The Vector Field Canopy

Mike R. Jeffrey
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Mike R. Jeffrey: University of Bristol, Department of Engineering Mathematics

Chapter Chapter 5 in Hidden Dynamics, 2018, pp 91-101 from Springer

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

Date: 2018
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DOI: 10.1007/978-3-030-02107-8_5

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