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This article is mainly historical, except for the discussion of integrability and characteristic exponents in Sect. 2. After summarising the achievements of Henri Poincaré, we discuss his theory of critical exponents. The theory is applied to the case of three degrees-of-freedom Hamiltonian systems in (1:2:n)-resonance (n4). In addition we discuss Poincaré's mathematical physics, in particular the theory of partial differential equations, rotating fluid masses and relativity. Attention is given to the priority question of Special Relativity.

Aspects of Poincaré’s Program for Dynamical Systems and Mathematical Physics