A period-doubling bifurcation for the Duffing equation
The purpose of this paper is to give a rigorous proof of existence of the period-doubling bifurcation (doubling of the rank of a subharmonic) generated by a Duffing equation with a periodic external force. This type of two-dimensional non-autonomous ordinary differential equation was the object of many studies via the association of the qualitative approach of dynamical systems with the numerical one. In particular, it is the case of the Ueda (here quoted) and Kawakami (not quoted) results on the structure of the parameter (or bifurcation) space, and that of the phase plane. A main theorem is formulated at the beginning. Then, making use of the expansion theory by generalized eigenfunctions due to Titshmarsh-Kodaira, properties of the Lyapunov exponent are shown. From them, and an asymptotic analysis with respect to the amplitude of the external force, basic bifurcation point properties, related to a reformulation of the initial equation, are given. It is worth noting that the problem of the existence of subharmonic solutions (so in particular those resulting from a period-doubling) was solved long ago by \textit{I. G. Malkin} (not known by the authors) in [Some problems in the theory of nonlinear oscillations, State Publ. House of Technical and Theoretical Literature, Moscow (1956; Zbl 0070.08703); English translation AEC tr 3766 (book 1), US Atomic Energy Commission (1959)]. This is made by the Poincaré method of the ``small parameter''. Even if Malkin's text does not mention explicitly the bifurcation problem, it can be obtained from the inequalities ensuring the existence of subharmonics.
two-dimensional non-autonomous ordinary differential equation
scientific article; zbMATH DE number 1117475
scientific article