The Vector Field Canopy

  • Mike R. Jeffrey


If a problem is specified only piecewise as
$$\displaystyle{\dot{\mathbf{x}} = \mathbf{f}\left (\mathbf{x}\right ) = \left \{\mathbf{f}^{K}(\mathbf{x})\;\;\mathrm{if}\;\;\mathbf{x} \in \mathcal{R}_{ K}\right \}_{K=\kappa _{1}\kappa _{2}\ldots \kappa _{m}}}$$
in terms of constituent vector fields fK, we may need to extend this across the discontinuity surface to form a combination \(\mathbf{f}(\mathbf{x};\boldsymbol{\lambda })\). There is an expression for \(\mathbf{f}(\mathbf{x};\boldsymbol{\lambda })\) that provides a series expansion in the switching multiplier s \(\boldsymbol{\lambda } = (\lambda _{1},\ldots,\lambda _{m})\). We call this the canopy of the constituent vector fields.


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Authors and Affiliations

  • Mike R. Jeffrey
    • 1
  1. 1.Department of Engineering MathematicsUniversity of BristolBristolUK

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