Journal of Dynamics and Differential Equations

, Volume 1, Issue 1, pp 75-94

First online:

Slow-motion manifolds, dormant instability, and singular perturbations

  • G. FuscoAffiliated withDipartimento di Metodi Matematici, Universitá di Roma
  • , J. K. HaleAffiliated withSchool of Mathematics, Georgia Institute of Technology

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If the coexistence of two phases at the transition temperature is kept under observation for a long time, then one observes that the system is not exactly in equilibrium and a very slow evolution driven by surface tension is taking place. Theoretically, one should eventually see a spatially homogeneous state, but the time for settling down is so long that what one actually observes is “motion towards a stable state.” The complexity of the spatial distribution of the two phases keeps decreasing but appears to be stable for very long periods of time with intermittent periods of fast motion when there are small inclusions of one of the two regions embedded in the other phase. For a simple reaction diffusion model, it is shown that this phenomenon can be explained by investigating the flow on the attractor and the unstable manifolds of equilibria.

Key words

Singular perturbations transition layers metastability integral manifolds