Odd harmonic periodic solutions of systems with odd nonlinearities

The connection between odd nonlinearities in dynamical systems and oscillations with no even harmonics is studied. It is found that when using harmonic balance techniques, one can often ignore even harmonics with no loss of generality as long as the nonlinearity is odd. It is shown that results on nonexistence of oscillations which ignore even harmonics are often valid results on nonexistence of general oscillations, with or without nontrivial even harmonics. It is also determined that if an isolated oscillation with nontrivial even harmonics is found by using a standard method, then there is also an oscillation with no even harmonics. The techniques used are harmonic balance-Galerkin methods associated with the describing function method.