Self-oscillating systems with controlled phase of external force

The purpose of this work is to study self-oscillatory systems under adaptive external action. This refers to the situation when the phase of the external action additionally depends on the dynamical variable of the oscillator. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with three-dimensional phase space. Methods. Methods of charts of dynamical regimes and charts of Lyapunov exponents are used, as well as the construction of phase portraits and stroboscopic sections. Results. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with a three-dimensional phase space. The pictures of characteristic dynamical regimes are described. Scenarios for the development of multidimensional chaos are described. Illustrations are given of the influence of the control parameter, which is responsible for the degree of dependence of the phase on the oscillator variable, on the dynamics of the system at different frequencies of action. Conclusion. The taling into account of the dependence of the phase on a dynamical variable leads to an extension of the tongues of subharmonic resonances, which are weakly expressed in the classical van der Pol oscillator. This is especially noticeable for even resonances of periods 2 and 4. For the generator of quasi-periodic oscillations in the non-autonomous case, three-frequency tori are observed, their regions begin to dominate with an increase in the adaptivity parameter, displacing the tongues of resonant two-frequency tori. A variety of multidimensional chaos characterized by an additional Lyapunov exponent close to zero is discovered, the possibility of developing hyperchaos as a result of destruction is shown.