Chaotic Behaviour in Simple Dynamical Systems

In this paper a description is given of the chaotic behaviour generated by a transversal homoclinic point of a plane map. A proof of Smale’s theorem via the shadowing property of hyperbolic sets is provided. The result is related to certain plane periodic systems of ODE’S like the periodically perturbed pendulum equation. To this end the so-called method of Melnikov is derived.

[1]  P. Holmes,et al.  A nonlinear oscillator with a strange attractor , 1979, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[2]  R. Robinson,et al.  Stability theorems and hyperbolicity in dynamical systems , 1977 .

[3]  D. Stoffer.  Transversal homoclinic points and hyperbolic sets for non-autonomous maps I , 1988 .

[4]  Eduard L. Stiefel,et al.  Methoden der analytischen Störungsrechnung und ihre Anwendungen , 1978 .

[5]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.