The simplest oscillators with adaptive properties

We study the behavior of a non-autonomous oscillator with phase of the external force having the property of adaptability, i.e. depends on the dynamic variable. In the frame of this work we consider the case when dependence of the phase of external force is polynomial including terms of the third degree. Such dependence of the phase of the external action leads to the appearance of the complex chaotic oscillations in the dynamics of the oscillator. In the parameter space a hierarchy of various periodic and chaotic oscillations is observed. It is shown that in the dynamics of the system oscillation modes are observed, similar to the modes of a non-autonomous oscillator with a potential in the form of a periodic function. In the case of a linear adaptation function, the structure of the parameter plane has characteristic cascades of period doubling bifurcations. When nonlinearities (the second and the third terms of polynomial function) are taken into account, structures are destroyed.