On the periodic motion of a generalized oscillator in two dimensions

Abstract

We investigate the system of two nonlinear differential equations

with nonlinearities k (x, y) and l (x, y) . We consider the functions k (x, y) and l (x, y) to be constant on each quadrant of ℝ² , which means that k and l are equal to different constants k₁ , k₂ , k₃ , k₄ and l₁ , l₂ , l₃ , l₄ , respectively, for each of four quadrants. We formulate sufficient conditions on constants k_i , l_j , i, j  ∈  {1, 2, 3, 4} , such that the problem has a nontrivial T -periodic solution.

Keywords and phrases:

• Lissajous curves
• Fučík spectrum
• asymmetric nonlinearities
• nontrivial periodic solutions