Chaos in the Duffing equation
The authors study the equation (1) \(z''+a^ 2g_ 0(z)=f(t)\), with \(f\) \(T\)-periodic, under the assumption that the equation \(z''+g_ 0(z)=0\) has either (i) an orbit homoclinic to a saddle point, or (ii) a heteroclinic orbit joining saddle points. In the homoclinic (resp. heteroclinic) case, they prove that if \(f'(t)\) (resp. \(f(t))\) has a simple zero, then for sufficiently large \(a\), equation (1) has a transverse homoclinic (resp. heteroclinic) orbit. This result follows from an abstract theorem the authors prove about existence of bounded solutions of the singularly perturbed equation \(\varepsilon x'+g(x)=\varepsilon^ 2h(t,x,\varepsilon)\), to which (1) can be converted by a change of variables. The proof of this theorem uses exponential dichotomies and scales of Banach spaces.
scientific article; zbMATH DE number 166901