Linear Lumping in Systems with Timescale Separation

3 Linear Lumping in Systems with Timescale Separation 

We now briefly provide a formalised framework for linear lumping in systems with timescale separation which is based on a similar approach to that presented in Sect. 6.3. We start with the initial value problem 

If a gap in timescales exists as discussed in Sect. 6.3, then the lumping matrix can be partitioned as 

The variables in the slow subspace S are therefore decoupled from those in the fast subspace, and therefore, the lumping allows the definition of a reduced set of variables S describing the longer timescale dynamics. The connections with the slow manifold methods described in Sect. 6.5 also become clear since the calculation of the points on the manifold involves solving the following algebraic set of equations  

We return to a discussion of numerical methods for solving such relationships in Sect. 7.10. 

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