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We study properties of evolution equations which are first order in time andarbitrary order in space (FTNS). Following Gundlach and Mart\'in-Garc\'ia(2006) we define strong and symmetric hyperbolicity for FTNS systems andexamine the relationship between these definitions, and the analogous conceptsfor first order systems. We demonstrate equivalence of the FTNS definition ofstrong hyperbolicity with the existence of a strongly hyperbolic first orderreduction. We also demonstrate equivalence of the FTNS definition, up to N=4,of symmetric hyperbolicity with the existence of a symmetric hyperbolic firstorder reduction.

Hyperbolicity of High Order Systems of Evolution Equations