Two Methods for the Numerical Detection of Hopf Bifurcations
T. J. Garratt, Gerald Moore, Alastair A. Spence · Birkhäuser Basel eBooks · 1991
This paper is concerned with the detection of Hopf bifurcations in the parameter dependent nonlinear system 1 $$\frac{{dx}}{{dt}} = f\left( {x,\lambda } \right),f:{R^n} \times R \mapsto {R^n},x \in {R^n},\lambda \in R.$$ The set $$\Gamma : = \left\{ {\left( {x,\lambda } \right) \in {R^{n + 1}}:f\left( {x,\lambda } \right) = 0} \right\}$$ represents the steady state solutions of (1) and it is often important to determine the (linearised) stability of a branch of Γ. If µ i denotes the eigenvalue with largest real part of the Jacobian matrix A:= fx(x,λ), then a steady state solution is stable (unstable) if Re(µ 1) is negative (positive). Also in applications it is desirable to detect a point of Γ where µ i is complex and Re(µ 1) changes sign as A varies, called a Hopf bifurcation point. If n is small in (1) it is certainly simplest and probably best to find all the eigenvalues of A using the QR algorithm. However for systems arising from spatial discretisations of p.d.e.’s n is typically very large with the Jacobian matrix sparse, and the direct application of the QR algorithm for a nonsymmetric matrix will be very expensive or may not even be feasible. Of course we need not find all the eigenvalues of A since only the eigenvalues which determine the stability are wanted. This paper is about the application of iterative methods for the calculation of the eigenvalues of A with largest real part.