Coexistence of low-amplitude quasiharmonic oscillations and giant-amplitude spike oscillations in systems with one degree of freedom

Oscillations that occur in a number of physical, chemical, and biological systems described by two first-order nonlinear differential equations are studied. It is proven analytically and numerically that there exists a region of parameters in which these simple systems can simultaneously support low-amplitude quasiharmonic oscillations and relaxation oscillations in the form of narrow spikes. It is established that the limit cycles of these spike oscillations can contain regions that correspond not only to slow and fast motions with the characteristic times tau(theta) and tau(eta), respectively, but also a region of superfast motion with the characteristic time tau(s) similar to tau(theta)(2)/tau(eta) << 2 tau(theta). The spike has a giant amplitude proportional to the ratio tau(eta)/tau(theta) >> 1 and a duration of an order of tau(eta).