Hidden Dynamics pp 171-200

# Linear Switching (Local Theory)

• Mike R. Jeffrey
Chapter

## Abstract

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

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