Giant noise and chaotic transitions' between the states that are close to small-amplitude quasi-harmonic and relaxation self-oscillations

Simple dynamical systems described by two first-order equations are analyzed. These systems allow the simultaneous excitation of stable small-amplitude quasi-harmonic and relaxation self-oscillations in the form of narrow spikes of giant amplitude. It is shown that random variation of the parameters of these systems may lead to the transformation of small noise into explosive noise of giant amplitude. A new type of chaotic oscillations is discovered and analyzed that develops under the periodic variation of the bifurcation parameter of the system or when the set of governing equations is supplemented with a certain additional equation describing small changes of this bifurcation parameter. It is established that these chaotic oscillations are associated with random transitions between the states that are close to small-amplitude quasi-harmonic or spike relaxation self-oscillations and that the maximum fractal dimension of these oscillations is about 2.4.