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Pietro Majer
Pietro Majer
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The paper

M. L. Cartwright, J. E. Littlewood. On non-linear dierential equations of the second order. I.On non-linear differential equations of the second order. I. The equation $y''-k(1-y^2)y+y=b\lambda k\cos(\lambda t+a)$ J.London Math. Soc. 20, (1945)

was not only written during the war, but also was stimulated by the war. Subsequently it played an important role in prehistory of hyperbolic dynamics.

In 1960 Stephen Smale conjectured that Morse-Smale systems are the only structurally stable systems. It was pointed out to Smale that his conjectures are likely to be false. Rene Thom argued that hyperbolic automorphism does not lie in the closure of Morse- Smale systems. Norman Levinson wrote to Smale with a reference to the above paper in which Cartwright and Littlewood studied certain differential equation of second order with periodic forcing. This work arose from war-related studies involving radio waves. The equation leads to a flow on R3. According to Levinson this flow has infinitely many periodic orbits; this phenomenon is robust which can be seen from the paper and also it was directly proved for a dierent equation in his own work. This led Smale to discovery of the famous horseshoe and subsequent explosive development in smooth dynamics.

The paper

M. L. Cartwright, J. E. Littlewood. On non-linear dierential equations of the second order. I. The equation $y''-k(1-y^2)y+y=b\lambda k\cos(\lambda t+a)$ J.London Math. Soc. 20, (1945)

was not only written during the war, but also was stimulated by the war. Subsequently it played an important role in prehistory of hyperbolic dynamics.

In 1960 Stephen Smale conjectured that Morse-Smale systems are the only structurally stable systems. It was pointed out to Smale that his conjectures are likely to be false. Rene Thom argued that hyperbolic automorphism does not lie in the closure of Morse- Smale systems. Norman Levinson wrote to Smale with a reference to the above paper in which Cartwright and Littlewood studied certain differential equation of second order with periodic forcing. This work arose from war-related studies involving radio waves. The equation leads to a flow on R3. According to Levinson this flow has infinitely many periodic orbits; this phenomenon is robust which can be seen from the paper and also it was directly proved for a dierent equation in his own work. This led Smale to discovery of the famous horseshoe and subsequent explosive development in smooth dynamics.

The paper

M. L. Cartwright, J. E. Littlewood. On non-linear differential equations of the second order. I. The equation $y''-k(1-y^2)y+y=b\lambda k\cos(\lambda t+a)$ J.London Math. Soc. 20, (1945)

was not only written during the war, but also was stimulated by the war. Subsequently it played an important role in prehistory of hyperbolic dynamics.

In 1960 Stephen Smale conjectured that Morse-Smale systems are the only structurally stable systems. It was pointed out to Smale that his conjectures are likely to be false. Rene Thom argued that hyperbolic automorphism does not lie in the closure of Morse- Smale systems. Norman Levinson wrote to Smale with a reference to the above paper in which Cartwright and Littlewood studied certain differential equation of second order with periodic forcing. This work arose from war-related studies involving radio waves. The equation leads to a flow on R3. According to Levinson this flow has infinitely many periodic orbits; this phenomenon is robust which can be seen from the paper and also it was directly proved for a dierent equation in his own work. This led Smale to discovery of the famous horseshoe and subsequent explosive development in smooth dynamics.

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Andrey Gogolev
Andrey Gogolev
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The paper

M. L. Cartwright, J. E. Littlewood. On non-linear dierential equations of the second order. I. The equation $y''-k(1-y^2)y+y=b\lambda k\cos(\lambda t+a)$ J.London Math. Soc. 20, (1945)

was not only written during the war, but also was stimulated by the war. Subsequently it played an important role in prehistory of hyperbolic dynamics.

In 1960 Stephen Smale conjectured that Morse-Smale systems are the only structurally stable systems. It was pointed out to Smale that his conjectures are likely to be false. Rene Thom argued that hyperbolic automorphism does not lie in the closure of Morse- Smale systems. Norman Levinson wrote to Smale with a reference to the above paper in which Cartwright and Littlewood studied certain differential equation of second order with periodic forcing. This work arose from war-related studies involving radio waves. The equation leads to a flow on R3. According to Levinson this flow has infinitely many periodic orbits; this phenomenon is robust which can be seen from the paper and also it was directly proved for a dierent equation in his own work. This led Smale to discovery of the famous horseshoe and subsequent explosive development in smooth dynamics.