The main goal of this paper is to study the interaction between \(1:3\) internal resonance and two external resonances for a class of weakly nonlinear oscillators with two important degrees of freedom. The examined oscillators possess symmetric restoring force, involving linear and cubic displacement terms.
First of all, the authors apply the method of multiple time scales which results in four slow-flow equations for amplitudes and phases of approximate solutions. Several types of constant solutions of these equations are identified and determined for both single-mode and two-mode forcing conditions. The first section includes also a procedure revealing the stability properties of all types of located constant solutions. In a particular case (constant solutions for single-mode forcing) the solutions are found in closed form.
The second section examines numerically a mechanical oscillator. This verifies the coexistence of both periodic motions with large period and quasiperiodic and chaotic motions which exhibit torus doubling, boundary crisis and transient chaos.