4to (254 x 187 mm), pp 88; a very good copy, disbound. £225
First edition of Bendixson’s paper on what is known as the Poincaré-Bendixson theorem. The theorem states that any orbit which stays in a compact region of the state space of a 2-dimensional planar continuous dynamical system either approaches a fixed point or a periodic orbit. Thus chaotic behaviour can only arise in continuous dynamical systems whose phase space has 3 or more dimensions. ‘Bendixson is probably best remembered for the Poincaré- Bendixson theorem. We shall say a little about how Bendixson came to prove this result. This came about because of his work in real analysis. In this area he first studied uniform convergence of series of real functions and took an important step towards giving precise conditions when the limit function of continuous functions is continuous. In examining periodic solutions of differential equations Bendixson used methods based on continued fractions. These methods had first been used by Legendre to prove that e and π are irrational.
‘The analysis problem which intrigued Bendixson more than all others was the investigation of integral curves to first order differential equations, in particular he was intrigued by the complicated behaviour of the integral curves in the neighbourhood of singular points. This important problem was first studied by Briot and his friend Bouquet and, before Bendixson worked on it, had recently been investigated by Poincaré. Poincaré had obtained a qualitative description of the integral curves but it was Bendixson who gave a quantitative description near the singular points’ (MacTutor History of mathematics).
GBP 225.00
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