Alexander, Murray E.; Moghadas, Seyed M.

$\cal{O}(\ell)$ shift in Hopf bifurcations for a class of nonstandard numerical schemes

Electron. J. Differ. Equ. 2005, 9-19, electronic only (2005)


Summary: Quantitative aspects of models describing the dynamics of biological phenomena have been mostly restricted to results of numerical simulations, often by employing standard numerical methods. However, several studies have shown that these methods may fail to reproduce the actual dynamical behavior of the underlying continuous model when the integration time-step, model parameters, or initial conditions vary in their respective ranges. In this paper, a non-standard numerical scheme is constructed for a general class of positivity-preserving system of ordinary differential equations. A connection between the dynamics of the system and that of the scheme is established in terms of codimension-zero bifurcations. It is shown that when the continuous model undergoes a bifurcation with a simple eigenvalue passing through zero (pitchfork, transcritical or saddle-node bifurcation), the scheme exhibits a corresponding bifurcation at the same bifurcation parameter value. On the other hand, for a Hopf bifurcation there is in general an $\mathcal{O}(\ell)$ shift in the bifurcation parameter value for the numerical scheme, where $\ell$ is the time-step. Partial results for the bifurcations of codimension-1 and higher are also discussed. Finally, the results are detailed for two examples: predator-prey system of Gause-type and the Brusselator system representing an autocatalytic oscillating chemical reaction.

Mathematics Subject Classification

65P30, 65C20, 37M05


finite-difference methods, codimension-zero bifurcations, quantitative analysis, scheme failure