvalues of the Jacobian matrix [7]. In the MATLAB package cl_matcontL [8], this subspace is computed efficiently using the Continuation of Invariant Subspaces (CIS) algorithm [9].…”

Bifurcation Tracking for High Reynolds Number Flow Around an Airfoil

“…This is still an active area of research for the types of flow of interest in this work. Finally, the work of Bindel et al [2014] implements the Continuation of Invariant Subspaces (CIS) method using a Matlab code to track simple bifurcations. This uses a projection method to reduce the size of the problem but requires a good approximation to the least stable subspace.…”