Mean square stabilisation of complex oscillatory regimes in nonlinear stochastic systems

ABSTRACT

A problem of stabilisation of the randomly forced periodic and quasiperiodic modes for nonlinear dynamic systems is considered. For this problem solution, we propose a new theoretical approach to consider these modes as invariant manifolds of the stochastic differential equations with control. The aim of the control is to provide the exponential mean square (EMS) stability for these manifolds. A general method of the stabilisation based on the algebraic criterion of the EMS-stability is elaborated. A constructive technique for the design of the feedback regulators stabilising various types of oscillatory regimes is proposed. A detailed parametric analysis of the problem of the stabilisation for stochastically forced periodic and quasiperiodic modes is given. An illustrative example of stochastic Hopf system is included to demonstrate the effectiveness of the proposed technique.

Keywords:

• Stabilisation
• oscillatory regimes
• invariant manifolds
• cycles, tori
• feedback

Acknowledgements

This work was partially supported by RFBR (13-08-00069) and by the Government of the Russian Federation (Act 211, contract 02.A03.21.0006).

Disclosure statement

No potential conflict of interest was reported by the authors.