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Hopf bifurcation of spike solutions for the shadow Gierer–Meinhardt model
| Content Provider | Semantic Scholar |
|---|---|
| Author | Ward, Michael J. Wei, Juncheng |
| Copyright Year | 2003 |
| Abstract | In the limit of small activator diffusivity, the stability of a one-spike solution to the shadow Gierer–Meinhardt activator-inhibitor system is studied for various ranges of the reaction-time constant $\tau$ associated with the inhibitor field dynamics. By analyzing the spectrum of the eigenvalue problem associated with the linearization around a one-spike solution, it is proved, for a certain parameter regime, that a complex conjugate pair of eigenvalues crosses into the unstable right half-plane $Re(\lambda) > 0$ as $\tau$ increases past a critical value $\tau_0$ . For this parameter regime, it is proved that there are exactly two eigenvalues in the right half-plane when $\tau > \tau_0$ and none when $0 \leq \tau . It is shown numerically that this critical value of $\tau$ represents the onset of an oscillatory instability in the height of the spike. For other parameter regimes, a similar Hopf bifurcation is confirmed numerically. Full numerical solutions to the shadow problem are computed for a spike that is initially centred at the origin of a radially symmetric domain. Different types of large-scale oscillatory motions for the height of a spike are observed numerically for values of $\tau$ well beyond $\tau_0$ . |
| Starting Page | 677 |
| Ending Page | 711 |
| Page Count | 35 |
| File Format | PDF HTM / HTML |
| DOI | 10.1017/S0956792503005278 |
| Volume Number | 14 |
| Alternate Webpage(s) | http://www.math.ubc.ca/~ward/papers/hopf_shadejam.pdf |
| Alternate Webpage(s) | http://www.math.ubc.ca/people/faculty/ward/papers_ps/hopf_shadejam.ps |
| Alternate Webpage(s) | https://doi.org/10.1017/S0956792503005278 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
