The saddle case of a nonsmooth Rayleigh–Duffing oscillator

The Focus Case of a Nonsmooth Rayleigh-Duffing Oscillator

“…It can be applied in several mechanical problems, such as the ship roll damping [1], the oscillation of pipes in heat exchangers [10], the planar flow-induced oscillation [16]. A nonsmooth Rayleigh-Duffing equation ẋ = y, y = −cx − dx 3 − (a + b|y|)y (1.1) for bd = 0 has been researched in [1,13,20] for its dynamics. With the scaling where µ 1 and µ 2 are real.…”

“…So it is sufficient to study their dynamics in the half plane x ≥ 0. A general introduction for the oscillator model (1.1) which describes some mechanical problems was given in [1,13,20] and the references therein. Moreover, Chen and Wang in [20] made a study for the saddle case of system (1.1), i.e., system (1.2a).…”

“…Similar mistakes have been made in the study of a nonsmooth Rayleigh-Duffing oscillator [20]. For the saddle case ofẋ = y,ẏ = µ 1 x + x 3 − (µ 2 + |y|)y, (6.13) where µ 1 , µ 2 are real, the closed orbits in [20,Lemma 3.5] will not be unique.…”

“…Similar mistakes have been made in the study of a nonsmooth Rayleigh-Duffing oscillator [20]. For the saddle case ofẋ = y,ẏ = µ 1 x + x 3 − (µ 2 + |y|)y, (6.13) where µ 1 , µ 2 are real, the closed orbits in [20,Lemma 3.5] will not be unique. Under the condition of [20,Lemma 3.5] the origin is unstable and the 2-saddle loop is internally unstable if it exists.…”

“…For the saddle case ofẋ = y,ẏ = µ 1 x + x 3 − (µ 2 + |y|)y, (6.13) where µ 1 , µ 2 are real, the closed orbits in [20,Lemma 3.5] will not be unique. Under the condition of [20,Lemma 3.5] the origin is unstable and the 2-saddle loop is internally unstable if it exists. Then system (6.13) exhibits at least two limit cycles for some parameters.…”