We classify generic instabilities of wave trains in reaction-diffusion systems on the real line as the wavenumber and system parameters are varied. We find three types of robust instabilities: Hopf with nonzero modulational wavenumber, sideband and spatio-temporal period-doubling. Near a fold, the only other robust instability mechanism, we show that all wave trains are necessarily unstable. We also discuss the special cases of homogeneous oscillations and reflection symmetric, stationary Turing patterns.
Bibliographical noteFunding Information:
This work was partially supported by the National Science Foundation through grant NSF DMS-0504271 (A. Scheel), and the Priority Program SPP 1095 of the DFG as well as the NDNS+ cluster of the NWO (J. Rademacher).
- Classification of instabilities
- Reaction-diffusion systems
- Wave trains