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We prove that exists a Lindstedt series that holds when a Hamiltonian isdriven by a perturbation going to infinity. This series appears to be dual to astandard Lindstedt series as it can be obtained by interchanging the role ofthe perturbation and the unperturbed system. The existence of this dual seriesimplies that a dual KAM theorem holds and, when a leading order Hamiltonianexists that is non degenerate, the effect of tori reforming can be observedwith a system passing from regular motion to fully developed chaos and back toregular motion with the reappearance of invariant tori. We apply these resultsto a perturbed harmonic oscillator proving numerically the appearance of torireforming. Tori reforming appears as an effect limiting chaotic behavior to afinite range of parameter space of some Hamiltonian systems. Dual KAM theorem,as proved here, applies when the perturbation, combined with a kinetic term,provides again an integrable system.

Dual Lindstedt series and KAM theorem