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We consider quasi-periodically systems in the presence of dissipation andstudy the existence of response solutions, i.e. quasi-periodic solutions withthe same frequency vector as the forcing term. When the dissipation is largeenough and a suitable function involving the forcing has a simple zero,response solutions are known to exist without assuming any non resonancecondition on the frequency vector. We analyse the case of non-simple zeroesand, in order to deal with the small divisors problem, we confine ourselves totwo-dimensional frequency vectors, so as to use the properties of continuedfractions. We show that, if the order of the zero is odd (if it is even, ingeneral no response solution exists), a response solution still exists providedthe inverse of the parameter measuring the dissipation belongs to a set givenby the union of infinite intervals depending on the convergents of the ratio ofthe two components of the frequency vector. The intervals may be disjoint andas a consequence we obtain the existence of response solutions in a set with"holes". If we want the set to be connected we have to require somenon-resonance condition on the frequency: in fact, we need a condition weakerthan the Bryuno condition usually considered in small divisors problems.

Response solutions for strongly dissipative quasi-periodically forced systems with arbitrary nonlinearities and frequencies