On the phenomenon of bifurcation space symmetrization as mechanism for bursting oscillations generation

Abstract

Bursting oscillations in a system with low-frequency excitation by introducing a new approach of time-varying asymmetric potential is considered. The paper shows that this complex type of oscillations occurs as a combination of double imperfect and double saddle-node bifurcation during each cycle. Furthermore, such a mechanism of generating bursting oscillations is reflected in the complete symmetrization of the bifurcation diagram, which represents a new result. In line with this is the original representation of the catastrophe surface. Such a new shape of catastrophe surface and its connection with bursting oscillations are analyzed in detail. A new way of explanation of bursting oscillations based on the analogy with the motion of a particle in extended potential which alternates from bistable to the monostable case is also provided. With this new concept, it is shown why there is no lower limit of the excitation frequency to generate bursting oscillations. Also, the authors believe that with such an approach, subharmonic oscillations, as they have been treated in this class of nonlinear oscillations, should be observed in a new way. The approach is illustrated on a simple real mechanical model, the functionality and effectiveness of which are strongly dependent on parameters of systems and the relations between them.

Appendix

Fourier series of the function $\sin (\varphi_{0} \sin (2\pi \tau )$:

$$\begin{aligned} \Phi \left( {\tau ,T} \right) = & \sin (\varphi _{0} \sin (2\pi \tau ) \approx 2J_{1} \left( {\varphi _{0} } \right)\sin \left( {2\pi \tau } \right) + 2J_{3} \left( {\varphi _{0} } \right)\sin \left( {6\pi \tau } \right) \\ & + {\mkern 1mu} {\mkern 1mu} \frac{{2\left[ {\varphi _{0} \left( {\varphi _{0}^{2} - 48} \right)J_{1} \left( {\varphi _{0} } \right) - 12\left( {\varphi _{0}^{2} - 16} \right)J_{2} \left( {\varphi _{0} } \right)} \right]}}{{\varphi _{0}^{3} }}\sin \left( {10\pi \tau } \right) + ... \\ \end{aligned}$$

(5)

where $J_{n} (\varphi_{0} )$ stands for Bessel functions of the first kind.