Linear Switching (Local Theory)

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

Chapter Chapter 8 in Hidden Dynamics, 2018, pp 171-200 from Springer

Abstract: Abstract In this section we take a look at systems that depend only linearly on the switching multiplier switching multiplier s λ = ( λ 1 , … , λ m ) $$\boldsymbol{\lambda } = (\lambda _{1},\ldots,\lambda _{m})$$ and are therefore expressible in the form x ̇ = f ( x ; λ ) = a ( x ) + B ̲ ̲ ( x ) λ , $$\displaystyle{ \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x};\boldsymbol{\lambda }) = \mathbf{a}(\mathbf{x}) + \underline{\underline{B}}(\mathbf{x})\boldsymbol{\lambda }\;, }$$ where B ̲ ̲ $$\underline{\underline{B}}$$ is an n × m matrix. In relation to ( 5.3 ), the quantities in (8.1) are a = 1 2 ∑ j = 1 m ∑ κ j = ± f κ 1 … κ j … κ m , B j = 1 2 ∑ j = 1 m ∑ κ j = ± κ j f κ 1 … κ j … κ m , $$\displaystyle{\mathbf{a} = \mbox{ $\frac{1} {2}$}\sum _{j=1}^{m}\sum _{ \kappa _{j}=\pm }\mathbf{f}^{\kappa _{1}\ldots \kappa _{j}\ldots \kappa _{m} }\;,\qquad \mathbf{B}_{j} = \mbox{ $\frac{1} {2}$}\sum _{j=1}^{m}\sum _{ \kappa _{j}=\pm }\kappa _{j}\mathbf{f}^{\kappa _{1}\ldots \kappa _{j}\ldots \kappa _{m} }\;,}$$ where B j is the j th column of B ̲ ̲ $$\underline{\underline{B}}$$ .

Date: 2018
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DOI: 10.1007/978-3-030-02107-8_8

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